Real Analysis, focusing on limits, continuity, and differentiability of
functions. Limits, Continuity, and Differentiability of Functions: Multiple
Choice Questions with Solutions
1. What is the limit of f(x) = (x^2 - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) 2 d) Undefined
Answer: c) 2
Solution: Apply L'Hôpital's rule or factor the numerator: lim(x→1) (x+1) = 2
2. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
. For which value of k is f(x) = |x - 3| / (x - 3) continuous at x = 3?
a) -1 b) 0 c) 1 d) No value of k
Answer: d) No value of k
Solution: The function has a jump discontinuity at x = 3 for all k
3. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f(x) continuous at x = 0?
a) Yes b) No c) Cannot be determined d) Continuous but not differentiable
Answer: a) Yes
Solution: lim(x→0) x^2 sin(1/x) = 0, which equals f(0)
4. What is the derivative of f(x) = e^(sin x) at x = π?
a) -1 b) 0 c) 1 d) e
Answer: b) 0
Solution: f'(x) = e^(sin x) cos x, and cos π = -1
5. For which value of a is f(x) = |x - a| differentiable at x = a?
a) -1 b) 0 c) 1 d) No value of a
Answer: d) No value of a
Solution: The absolute value function always has a corner at x = a
6. What is the limit of (sin x) / x as x approaches 0?
a) 0 b) 1 c) π/2 d) Undefined
Answer: b) 1
Solution: This is a well-known limit, provable using squeeze theorem
7. If f(x) = x^3 + 2x^2 - 5x + 1, what is f'(2)?
a) 15 b) 17 c) 19 d) 21
Answer: c) 19
Solution: f'(x) = 3x^2 + 4x - 5, so f'(2) = 3(4) + 4(2) - 5 = 19
8. Which of the following functions is continuous everywhere?
a) floor(x) b) ceiling(x) c) e^x d) 1/x
Answer: c) e^x
Solution: e^x is continuous for all real x
9. What is the limit of (1 + 1/n)^n as n approaches infinity?
a) 1 b) 2 c) e d) ∞
Answer: c) e
Solution: This is the definition of e
10. If f(x) = x^2 and g(x) = sin x, what is (f ∘ g)'(π/2)?
a) 0 b) 1 c) -1 d) π/2
Answer: a) 0
Solution: (f ∘ g)'(x) = 2sin(x)cos(x), and cos(π/2) = 0
11. Which of the following is an indeterminate form?
a) 0/0 b) ∞/∞ c) 0 · ∞ d) All of the above
Answer: d) All of the above
Solution: These are all classic indeterminate forms
12. What is the derivative of tan x?
a) sec x b) sec^2 x c) csc^2 x d) cot x
Answer: b) sec^2 x
Solution: Use quotient rule on sin x / cos x
13. If f(x) is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), which theorem guarantees
a point c in (a,b) where f'(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: b) Rolle's Theorem
Solution: This is precisely the statement of Rolle's Theorem
14. What is the limit of (1 - cos x) / x^2 as x approaches 0?
a) 0 b) 1/2 c) 1 d) ∞
Answer: b) 1/2
Solution: Use L'Hôpital's rule twice or Taylor series expansion
15. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by the Mean Value Theorem on [0,2]?
a) 0 b) 1 c) √3 d) 2
Answer: b) 1
Solution: f'(c) = (f(2) - f(0)) / (2 - 0) = 6, so c = 1
16. Which of the following functions is not continuous at x = 0?
a) sin(1/x) b) x sin(1/x) c) x^2 sin(1/x) d) All are continuous
Answer: a) sin(1/x)
Solution: sin(1/x) oscillates rapidly near 0, while the others approach 0
17. What is the derivative of ln|x|?
a) 1/x b) -1/x c) 1/|x| d) x
Answer: c) 1/|x|
Solution: Use the chain rule, noting that d/dx |x| = x/|x|
18. If f(x) = x^2 + 2x + 1 and g(x) = √x, what is (g ∘ f)(2)?
a) 3 b) 5 c) 7 d) 9
Answer: a) 3
Solution: f(2) = 9, then g(9) = 3
19. What is the limit of (x^n - 1) / (x - 1) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: This is the definition of the derivative of x^n at x = 1
20. If f(x) is differentiable at x = a, which of the following must be true?
a) f(x) is continuous at x = a
b) f'(a) exists
c) f(x) has a tangent line at x = a
d) All of the above
Answer: d) All of the above
Solution: Differentiability implies continuity and existence of derivative
21. What is the derivative of arcsin x?
a) 1/√(1-x^2) b) -1/√(1-x^2) c) 1/√(1+x^2) d) -1/√(1+x^2)
Answer: a) 1/√(1-x^2)
Solution: Use implicit differentiation on sin(arcsin x) = x
22. For which value of k is f(x) = kx + |x| differentiable at x = 0?
a) -1 b) 0 c) 1 d) No value of k
Answer: c) 1
Solution: Left and right derivatives must match: k + 1 = k - 1, so k = 1
23. What is the limit of (sin x) / x as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: sin x oscillates between -1 and 1, while x grows without bound
24. If f(x) = e^x and g(x) = ln x, what is (f ∘ g)'(e)?
a) e b) 1 c) 1/e d) e^e
Answer: b) 1
Solution: (f ∘ g)'(x) = e^(ln x) · (1/x) = 1 for all x > 0
25. Which of the following is an example of a function that is continuous but nowhere differentiable?
a) x^2 b) |x| c) sin x d) Weierstrass function
Answer: d) Weierstrass function
Solution: The Weierstrass function is a classic example of such a function
26. What is the limit of (1 - x^n) / (1 - x) as x approaches 1?
a) 0 b) 1 c) n d) ∞
Answer: c) n
Solution: Use L'Hôpital's rule or factor the numerator
27. If f(x) = x^3 - 3x^2 + 2x, at which point(s) does f'(x) = 0?
a) x = 0 b) x = 1 c) x = 2 d) x = 0 and x = 2
Answer: d) x = 0 and x = 2
Solution: f'(x) = 3x^2 - 6x + 2 = (3x - 1)(x - 2)
28. What is the derivative of sec x?
a) tan x sec x b) -csc x cot x c) sec x tan x d) csc x cot x
Answer: c) sec x tan x
Solution: Use quotient rule on 1 / cos x
29. If f(x) is continuous on [a,b], which theorem guarantees that f(x) attains its maximum and
minimum values on this interval?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: d) Extreme Value Theorem
Solution: This is the statement of the Extreme Value Theorem
30. What is the limit of (e^x - 1) / x as x approaches 0?
a) 0 b) 1 c) e d) ∞
Answer: b) 1
Solution: This is the definition of the derivative of e^x at x = 0
31. If f(x) = x^4 - 2x^2 + 1, what is the value of f''(1)?
a) 8 b) 12 c) 16 d) 20
Answer: c) 16
Solution: f'(x) = 4x^3 - 4x, f''(x) = 12x^2 - 4, so f''(1) = 12 - 4 = 8
32. Which of the following functions is both even and odd?
a) x^2 b) x^3 c) sin x d) f(x) = 0
Answer: d) f(x) = 0
Solution: Only the zero function satisfies f(-x) = f(x) and f(-x) = -f(x)
33. What is the limit of (x^2 - 1) / (x - 1) as x approaches -1?
a) -2 b) -1 c) 0 d) 1
Answer: a) -2
Solution: Factor numerator: lim(x→-1) (x + 1) = -2
34. If f(x) = x^2 and g(x) = √x, what is (f ∘ g)'(4)?
a) 1/4 b) 1/2 c) 1 d) 2
Answer: b) 1/2
Solution: (f ∘ g)'(x) = 2√x · (1/(2√x)) = 1, so (f ∘ g)'(4) = 1/2
35. What is the derivative of x^x?
a) x^x b) x^x(1 + ln x) c) x^(x-1) d) e^x
Answer: b) x^x(1 + ln x)
Solution: Use logarithmic differentiation: y = x^x, ln y = x ln x
36. If f(x) is differentiable and f'(x) > 0 for all x, which of the following must be true?
a) f(x) is increasing b) f(x) is decreasing c) f(x) is constant d) None of the above
Answer: a) f(x) is increasing
Solution: Positive derivative implies increasing function
37. What is the limit of (sin x) / x as x approaches π?
a) 0 b) 1/π c) -1/π d) 1
Answer: b) 1/π
Solution: This limit is not indeterminate; evaluate directly
38. If f(x) = x^3 - 3x + 2, what is the value of c guaranteed by Rolle's Theorem on [-2,2]?
a) -1 b) 0 c) 1 d) No such c exists
Answer: b) 0
Solution: f(-2) = f(2), so Rolle's Theorem applies. f'(x) = 3x^2 - 3, f'(0) = 0
39. What is the derivative of arctan x?
a) 1/(1+x^2) b) -1/(1+x^2) c) 1/(1-x^2) d) -1/(1-x^2)
Answer: a) 1/(1+x^2)
Solution: Use implicit differentiation on tan(arctan x) = x
40. Which of the following is an example of a function that is everywhere continuous but
differentiable nowhere?
a) |x| b) x^2 c) Weierstrass function d) floor(x)
Answer: c) Weierstrass function
Solution: The Weierstrass function has this property
41. What is the limit of (1/x) sin(1/x) as x approaches ∞?
a) 0 b) 1 c) -1 d) Does not exist
Answer: a) 0
Solution: 1/x approaches 0, and sin(1/x) is bounded between -1 and 1
42. If f(x) = e^x and g(x) = ln x, what is (g ∘ f)(0)?
a) 0 b) 1 c) e d) undefined
Answer: a) 0
Solution: f(0) = 1, then g(1) = ln 1 = 0
43. What is the derivative of csc x?
a) -csc x cot x b) sec x tan x c) -csc^2 x d) cot x csc x
Answer: a) -csc x cot x
Solution: Use quotient rule on 1 / sin x
44. If f(x) is continuous on [a,b] and f(a) < 0 < f(b), which theorem guarantees a point c in (a,b) where
f(c) = 0?
a) Mean Value Theorem b) Rolle's Theorem c) Intermediate Value Theorem d) Extreme Value
Theorem
Answer: c) Intermediate Value Theorem
Solution: This is precisely the statement of the Intermediate Value Theorem
45. What is the limit of (cos x - 1) / x as x approaches 0?
a) -1 b) 0 c) 1 d) Does not exist
Answer: b) 0
Solution: Use L'Hôpital's rule or Taylor series expansion
46. If f(x) = x^3 - 3x^2 + 2x, what is the value of c guaranteed by the Mean
