Calculus -1

Zhejiang University of Technology MATH 011: Calculus I

Mid Term # 1 Max Marks: 30

Question 1 [10 marks] Choose the correct answer:

(i) The domain of the quadratic function x2 − 3x such that f(x) < 0 is:

(a) (0, 3) (b) [0, 3] (c) (−∞,∞) (d) [0,∞)

(ii) Which of the following conics is the graph of a valid function?

(a) the circle (x− 2)2 + (y − 3)2 = 49 (b) the ellipse x

2

16 + y

2

9 = 1

(c) the parabola y = 8x2

(d) the ellipse x 2

4 − y

2

5 = 1

(iii) Let f(x) = √ x and g(x) = x + 1. The domain of the composite function (g ◦f)(x) = g(f(x)) is:

(a) [−1,∞) (b) [0,∞) (c) R (d) R+

(iv) Consider a function f : X → X, where X = {3, 7, 9, 11}. If f(3) = 7, f(7) = 9, f(9) = 11 and f(11) = 3, what is the value of x that satisfies (f ◦f)(x) = 9?

(a) 3 (b) 7 (c) 9 (d) 11

(v) The left and right hand limits, x → 1− and x → 1+, respectively, for f(x) = 1 x3−1 are:

(a) 1/2, 5 (b) 0.25, 1.5 (c) ∞,−∞ (d) −∞,∞

(vi) It can be shown that the inequality 1 − x 2

6 < x sin x

2−2 cos x < 1 holds for all values of x close to zero. As x

approaches zero, what does this tell you about x sin x 2−2 cos x? It approaches:

(a) −∞ (b) +∞ (c) 0

1

(d) 1

(vii) The domain of f(x) is [−2, 2], with f(−2) = f(2) = 0. Which of the following statements about this function is false?

(a) limx→−2+ f(x) = 0 (b) limx→+2− f(x) = 0 (c) limx→−2− f(x) and limx→+2+ f(x) do not exist (d) Ordinary two-sided limits exist at both −2 and 2

(viii) The domain of f(x) is [−1, 1), with f(0) = 1. Which of the following statements about this function is false?

(a) limx→−1− f(x) does not exist (b) limx→1+ f(x) does not exist (c) limx→1− f(x) does not exist (d) limx→0− f(x) = 1

(ix) The vertical and horizontal lines through the point (−1, 4/3) are, respectively:

(a) x = 4/3 and y = −1 (b) x = −1 and y = 4/3 (c) x = −4/3 and y = 1 (d) x = 1 and y = −4/3

(x) For what values of k, respectively, will the lines 2x + ky = 3 and 4x + y = 1 be perpendicular and parallel to each other?

(a) k = 4 and k = 2 (b) k = −4 and k = −2 (c) k = −8 and k = 1/2 (d) k = −8 and k = 2

(xi) The distance from the point (3,−2) to the line 3x− 4y + 2 = 0 is:

(a) 19/25 (b) 3/5 (c) 3/25 (d) 19/5

(xii) The lines 2x− 3y + 7 = 0 and 3x + 7y − 2 = 0 meet at point

(a) (−43/23, 25/23) (b) (−43/5, 139/35) (c) (−1/11, 25/77) (d) (47/17,−107/119)

(xiii) The derivative with respect to x of the function x2ex is

2

(a) ex(2x + x2) (b) 2xex

(c) 2x2ex

(d) 2x + ex

(xiv) The derivative with respect to x of the function sin2 x + cos2 x is

(a) 0 (b) 1 (c) ex

(d) ejx

(xv) The derivative of f(x) = |x| at x = 0 is

(a) 0 (b) 1 (c) −1 (d) does not exist

(xvi) Let f(x) = √ x for x > 0. The tangent line to the curve y =

√ x at x = 4 is

(a) 4x + 1 (b) 1

2 x−1/2

(c) y = 1 4 x + 1

(d) 1 2

√ x + 1

(xvii) The integral ∫ x cos xdx evaluates to

(a) −cos x + sin x + C (b) x sin x + cos x + C (c) sin x + C (d) x sin x− cos x + C

(xviii) The integral ∫

1√ x dx evaluates to

(a) 2 √ x + C

(b) ln|x| + C (c) − 1

2 3 √ x

+ C

(d) 2 3

3 √ x + C

(xix) The integral ∫

1√ 16+x2

dx evaluates to

(a) tan−1(x 4 ) + C

(b) 1 4

tan−1(x 4 ) + C

(c) sin−1(x 4 ) + C

(d) 1 4

sin−1(x 4 ) + C

(xx) Let F ′(x) = f(x). Which of the following is true?

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(a) F (x) is the only antiderivative of f(x) (b) f(x) is an antiderivative of F (x) (c) G(x) = F (x) + C is also an antiderivative of f(x), for any arbitrary constant C (d) G(x) = F (x) + C is an antiderivative of f(x) only for some fixed C

Question 2 [4 marks]

(a) [1 marks] At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lb/in2. Below the surface, the water pressure increases by 4.34 lb/in2 for every 10 ft of descent.

(i) Express the water pressure as a function of the depth below the ocean surface. (ii) At what depth is the pressure 100 lb/in2?

(b) [2 marks] Find the domains of the following functions:

(i) cos x 1−sin x

(ii) tan x 1−exp|x|

(c) [1 mark] Let f : R → R be a function such that f(0) = 1 and for any x, y ∈ R, f(xy + 1) = f(x)f(y) − f(y) −x + 2 holds. Find f(x).

Question 3 [4 marks]

(a) [2 marks] Evaluate the following limits:

(i) limt→1 t2+t−2 t2−1

(ii) limx→−1 √ x2+8−3 x+1

(b) [2 marks] Using the squeeze (or sandwich) theorem, show that

(i) limθ→0 sin θ θ

= 1

(ii) limx→0+ √ xesin(π/x) = 0

Question 4 [4 marks]

(a) [2 marks] Show that the points A(2,−1), B(1, 3) and C(−3, 2) are consecutive vertices of a square by analyzing the slopes of the sides involved. Then find the fourth vertex by computing lines AD and CD and obtaining their point of intersection.

(b) [2 marks] Prove that the diagonals of a rhombus are perpendicular to each other by computing and analyzing their slopes.

Question 5 [4 marks]

(a) [2 marks] Differentiate the following functions:

(i) (x2 − 2 x3

)2

(ii) x 2+2x+2

2x3+x−1

(b) [2 marks] Find the first and second derivatives of the functions below:

(i) p = (q 2+3 12q

)(q 4−1 q3

)

4

(ii) y = sec x

Question 6 [4 marks]

(a) [2 marks] Show that ∫

1√ 1−x2

dx = sin− 1x + C.

(b) [2 marks] Evaluate the following integrals:

(i) ∫

1√ 4+25x2

dx

(ii) ∫

sin2 xdx

5