CALCULUS II

Exam I, September 27, 2010

You must show all your work. Problems (1), (5), (6-a) are 15 pointseach; the rest are 10 points each.

(1) Consider the cycloid x = r(θ − sin θ), y = r(1− cos θ). Find the areaof the surface obtained by rotating one arch of the cycloid about the

x-axis.

(2) Consider the two polar equations: the circle r = 3 cos θ and the cardioid

r = 1 + cos θ.

(a) Find the polar curves on the same plane.

(b) Find the area of the region that lies inside the circle r = 3 cos θ

and outside the cardioid r = 1 + cos θ.

(3) Define the convergence and the sum of the series∑∞

i=1 ai.

(4) Find the values of x for which the series∑∞

n=0(cosn x)/2n converges,

and then find the sum of the series for those values of x.

(5) Test the p-series∑∞

n=1 1/np for the convergence or divergence.

(6) Determine whether the following series is convergent or divergent. (Name

the test(s) that you are using and check the conditions for the test(s)

carefully.)

(a)∑∞

n=1(ln n)/n3

(b)∑∞

n=1(−1)nnn/n!(c)

∑∞n=1(−1)n(arctan n)/n2

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CALCULUS II

Exam II, November 8, 2010

You must show all your work. Problems (2), (3), (4) and (5) are worth 15points each; the rest are worth 10 points each.

(1) Find the values of x for which the series∑∞

n=0(cosn x)/2n converges,

and then find the sum of the series for those values of x.

(2) Find the power series for f(x) = tan−1 x and the interval of conver-gence.

(3) Find the MacLaurin series of sinx, and show that sinx equals itsMacLaurin series for all values of x (assuming that the power series∑

xn/n! converges for all x).

(4) For the points A(6,−3, 0), B(3,−7, 1), C(3, 7,−1) and D(4, 5,−3),find the shortest distance between the lines AB and CD.

(5) The points A,B and C have position vectors a,b and c respectively,relative to an origin O, where

a = 3i + 4j + 5k, b = 4i + 6j + 7k, c = i + 5j + 3k.

(a) Find (b− a)× (c− a).(b) Hence, or otherwise, find the area of 4ABC and the volume of

tetrahedron OABC.

(c) Find an equation of the plane ABC in the Cartesian form ax+by + cz = d.

Given that the point D has position vector (2i + j + 2k),

(6) Find

lim(x,y)→(0,0)

3x2yx2 + y2

if it exists.

(7) (a) Define “z = f(x, y) is differentiable at (a, b)”.(b) Find the indicated partial derivative: u = erθ sin θ; ∂3u/∂r2∂θ.

(8) Find the linear approximation of the function f(x, y) =√

20− x2 − 7y2at (2, 1) and use it to approximate f(1.95, 1.08).

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CALCULUS II

Final Exam, December 17, 2010

You must show all your work. Each problem is worth of 10 points.

(1) Find the linear approximation of the function f(x, y) =√

20− x2 − 7y2at (2, 1) and use it to approximate f(1.95, 1.08).

(2) The pressure P (in kilopascals), volume V (in liters), and tem-perature T (in kelvins) of a mole of an ideal gas are related bythe equation PV = 8.31T . Find the rate at which the pressureis changing when the temperature is 300 K and increasing ata rate of 0.1 K/s and the volume is 100 L and increasing at arate of 0.2 L/s.

(3) Find ∂z/∂x and ∂z/∂y if x3 + y3 + z3 + 6xyz = 1.

(4) Find the directional derivative of the function f(x, y, z) = x sin(yz)at the point (1, 3, 0) in the direction of the vector v = i+2j−k.

(5) Find the equations of the tangent plane and normal line at thepoint (−2, 1,−3) to the ellipsoid

x2

4+ y2 +

z2

9= 3

(6) Find the local maximum and minimum values and saddle pointsof f(x, y) = x4 + y4 − 4xy + 1.

(7) Find the extreme values of the function f(x, y) = x2 + 2y2 onthe circle x2 + y2 = 1.

(8) Evaluate the iterated integral∫ 1

0

∫ 1x

sin(y2) dydx.

(9) Use a double integral to find the area enclosed by one loop ofthe four-leaved rose r = cos 2θ.

(10) Evaluate∫∫∫

E

√x2 + z2 dV , where E is the region bounded by

the paraboloid y = x2 + z2 and the plane y = 4.

(11) Evaluate∫∫∫

Be(x

2+y2+z2)3/2 dV , where B is the unit ball:

B = {(x, y, z) | x2 + y2 + z2 ≤ 1}.

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CALCULUS II

Exam I, September 26, 2011

You must show all your work.

(1) [10] A circle of radius r is rolling along the x-axis. A point P on thecircumference of the circle is traced out and makes a cycloid. Whenone position of P is the origin, show that parametric equations for thecycloid are x = r(θ − sin θ), y = r(1− cos θ).

(2) [25=8+8+9] Consider the cycloid x = r(θ − sin θ), y = r(1− cos θ).(a) Find the area under one arch of the cycloid.(b) Find the length of one arch of the cycloid.(c) Find the area of the surface obtained by rotating one arch of the

cycloid about the x-axis.

(3) [25=5+10+10] Consider the two polar equations: the circle r = 3 cos θand the cardioid r = 1 + cos θ.(a) Draw the polar curves on the same plane.(b) Find the area of the region that lies inside the circle r = 3 cos θ

and outside the cardioid r = 1 + cos θ.(c) Find the length of r = 1 + cos θ.

(4) [10] A curve is defined parametrically by the equations x = 4 cosh φ andy = 3 sinh φ. Let P be the point where φ = ln 1

2. Find the equation of

the tangent to the curve at the point P .

(5) [20=5+5+5+5] Let P be the point (−6, 4) on the ellipsex2

102+

y2

52= 1.

(a) Find the equation of the tangent to the ellipse at P .(b) Let the tangent meet the tangents at the ends of the major axis

at M1 and M2 respectively. Find the coordinates of M1 and M2.(c) Find the eccentricity and show that F = (−5√3, 0) is a focus of

the ellipse.(d) Show that M1F and M2F are perpendicular to each other.

(6) [10] Define the convergence and the sum of the series∑∞

i=1 ai.

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CALCULUS II

Exam II, November 7, 2011

You must show all your work.

(1) [10] Consider the two polar equations: the circle r = 3 cos θ and thecardioid r = 1 + cos θ. Find the area of the region that lies insidethe circle r = 3 cos θ and outside the cardioid r = 1 + cos θ.

(2) [10] Simply write the numbers of the statements that are TRUE amongthe following four statements. No reason is necessary. (Write onlyyour answer; for example, the answer is (a) and (c).)(a) If limn→∞ an = 0 then the infinite series

∑an converges.

(b) If the infinite series∑

an diverges then the series∑ |an| also

diverges.(c) If the infinite series

∑cn6n converges then the series

∑cn(−2)n

also converges.(d)

∑∞n=0

(−1)nn! =

1e .

(3) [10] Find one series which is convergent among the following, andthen give a reason why.(a)

∑∞n=2

1ln n (b)

∑∞n=1

3n2+4n+5n3+1

(c)∑∞

n=1(−1)n ln nn (d)∑∞

n=1 sin1n

(4) [10] Find the values of x for which the series∑∞

n=0(cosn x)/2n con-

verges, and then find the sum of the series for those values of x.

(5) [10] Find a power series for f(x) = tan−1 x and the interval of con-vergence.

(6) [10] In the power series of f(x) = e−x2 sinx about a = 0, write thecoefficient of x5.

(7) [15] For the points A(6,−3, 0), B(3,−7, 1), C(3, 7,−1) and D(4, 5,−3),find the shortest distance between the lines AB and CD.

(8) [5] Find the volume of the parallelepiped determined by three vectors〈2, 3,−1〉, 〈1, 2, 0〉 and 〈−2, 1, 0〉.

(9) [20] Let r(t) be a vector function in R3. Prove

κ(t) =|r′(t)× r′′(t)|

|r′(t)|3 .

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CALCULUS II

Final Exam, December 12, 2011

(1) [5 points] A curve is defined parametrically by the equations x =4 cosh φ and y = 3 sinhφ. Let P be the point where φ = ln 12 . Findthe equation of the tangent to the curve at the point P .

(2) [10 points] Find a power series for f(x) = tan−1 x and the intervalof convergence.

(3) [5 points] Show that the following limit does not exist.

lim(x,y)→(0,0)

xy2

x2 + y4

(4) [5 points] Find all the first partial derivatives of

f(x, y) = arctanx + y1− xy .

(5) [5 points] Find the linear approximation of the function f(x, y) =ln(x− 3y) at (7, 2) and use it to approximate f(6.9, 2.06).

(6) [5 points] Find ∂x/∂y at (1, 1, 0) if x3 + y3 + z3 + 6xyz = 2.

(7) [10 points] Let Π be the tangent plane to the level surface x2y+z2 = 1at the point (1, 1, 0). Find the distance from the point (0, 0, 1) tothe plane Π.

(8) [5 points] Find the equation of the normal line at the point (−2, 1,−3)to the ellipsoid

x2

4+ y2 +

z2

9= 3

(9) [10 points] Find the local maximum and minimum values and saddlepoints of f(x, y) = x4 + y4 − 4xy + 1.

(10) [10 points] Use a double integral to find the area enclosed by oneloop of the polar curve r = cos2 θ.

(11) [10 points] Use cylindrical coordinates to evaluate the triple integral∫∫∫

S

√x2 + z2 dV where S is the solid enclosed by the paraboloid

y = x2 + z2 and the plane y = 4.

(12) [10 points] Use spherical coordinates to find the volume of the solidS that lies above the cone z =

√x2 + y2 and below the sphere

x2 + y2 + z2 = z.

(13) [10 points] Use a change of variables to evaluate∫∫

Re

x+yx−y dA, where

R is the trapezoidal region with vertices (1, 0), (2, 0), (0,−2), and(0,−1).

1

CALCULUS II

Exam I, September 26, 2012

You must show all your work.

(1) [20=10+10] Consider a cycloid x = rθ − r sin θ, y = r − r cos θ.(a) Find the length of one arch of the cycloid.(b) Find the area of the surface obtained by rotating one arch of the

cycloid about the x-axis.

(2) [20=3+7+10] Consider the parametric equations x = rθ− d sin θ, y =r − d cos θ where 0 < d < r.(a) Sketch the parametric curve, called a trochoid. (Remark that if

r = d, then the curve will be a cycloid.)(b) Find the slope of the tangent line to the trochoid when θ = π/2,

and show that the trochoid does not have a vertical tangent atany point.

(c) Find the area under one arch of the trochoid.

(3) [30=5+5+10+10] Consider the three polar equations: r1 = 1 + 2 sin θ,r2 = sin 2θ and r3 = cos 2θ.(a) Draw the polar curve r1.(b) Write down the expression for the length of the inner loop of the

curve r1 (Do not evaluate it).(c) Find the area of the region enclosed by the inner loop of the curve

r1.(d) Find the area of the region that lies inside both curves r2 and r3.

(4) [5] Find the cosine of the angle between a diagonal of a cube and oneof its edges.

(5) [15] Given the points A(6,−3, 0), B(3, 7,−1), C(3,−7, 1) and D(4, 5,−3),find the shortest distance between the lines AB and CD.

(6) [10] Reparametrize the curve

r(t) = 2ti + (1− 3t)j + (5 + 4t)kwith respect to arc length s measured from the point where t = 0 inthe direction of increasing t. Namely, find r(s).

1

CALCULUS II

EXAM II, NOVEMBER 7, 2012

You must show all your work.The marks for each question are indicated in square brackets [ ].The total will be 90.

(1) Given f(x, y) = 4x2 + y2,(a) Sketch the level curves f(x, y) = 0 and f(x, y) = k for some

k > 0. [3](b) Sketch the graph of z = f(x, y). [4]

(2) Show that the following limit does not exist. [10]

lim(x,y)→(0,0)

xy2

x2 + y4

(3) For a three variable function f , define its partial derivative fz. [5]

(4) Suppose that z = f(x, y) is a differentiable function of x and y,where x = g(s, t) and y = h(s, t) are differentiable functions of s andt. What is the Chain Rule? [5]

(5) Define the directional derivative of z = f(x, y) at (x0, y0) in thedirection of a unit vector u = 〈a, b〉. [5]

(6) Define the gradient ∇F of w = F (x, y, z). [5]

(7) Let z = f(x, y) be a differentiable function at (a, b). Suppose (a, b)is a critical point of f . Then state the Second Derivatives Test. [5]

(8) To find the maximum and minimum values of f(x, y, z) subject tothe constraint g(x, y, z) = k, we may use the method of Lagrangemultipliers. Explain this method in detail. [5]

(9) Find all the first partial derivatives of [5]

f(x, y, z) = xy sin−1(yz).

(10) Find the linear approximation L(x, y) of the function f(x, y) = 1 +x ln(xy − 5) at (2, 3). [10]

1

2 EXAM II, NOVEMBER 7, 2012

(11) Use the chain rule to find the derivative of the function [5]

w = xeyz , x = t2, y = 1− t, z = 1 + 2t.

(12) [15 points] Let Π be the tangent plane to the level surface x2y+z2 = 1at the point (1, 1, 0).(a) Find an equation of the plane Π. [5](b) Find the distance from the point (0, 0, 1) to the plane Π. [5](c) Find an equation of the normal line at the point (1, 1, 0) to level

surface x2y + z2 = 1. [5]

(13) Find the local maximum and minimum values and saddle points off(x, y) = y cos x. [8]

CALCULUS II

FINAL EXAM, DECEMBER 14, 2012

(1) [15 points] Let r(t) be a vector function in R3. Define the curvatureκ of the curve r(t). Prove

κ(t) =|r′(t)× r′′(t)|

|r′(t)|3 .

(2) [10 points] Evaluate the line integral∫

C

−xx2 + y2

dx +y

x2 + y2dy

where the curve C is given by x = cos t, y = sin t, 0 ≤ t ≤ π2 .(3) [15 points] Evaluate the double integral∫∫

Re|x−y| dA

where R = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.(Hint: Use

∫∫R1+R2

f(x, y) dA =∫∫

R1f(x, y) dA+

∫∫R2

f(x, y) dA.)

(4) [15 points] Evaluate the double integral∫∫

Rsin(y2) dA

where R = {(x, y) | 0 ≤ x ≤ √π4 , x ≤ y ≤√

π4 }. (Hint: Express R

as a region of type II.)

(5) [15 points] Use a change of variables to evaluate∫∫

Re

x+yx−y dA

where R is the trapezoidal region with vertices (1, 0), (2, 0), (0,−2),and (0,−1).

(6) [15 points] Evaluate the triple integral∫ 1

0

∫ 1y

∫ y0

ex

2xdzdxdy

Hint: Find a new expression of

B = {(x, y, z) | 0 ≤ y ≤ 1, y ≤ x ≤ 1, 0 ≤ z ≤ y}.

(7) [15 points] Using the spherical coordinates, evaluate the triple inte-gral

∫ 2−2

∫ √4−y2

0

∫ √4−x2−y2

−√

4−x2−y2y2

√x2 + y2 + z2 dzdxdy

1

CALCULUS II

Exam I, October 4, 2013

You must show all your work.

(1) Consider the cycloid x = rθ − r sin θ, y = r − r cos θ.(a) Find the area of one arch of the cycloid. [10]

(b) Find the length of one arch of the cycloid. [10]

(c) Find the area of the surface obtained by rotating one arch of the

cycloid about the x-axis. [10]

(2) Consider the two polar equations: r1 = 1 + sin θ and r2 = 3 sin θ.

(a) Find the area of the region that lies outside r1 and inside r2. [10]

(b) Find the length of r1. [15]

(c) Find the slope of the tangent line for r1 when θ = π/3. [10]

(3) Find the cosine of the angle between a diagonal of a cube and a diagonal

of one of its faces. [5]

(4) Find the angle between two vectors a and b when a · b = √3 anda× b = 〈1, 2, 2〉. [5]

(5) Two planes π1 and π2 have equations:

π1 : 3x− 2y + 2z = 4, π2 : 〈x, y, z〉 · 〈1, 2, 3〉 = 11(a) Find parametric equations for the line of intersection of the two

planes. [10]

(b) Find the cosine of the angle between the planes. [5]

(6) Draw the surface x2 + y2 − z2 = 1 with a full explanation. [10]

1

CALCULUS II

EXAM II, NOVEMBER 15, 2013

You must show all your work. The total marks will be 100.The marks for each question are indicated in square brackets [ ].

(1) Consider two polar equations: r1 = 1 + sin θ and r2 = 3 sin θ. Findthe area of the region that lies outside r1 and inside r2. [15]

(2) For the space curve given by the vector function r(t) = 〈x(t), y(t), z(t)〉,show that its curvature κ(t) is [20]

κ(t) =|r′(t)× r′′(t)|

|r′(t)|3 .

(3) Using the formula in Problem (2), find the curvature κ(x) of theparabola y = f(x) = x2. Sketch (draw roughly) the graphs of bothy = f(x) and y = κ(x) in the same xy-plane. [10+5]

(4) Show that the following limit does not exist. [10]

lim(x,y)→(0,0)

x2 + sin2 y2x2 + y2

(5) Find (i) an equation of the tangent plane to the elliptic paraboloidz = 2x2 + y2 at the point (1, 1, 3), and find (ii) an equation of theline perpendicular to the surface z = 2x2 + y2 at (1, 1, 3). [10+5]

(6) Suppose that z = f(x, y) is a differentiable function of x and y,where x = g(s, t) and y = h(s, t) are differentiable functions of s andt. In this case, state the Chain Rule? [5]

(7) Answer the following:(a) Show that the function u(x, t) = sin(x− at) satisfies the follow-

ing equation [5]

∂2u

∂t2= a2

∂2u

∂x2.

(b) Find ∂∂z (y)(=∂y∂z ) if yz + x ln y = z

2. [5](c) Find Duf(1, 2) if f(x, y) = x3 − 3xy + 4y2 and u is the unit

vector given by the angle θ = π/6. [10]

1

CALCULUS II

FINAL EXAM, DECEMBER 18, 2013

(1) For the space curve given by the vector function

r(t) = 〈x(t), y(t), z(t)〉,prove that its curvature κ(t) is [10]

κ(t) =|r′(t)× r′′(t)|

|r′(t)|3 .

(2) Find the local maximum and minimum values and saddle point(s)of the function f(x, y) = sin x sin y with domain −π < x, y < π.

[20]

(3) Find the extreme values of f(x, y) = e−xy on the region de-scribed by the inequality x2 + 4y2 ≤ 1. [20]

(4) Find the volume of solid bounded by the paraboloid z = x2+y2,the xy-plane and the cylinder x2 + y2 = 2y. [15]

(5) Using the cylindrical coordinates, evaluate the triple integral∫∫∫

B

√x2 + z2 dV

where B is bounded by the paraboloid y = x2 + z2 and theplane y = 4. [15]

(6) Using the spherical coordinates, evaluate the triple integral [20]

∫ 20

∫ √4−y2

0

∫ √4−x2−y2

−√

4−x2−y2y2

√x2 + y2 + z2 dzdxdy

1

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