AP Calculus AB Questions

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Home>Mathematics homework help>Calculus homework help>AP Calculus AB Questions

1. 

Find the limit as h goes to 0 of the quotient of the quantity 5 times the square of the quantity x plus h, minus 2 minus the quantity 5 times x squared minus 2, and h . (5 points)

 

·

· 5x2 + h2 + 7

·

· 10x

·

· 3x2 - 2

·

· 10x - 2

2. 

Given f of x equals 5 times x minus 6 for x less than 2, equals x squared for x between 2 and 4 inclusive, and equals 2 times x plus 8 for x greater than 4 find f '(x) and give its domain. (5 points)

 

· 5 for x less than 2, equals 2x for x between 2 and 4, and equals 2 for x greater than 4 Domain: All real numbers, x ≠ 2, 4.

· 5 for x less than or equal to 2, equals 2x for x between 2 and 4 including 4, and equals 2 for x greater than 4 Domain: All real numbers.

· 5 for x less than 2, equals 2x for x between 2 and 4 including 4, and equals 2 for x greater than 4 Domain: All real numbers, x ≠ 2.

· 5 for x less than 2, equals 2x for x greater than or equal to 2 and less than 4, and equals 2 for x greater than 4 Domain: All real numbers, x ≠ 4.

3. 

Use a graphing calculator to graph f of x equals the quotient of the quantity 2 times x squared minus 8 and the quantity x squared minus 6 and then select the response which is true. (5 points)

 

· f increases without bound

· f has no tangent line

· f has a horizontal tangent line

· none of these responses

4. 

Find dy, dx for y = 3cos(x) + sec(x). (5 points)

 

·

· -3sin(x) + sec(x)tan(x)

·

· 3cos(x) - sec(x)tan(x)

·

· 3sin(x) - sec2(x)

·

· 3cos(x) - sec(x)

5. 

Differentiate y equals the quotient of the quantity 1 plus cosine x and the quantity 1 minus cosine x . (5 points)

 

· -1

· -2csc(x)

· 2csc(x)

· the quotient of negative 2 sine x and the square of the quantity 1 minus cosine x

6. 

Which one of the following statements is true? (5 points)

 

·

· If f and g are differentiable, then the derivative with respect to x of the product of f of x and g of x equals the product of f prime of x and g prime of x .

·

· the derivative with respect to x of the absolute value of the quantity x squared plus x equals the absolute value of the quantity 2 times x plus 1

·

· If f and g are differentiable, then the derivative with respect to x of the square root of f of x equals the quotient of f of x and 2 times the square root of f prime of x .

·

· None of these statements are true.

7. 

Is the following true or false?

the derivative with respect to x of the product of x cubed and e raised to the x power equals the product of x squared times e to the x power and the quantity 3 times x plus 2 (5 points)

 

·

· True

·

· False

8. 

FREE RESPONSE If h(x) = f[g(x)], use the table of values for f, g, f ' and g ' to find the value of h '(1). (5 points)

x

f(x)

g(x)

f '(x)

g '(x)

1

3

2

2

6

2

1

8

5

7

3

7

2

7

9

#8 (answer)___________________________

 

9. 

Determine the slope of the graph of x4 = ln(xy) at the point (1, e). (5 points)

·

·

· 3e

·

· e4

·

· 1

·

· 4

10. 

Find y' if x = sin(x + y). (5 points)

 

·

· the quotient of 1 minus the cosine of the quantity x plus y and the cosine of the quantity x plus y

·

· 1

·

· 0

·

· cos(x + y)

11. 

Use the graph of f(x) below to estimate the value of f '(3):

Graph is an inverted parabola with intercepts at x equals negative 3 and positive 3 and y-intercept at y = 9. (4 points)

 

·

· -5

·

· -1

·

· 0

·

· 1

12. 

Given the graph of f '(x), the derivative of f(x), which of the following statements are true about the graph of f(x)? (4 points)

Graph is parabola with x intercepts at x equals 0 and 3 and vertex at x = 1

I.The graph of f is decreasing on the interval (0, 2). II.The graph of f is increasing on the interval (1, infinity). III.The graph of f has point of inflection at x = 0.

 

· I only

· II only

· III only

· None of these are true

13. 

Find the critical point of the function f(x) = x2 - 2x + 7. (4 points)

 

·

· (-1, 10)

·

· (-1, 6)

·

· (1, 0)

·

· (1, 6)

14. 

For what values of x is the graph of y equals 3 divided by the quantity 6 minus x concave upward? (4 points)

 

· No values of x

· x < 6

· x > -6

· x > 6

15. 

At what value(s) of x does f(x) = x4 - 18x2 have a critical point where the graph changes from decreasing to increasing? (4 points)

 

·

· 0 and -3 only

·

· 0 and 3 only

·

· -3, and 3 only

·

· 0 only

16. 

The maximum value of f(x) = 2x3 - 18x2 + 48x - 1 on the interval [0, 3] is (4 points)

 

· -1

· 39

· 35

· 71

17. 

In which interval is the function f(x) = x cubed divided by 3 - 4x2 + 12x + 1 increasing? (4 points)

 

·

· (-∞, 2) only

·

· (-∞, 2) U (6, ∞)

·

· (6, ∞) only

·

· (2, 6) only

18. 

What are the critical values on the graph of y = x3 - 12x + 1? (4 points)

 

·

· x = -2, 0, 2

·

· x = -2, 2

·

· x = -4, 4

·

· x = -3.42, 0, 3.42

19. 

Find the x-coordinate of the point(s) of inflection on the graph of y = x3 - 12x2 - 5. (4 points)

 

·

· x = -2

·

· x = 4

·

· x = 2

·

· x = 12

20. 

Which one of the following statements is true? (4 points)

 

· If f "(x) > 0 on the interval (a, b) then f(x) is concave down on the interval (a, b).

· If f '(x) > 0 on the interval (a, b) then f(x) is increasing on the interval (a, b).

· If f '(c) = 0, then x = c is a relative maximum on the graph of f(x).

· None of these are true.