Zhejiang University of Technology MATH 011: Calculus I

Final Exam Max Marks: 40

Question 1 [10 marks] Choose the correct answer:

(i) Let

f(x) =

{ 0 x ≤ 0 sin 1

x x > 0

Which of the following statements about f(x) is true?

(a) limx→0+ f(x) = 0 (b) limx→0− f(x) = 0 (c) limx→0 f(x) (d) limx→0+ f(x)

(ii) The tangent line to the function f(x) = 8√ x−2 at point (x,y) = (6, 4) is given by

(a) 3y = 6x + 2 (b) y = −1

2 x + 7

(c) 7y = x + 3 (d) y = 2x + 3

(iii) The function f(x) = x is defined on the interval (0, 1). Which of the following statement is true for f(x)?

(a) f(x) is not a continuous function (b) f(x) has a maximum at 1 (c) f(x) has a minimum at 0 (d) f(x) does not have a maximum or minimum value

(iv) The absolute maximum and minimum values of f(x) = 2 3 x− 5 defined on the interval −2 ≤ x ≤ 3 are,

respectively:

(a) (3,−3) and (−2,−19/3) (b) (−2,−19/3) and (3,−3) (c) (∞,∞) and (−∞,−∞) (d) do not exist

(v) The integral ∫

tan θ sec2 θdθ evaluates to

(a) 1 2

tan2 θ + C (b) sec2 θ + 3 tan2 θ sec2 θ + C

(c) sec 3 θ 3

+ C

(d) sec 2 θ 2

+ C

1

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(vi) The integral ∫

cos θ(tan θ + sec θ)dθ evaluates to

(a) sec2 θ (b) sec θ tan θ (c) −sin θ + cos θ (d) −cos θ + θ + C

(vii) Which of the following functions have a derivative at every point?

(a) a continuous function (b) a smooth function (c) an absolute value function (d) a sawtooth function

(viii) The length of the curve y = 1 3 (x2 + 2)3/2 from x = 0 to x = 3 is

(a) 53/6 (b) 12 (c) 32/3 (d) 8

27 (10 √

10 − 1)

(ix) The polar coordinates corresponding to the Cartesian coordinates ( √

3,−1) and (− √

3, 1) are, respec- tively,

(a) ( √

2,π/4) and 2 √

2,−3π/4 (b) (3,π) and (3,π/2) (c) (2, 11π/6) and (2, 5π/6) (d) (5,π − arctan(4/3)) and (13,−arctan(12/5))

(x) A point (2, 7π/3) in polar coordinates has the equivalent Cartesian coordinates

(a) (1, √

3) (b) (−1,

√ 3)

(c) (1,− √

3) (d) (3

√ 3/2,−3/2)

(xi) Referred to origin O, points A and B have position vectors OA = î + 2ĵ − 2k̂ and OB = 2̂i− 3ĵ + 6k̂. The value of α for which angles ∠AOP and ∠POB are equal if OP = (1 + α)̂i + (2 − 5α)ĵ + (−2 + 8α)k̂ is

(a) 3/10 (b) 1/

√ 6

(c) 1/ √

3 (d) √

2

(xii) The position vector of A is −→ OA = î− ĵ + 2k̂. The acute angle between

−→ OA and the y axis is given by

(a) cos θ = 1/ √

6 (b) cos θ = 1/

√ 2

(c) cos θ = √

3/2 (d) cos θ = 1/2

2

(xiii) A square matrix of the second order partial derivatives of a scalar-valued function is a

(a) Jacobian matrix (b) Hessian matrix (c) orthogonal matrix (d) diagonal matrix

(xiv) The second order partial derivative ∂ 2f ∂x2

of the function x cos y + yex is

(a) yex

(b) −sin y + ex (c) cos y + yex

(d) −x sin y + ex

(xv) Given f(x,y,z) = ln(x + 2y + 3z), ∂f ∂z

is

(a) 1 x+2y+3z

(b) 2 x+2y+3z

(c) 3 x+2y+3z

(d) none of the above

(xvi) Given w = xy + e y

y2+1 , ∂

2w ∂x∂y

is

(a) y (b) 1 (c) ey

(d) 2yey

(xvii) The area of the region R bounded by y = x and y = x2 in the first quadrant is

(a) 1/3 (b) 1/2 (c) 1/6 (d) 3/2

(xviii) The area of the region R enclosed by the parabola y = x2 and the line y = x + 2 is

(a) 6 (b) 1/3 (c) 33 (d) 9/2

(xix) The double integral ∫ 3 0

∫ 2 0

(4 −y2)dydx evaluates to

(a) 1 (b) 1/6 (c) 16 (d) 3

3

(xx) The double integral ∫ 1 0

∫ y2 0

(3y3exy)dxdy evaluates to

(a) e− 2 (b) π

2

2 + 2

(c) 8 ln 8 − 16 + e (d) 3

2 ln 2

Question 2 [5 marks]

(a) [1 mark] Show that limh→0 cosh−1

h = 0.

(b) [1 mark] Evaluate limx→0 tan 3x sin 8x

.

(c) [1 mark] Differentiate y = (1 + tan4 t 12

)3.

(d) [2 marks] Find the critical points for the following functions and use the second derivative test to determine if they are local minima, maxima or points of inflexion. Assume the derivative is defined over the entire domain.

(i) y = x3 + x2 − 8x + 5 (ii) y = 13√

1−x2

Question 3 [5 marks]

(a) [2 marks] The temperature (in ◦C) of a cup of juice is observed to be T(t) = 25(1 − e−0.1t) where t is time. Evaluate the rate of change of T(t) and then use integration to find the change in temperature between times t = 1 and t = 5. Verify your answer by evaluating T(t) at t = 1 and t = 5.

(b) [2 marks] Find the area contained between the curves y = 3x−x2 and x + x2.

(c) [1 mark] Evaluate ∫

1√ 16−9×2

dx.

Question 4 [5 marks]

(a) [2 marks] Find the areas of the surfaces generated by revolving the curves below about the x axis:

(i) y = x3/9, 0 ≤ x ≤ 2 (ii) y =

√ x, 3

4 ≤ x ≤ 15

4

(b) [3 marks] Solve the following initial value problems:

(i) x2y′ + 2xy = ln x, y(1) = 2 (ii) tdu

dt = t2 + 3u, t > 0,u(2) = 4

(iii) xy′ = y + x2 sin x, y(π) = 0

Question 5 [5 marks]

(a) [1 mark] Where does the line −→r = (1, 1, 0) + t(2, 3, 4) meet the plane 2x + y −z = 0?

(b) [1 marks] (i) Find the point on the plane x – y + z = 2 that is closest to the point P : (1, 1,−1). (ii)

4

Find the distance from the point P to the plane.

(c) [3 marks] (i) Find a vector parametric form of the plane in 3-space that passes through the points (1, 4, 2), (0, 3, 0), and (−1, 1, 3). (ii) Write the plane −→r = (1, 4, 2) + s(−1,−1,−2) + t(−2,−3, 1) in the scalar form ax + by + cz = d. (iii) Find a parametric equation for the line in 3-space through the point (0,0,0) and that is perpendicular to the plane in (ii). (iv) Find the point on the plane that is closest to (0,0,0). (v) Find the distance from the point (0,0,0) to the plane.

Question 6 [5 marks]

Find all the second order partial derivatives of the following functions:

(i) f(x,y) = x + y + xy (ii) g(x,y) = x2y + cos y + y sin x (iii) f(x,y) = x3y5 + 2x4y (iv) w =

√ u2 + v2

(v) z = arctan x+y 1−xy

Question 7 [5 marks]

Evaluate the following double integrals:

(i) ∫ 1 0

∫ 1−x 0

(x2 + y2)dydx

(ii) ∫ 1 0

∫ 1−u 0

(v − √ u)dvdu

(iii) ∫ 2 0

∫ 2 x

2y2 sin xydydx

(iv) ∫ 1 0

∫ 1 y

(x2exy)dxdy

(v) ∫ 2 0

∫ 4−x2 0

xe2y

4−y dydx

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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 = 8×2

(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

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(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?

3

(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

2×3+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+25×2

dx

(ii) ∫

sin2 xdx

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Zhejiang University of Technology MATH 011: Calculus I

Mid Term # 2 Max Marks: 30

Question 1 [10 marks] Choose the correct answer:

(i) The area of the surface generated by revolving the curve y = 2 √ x, 1 ≤ x ≤ 2 about the x axis is:

(a) 8π 3

(3 √

3 − 2 √

2)

(b) π √

2 (c) π/2 (d) 61π

1728

(ii) The area of the surface generated by revolving y = tan x, 0 ≤ x ≤ π/4 about the x axis is:

(a) 2π ∫ π/4 0

tan x √

1 + sec4 xdx

(b) 2π ∫ π/4 0

x2 √

1 + sec4 xdx

(c) 2π ∫ π/4 0

sec2 xdx

(d) 2π ∫ π/4 0

sec x tan xdx

(iii) The lateral surface area of the cone generated by revolving the line segment y = x 2 , 0 ≤ x ≤ 4 about the

x axis is:

(a) 2 √

5 (b) 4π (c) 4π

√ 5

(d) π/4

(iv) The curve x = y3/4, 0 ≤ y ≤ 1 is revolved about the y axis. The resulting surface area is:

(a) π 2

(b) π 2

√ 5

(c) 49π 3

(d) π 9

( √

8 − 1)

(v) Which of the following functions is a solution of the differential equation y′′ + y = sin x?

(a) y = sin x (b) y = cos x (c) y = 1

2 x sin x

(d) y = −1 2 x cos x

(vi) For what values of P is a population modeled by the differential equation dP dt

= 1.2P(1− P 4200

) increasing?

(a) 0 < P < 4200 (b) P > 4200 (c) P = 0

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(d) P = 4200

(vii) Which of the following functions is a solution of the differential equation tdy dt

= y + t2 sin t?

(a) y = t cos t (b) y = t sin t (c) y = −t cos t− t (d) y = t cos t− t sin t

(viii) Which of the following differential equations is linear?

(a) y′ + x √ y = x2

(b) y′ −x = y tan x (c) u2et = t +

√ tdu dt

(d) dR dt

+ t cos R = e−t

(ix) The parametric equations x = sinh t,y = cosh t describe a particle moving along

(a) a parabola (b) an ellipse (c) the upper branch of a hyperbola (d) the lower branch of a hyperbola

(x) The motion of the particle described by the parametric equations x = 5 sin t,y = 2 cos t takes place on

(a) a parabola (b) an ellipse (c) the upper branch of a hyperbola (d) the lower branch of a hyperbola

(xi) A point P(r,θ) give by the polar coordinates (−2,−π/3) may also be represented by

(a) (−3,π) (b) (2,π/3) (c) (2, 2π/3) (d) (2, 7π/3)

(xii) A point P(r,θ) give by the polar coordinates (−2,π/3) may also be represented by

(a) (2,−2π/3) (b) (−2, 2π/3) (c) (2,−π/3) (d) (2,π/3)

(xiii) The nth term of the infinite sequence 1,−4, 9,−16, 25, … is

(a) (−1)n+1; n ≥ 1 (b) n2 − 1; n ≥ 1 (c) (−1)n+1n2; n ≥ 1

2

(d) 1+(−1)n+1

2 ; n ≥ 1

(xiv) The nth term of the infinite sequence 1, 5, 9, 13, 17, … is

(a) 4n− 3; n ≥ 1 (b) n− 4; n ≥ 1 (c) 3n+2

n! ; n ≥ 1

(d) n 3

5n+1 ; n ≥ 1

(xv) The first four terms of the infinite sequence an = 1−n n2

are

(a) 0,−1/4,−2/9,−3/16 (b) 1,−1/3, 1/5,−1/7 (c) 1/2, 1/2, 1/2, 1/2 (d) none of the above

(xvi) The first four terms of the infinite sequence an = 2n

2n+1 are

(a) 0,−1/4,−2/9,−3/16 (b) 1,−1/3, 1/5,−1/7 (c) 1/2, 1/2, 1/2, 1/2 (d) none of the above

(xvii) Given −→a = 3̂i + 4ĵ − k̂ and −→ b = î− ĵ + 3k̂, −→a •

−→ b is

(a) 23 (b) 0 (c) 13 (d) −4

(xviii) −→a and −→ b are two non-zero vectors. The vector −→r = (

−→ b • −→ b )−→a − (−→a •

−→ b ) −→ b is

(a) perpendicular to −→ b

(b) parallel to −→a (c) anti-parallel to −→a (d) parallel to

−→ b

(xix) The work done by a constant force of 40N, applied at an angle of 60◦ to the horizontal, results in a displacement of 3m. The work done is

(a) 60J (b) 90J (c) 30J (d) 120J

(xx) −→u = (2, 0,−1) and −→v = (1, 4, 7). Their cross product is

(a) (14,−15, 22) (b) (4,−15, 8)

3

(c) (3,−1, 0) (d) (−6, 2, 0)

Question 2 [4 marks]

(a) [2 marks] Find the exact lengths of the following curves by setting up and evaluating the corresponding definite integral.

(i) y = 1 + 6×3/2, 0 ≤ x ≤ 1 (ii) y = ln(sec x), 0 ≤ x ≤ π/4

(b) [2 marks] Find the centroid of the region bounded by the curves y = x3 −x and y = x2 − 1.

Question 3 [4 marks]

(a) [2 marks] For what values of r does the function y = erx satisfies the differential equation 2y′′ + y′−y = 0? If r1 and r2 are the values of r that you found, show that every member of the family of functions y = aer1x + ber2x is also a solution.

(b) [2 marks] Solve the following linear differential equations:

(i) xy′ + y = √ x

(ii) xy′ − 2y = x2,x > 0

Question 4 [4 marks]

(a) [2 marks] Eliminate the parameter to find a Cartesian equation of the following curves. Sketch the curve in the xy plane and indicate with an arrow the direction in which the curve is traced as the parameter increases.

(i) x = sin 1 2 θ,y = cos 1

2 θ,−π ≤ θ ≤ π

(ii) x = sin t,y = csc t, 0 < t < π/2 (iii) x = e2t,y = t + 1 (iv) x = cos(π − t),y = sin(π − t), 0 ≤ t ≤ π

(b) [2 marks] Identify the curve by finding a Cartesian equation for it:

(i) r cos θ = 2 (ii) r2 = 5 (iii) r = 2 cos θ (iv) r2 cos 2θ = 1

Question 5 [4 marks]

(a) [2 marks] Determine if the following sequences converge:

(i) xn = sin2 n√

n

(ii) xn = 2n+1 en

(iii) xn = n3

10n2+1

(iv) xn = tan−1 n

n

4

(b) [2 marks] Determine if the following series converge using either the integral test or the ratio and root test:

(i) ∑∞ n=1 cos(nπ)

(ii) ∑∞ n=1

3n

n(2n+1)

(iii) ∑∞ n=1

3n

n! (iv)

∑∞ n=1(

3n+2 2n−1 )

n

Question 6 [4 marks]

(a) [1 mark] Find the point on the plane x−y + z = 0 that is closest to the point P : (2, 3, 0). Also find the distance from the point P to the plane.

(b) [3 marks] (i) Find a vector parametric form of the plane in 3-space that passes through the points (3, 2, 1), (0, 0, 2), and (1,−2, 3). (ii) Write the plane r = (3, 2, 1) + s(−3,−2, 1) + t(−2,−4, 2) in the scalar form ax + by + cz = d. (iii) Find a parametric equation for the line in 3-space through the point (2, 5, 0) and that is perpendicular to the plane in (ii). (iv) Find the point on the plane that is closest to (2, 5, 0). (v) Find the distance from the point (2, 5, 0) to the plane.

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