Exam Papers in Calculus IV and Calculus for CN Yang Scholars 2006-2009

Exam questionsMAS212 / MATH1B

2006-2009

MAS212 is calculus of several variables for mathematical sciences students. MATH1Bis calculus of several variables plus something else for CN Yang scholars. The curriculumhas been changed and this something else was partial differential equations in 2007-2008and basics of linear algebra in 2009.

This brochure contains all final exams for MAS212, years 2006-2009 (the course didn’texist before that), all final exams for MATH1B, years 2007-2009 (the course didn’t existbefore that), mid-term exams, years 2008-2009 (earlier mid-terms have been lost).

2006

MAS212 Final Exam

Question 1 (20 marks)

Let f : E → Rm, for E = R− {0}, be a function given by x 7Ï (1, 1

x, 1x2 , . . . ,

1xm−1 ); that is, the

i-th coordinate of the image (1 ≤ i ≤ m) is given by 1xi−1 .

(i) (5) Find, provided it exists, each of the following limits:

limx→− 1

2

f (x);

limx→∞

f (x);limx→0

f (x).

(ii) (10) Find the equation for the tangent line to the curve f (E) ⊂ Rm at the point f (x),

for x ∈ E.

1

(iii) (5) Write down a formula, in terms of a definite univariate integral, for the lengthof the arc f ([1, 2]) of the curve f (R). Prove that this length is finite.


Let f : Rn → R

m be a function differentiable at a ∈ Rn.

(i) (10) Define φ : R2 → R

m by φ(x, y) = f (a+(x+x3)v+y2u), where v, u ∈ Rn. Derive

a formula for φ′(0, 0) as a function of f ′(a).

(ii) (10) Show that for any u, v ∈ Rn the following formula for directional derivatives

holds: Du+vf (a) = Duf (a) +Dvf (a).


Compute ∫∫∫∫

x2+y2+u2+v2≤1

exp(x2 + y2 − u2 − v2)dx dy du dv.

Hint: use 2-dimensional polar coordinates for two different circles: one in (u, v)-coordinates, another in (x, y)-coordinates.


(i) (10) For the ellipse C = {(x, y) ∈ R2 | x2

a2 + y2

b2 = 1}, compute

I =∮

C

(x + y)dx − (x − y)dy,

using Green’s Theorem (C is taken in the counter-clockwise direction).

(ii) (5) Which vector field R2 → R

2 is being integrated here? How does the value of Ichange if one integrates over C taken clockwise?

2


(i) (5) Use Divergence Theorem to express volume of a body given by a smoothbounded surface in R

3 as a surface integral.

(ii) (5) Consider the surface T (a torus) given by the parametric representation

x = (b + a cosψ) cosφ, 0 < a < b,

y = (b + a cosψ) sinφ,z = a sinψ.

(1)

Write down, in the variables ψ and φ, the expressions for the unit normal vector toT (pointing towards the exterior of T), and for the area element of T .

(iii) (15) Use the latter and the Divergence Theorem to find the volume of the torusgiven by (1).

2007

MAS212 Final Exam


Let f1, f2, f3, f4, f5 be functions of two variables. It is known that

(i) lim(x,y)→(0,0)

f1(x, y) does not exist,

(ii) f2 is differentiable at (0, 0) and grad f2(0, 0) 6= (0, 0),

(iii) f3 is differentiable at (0, 0) and (0, 0) is a degenerate critical point of the function f3,

(iv) f4 is differentiable at (0, 0) and (0, 0) is a non-degenerate local maximum of thefunction f4,

(v) f5 is differentiable at (0, 0) and (0, 0) is a non-degenerate saddle of the function f5.

3

Figure 1 consists of 5 diagrams showing level curves of functions f1, f2, f3, f4, f5 (in somerandom order). Determine which diagram corresponds to which function.


Find the general solution to the partial differential equation fx + 2fy = 1.


Find all critical points of the function f (x, y) = arctanx · cos y − x2 . For each critical

point determine whether it is degenerate or not. For each non-degenerate critical pointdetermine whether it is a local maximum, a local minimum, or a saddle.


Find the average value of the function f (x, y) = y in the area given by the inequality0 ≤ y ≤ 1 − x2.


Calculate

∫∫∫

x2+y2+z2≤1

e(x2+y2+z2)3/2dx dy dz.


(i) Calculate the curl of the vector field xj + sin2 yk.

(ii) Find the flux of the vector field sin 2yi + k through the half-sphere given by x2 +y2 + z2 = 1, z ≥ 0. The half-sphere is shown in Figure 2.

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Figure 1: Level curves of functions f1, f2, f3, f4, f5 from Question 1, MAS212 final exam2007

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Figure 3: Level curves of functions from MATH1B 2007 final, Question 1

MATH1B Final Exam


Figure 3 represents level curves of three functions in a neighbourhood of the origin.One of them has a non-degenerate local minimum at the origin, another one has adegenerate local minimum, and the third one does not have a critical point at the origin.Determine which is which.


Figure 4 represents the gradients of three functions in a neighbourhood of the origin.One of them has a non-degenerate local minimum at the origin, another one has a

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Figure 4: Gradients of functions from MATH1B 2007 final, Question 2

degenerate local minimum, and the third one does not have a critical point at the origin.Determine which is which.


Recall that given a function f : Rm → R, a point a ∈ R

m, and a vector v ∈ Rm, the

derivative of the function f at the point a along the vector v is

Dvf (a) = limt→0

f (a + vt) − f (a)t

.

Prove that Dcvf (a) = cDvf (a) for any constant c ∈ R.


Find the minimal and the maximal values of the function f (x, y, z) = x2 + y2 − z on the

ellipsoid x2 + y2

2 + z2

4 = 1.


Evaluate the integral ∫∫

D

x2

y2dxdy,

where D is the area bounded by the hyperbola xy = 1 and straight lines x = y, x = 2.

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Figure 5: Solid from MATH1B 2007 final, Question 6


Find the centre of mass of the solid of constant density occupying a region V ⊂ R3, where

V is given by the inequality x2 + y2 ≤ z ≤ 1 (see Figure 5).


⊲ What are components of the vector field x3i + y3j + z3k?

⊲ Find the flux of the vector field x3i + y3j + z3k through the unit sphere given byx2 + y2 + z2 = 1.

2008

MAS212 Final Exam


A function f (x, y) has a non-degenerate critical point at the origin. It is known that

8

one of the four diagrams in Figure 6 represents level curves of the function f (x, y) ina neighbourhood of the origin. Which diagram is it? What type of a non-degeneratecritical point is it?


Let f (x, y) =√

|xy|.(i) Find partial derivatives fx(0, 0) and fy(0, 0).

(ii) Prove that the function f (x, y) is not differentiable at (0, 0).


Find all critical points of the function f (x, y) = 83x

3 − 3xy + y2

2 + x. For each criticalpoint, determine whether it is degenerate or not. For each non-degenerate critical pointdetermine whether it is a local maximum, a local minimum, or a saddle.


Evaluate the integral∫∫∫

V

cos√

x2 + y2dxdydz,

where the solid V is given by inequalities x2 + y2 ≤(

π2)2

, 0 ≤ z ≤ 1 +√

x2 + y2.


Evaluate the work∮

C

(ex sin y − y)dx + (ex cos y + x)dy,

where C is the contour shown in Figure 7 (oriented counterclockwise). The contourconsists of eight line segments and two arcs of unit circles centered at (−1, 1) and (0, 1)respectively.

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Figure 6: Illustration to Question 1 from MAS212 final exam 2008.

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Figure 7: Contour for Question 5, MAS212 final 2008.


Given a function f (x, y, z) and a vector field v = Pi +Qj + Rk, prove that

curl (fv) = f curl v + (grad f ) × v.


Consider the vector field(

2yex + z3) i − y2exj + ln(

1 + x2 + y2) k.

(i) Find its divergence.

(ii) Evaluate its flux through the surface given by

z = sin(

π√

x2 + y2)

, x2 + y2 ≤ 1.

The surface is shown in Figure 8.

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Figure 8: Surface z = sin(

π√

x2 + y2)

oriented with normal pointing upwards.

MATH1B Final Exam


A function f (x, y) has a non-degenerate critical point at the origin. It is known thatone of the four diagrams in Figure 9 represents level curves of the function f (x, y) ina neighbourhood of the origin. Which diagram is it? What type of a non-degeneratecritical point is it?


Let f (x, y) = sin(xy).

(i) Find partial derivatives fx and fy .

(ii) Prove that the function f (x, y) is differentiable at (0, 0).


Find all critical points of the function f (x, y, z) = xy − z3 + 3z2. For each critical point,determine whether it is degenerate or not. For each non-degenerate critical point findits Morse index.

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Figure 9: Illustration to Question 1, MATH1B final 2008.

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Figure 10: Path for Question 5, MATH1B 2008 final.



V

cos√

x2 + y2 + z2dxdydz,

where V is the solid half sphere given by x2 + y2 + z2 ≤(

π2)2

, z ≥ 0.


Find the work of the vector field ex cos yi − ex sinyj along the path shown in Figure 10.The path initiates at (0, 0) and terminates at (0, 1). It consists of seven line segments andtwo arcs of unit circles centred at (−1, 1) and (0, 1) respectively.


Given a function f (x, y, z) and a vector field v = Pi +Qj + Rk, prove that

div (fv) = f div v + v · grad f .

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Figure 11: Gradients of the functions for Question 1, mid-term 2008


Solve the following boundary value problem:

fxx + fyy = 0 for x2 + y2 < 25, f (x, y) = x − y2 for x2 + y2 = 25.

MAS212 / MATH1B Mid-Term Test


Figure 11 represent gradients of some functions in a neighbourhood of the origin. Sketchlevel curves in each case. Attach this page to your answer booklet.

15


Given a function f (x, y) having continuous second-order partial derivatives, assume that

(a) f (x, y) satisfies the PDE fxx + fyy = 0;

(b) and all critical points of the function f (x, y) are non-degenerate.

Prove that all critical points of the function f (x, y) are saddles.


Find the maximal and the minimal value of the function f (x, y, z) = x2 − yz+ 2z2 on theellipsoid given by x2 + y2 + 5z2 = 1.


Reverse the order of integration and evaluate

∫

√ π2

0dx

∫ π2

x2

√y cos ydy.


Evaluate the area enclosed by the curve (x2 + y2)2 = (x + y)2.


Evaluate the integral

∫∫∫

V

zdxdydz, where V is the bounded region in R3 cut from the

solid parabolic cylinder y ≥ x2 by the planes z = 0 and y + z = 1. The region is shownin Figure 12.

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Figure 12: Region of integration for Question 6, mid-term 2008.

2009

MAS212 Final Exam


Each of the six diagrams in Figure 13 represents the graph of some function of twovariables whose second partial derivatives are continuous on R

2. Diagrams show thegraphs in a neighbourhood of (0, 0) together with respective tangent planes at (0, 0).

For each of the functions, one computes the determinant of the Hessian matrix atthe origin, that is, fxx(0, 0)fyy(0, 0) − f2

xy(0, 0). It turns out that this expression is positivefor two of the six functions, is zero for two of them, and is negative for two of them.Determine which diagrams represent positive Hessian, which ones represent negativeHessian, and which ones represent zero Hessian.


Let f (x, y) = xy2

x2 + y4 .

(i) Prove that the limit of f (x, y) as (x, y) approaches (0, 0) along any straight line is 0.

(ii) Prove that, nevertheless, lim(x,y)→(0,0) f (x, y) is not defined.

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Figure 13: Graphs of functions from Question 1, MAS212 and MATH1B Final Exam 2009


Assume that a function f : Rm → R satisfies the following conditions:

(a) all its second partial derivatives are continuous on Rm;

(b) f is harmonic, that is, fx1x1 + fx2x2 + · · · + fxmxm = 0 on Rm;

(c) f is Morse, that is, all its critical points are non-degenerate.

Prove that f doesn’t have a local maximum or a local minimum.Hint: recall how the trace of an m ×m matrix is determined by its eigenvalues.


18


V

dxdydz

1 + (x2 + y2)4 ,

where V is given by inequalities y ≥ 0 and 0 ≤ z ≤ x2 + y2 ≤ 4.


Express the area bounded by the curve x25 +y 2

7 = 1 as an integral in variables p, α, where

x = p5 cos5 α, y = p7 sin7 α.

You are not required to evaluate the integral, just write it down.


Find the work of the vector field

(

ex + 2y)

i +(

ln(1 + y17) + 3z)

j +(

cos(2009πz) + 5x)

k

over the circle x2 + y2 = 9, z = 1 oriented counterclockwise if viewed from top.


Find the flux of the vector field (ey − z)i + (sinx + 2z)j + x5yk across the torus given by

x = (b + a cosψ) cosφ,y = (b + a cosψ) sinφ,z = a sinψ,

where 0 ≤ φ, ψ ≤ 2π are parameters and 0 < a < b are some constants (see Figure 14).

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−6

−4

−2

0

2

4

6−6

−4−2

02

46

−6

−4

−2

0

2

4

6

Figure 14: Torus with a = 2, b = 5.

MATH1B Final Exam


Each of the six diagrams in Figure 13 represents the graph of some function of twovariables whose second partial derivatives are continuous on R

2. Diagrams show thegraphs in a neighbourhood of (0, 0) together with respective tangent planes at (0, 0).

For each of the functions, one computes the determinant of the Hessian matrix atthe origin, that is, fxx(0, 0)fyy(0, 0) − f2

xy(0, 0). It turns out that this expression is positivefor two of the six functions, is zero for two of them, and is negative for two of them.Determine which diagrams represent positive Hessian, which ones represent negativeHessian, and which ones represent zero Hessian.


Assume that a function f : Rm → R satisfies the following conditions:

(a) all its second partial derivatives are continuous on Rm;

(b) f is harmonic, that is, fx1x1 + fx2x2 + · · · + fxmxm = 0 on Rm;

(c) f is Morse, that is, all its critical points are non-degenerate.

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Prove that f doesn’t have a local maximum or a local minimum.Hint: recall how the trace of an m ×m matrix is determined by its eigenvalues.



V

dxdydz

1 + (x2 + y2)4 ,

where V is given by inequalities y ≥ 0 and 0 ≤ z ≤ x2 + y2 ≤ 4.


Find the work of the vector field

(

ex + 2y)

i +(

ln(1 + y17) + 3z)

j +(

cos(2009πz) + 5x)

k

over the circle x2 + y2 = 9, z = 1 oriented counterclockwise if viewed from top.


Let A be an n × n matrix. It is known that detA = 3. What is det(2A−1)?


Let An be the n × n matrix given by

An =

0 1 1 · · · 1 11 5 0 · · · 0 01 0 5 · · · 0 0...

......

. . ....

...1 0 0 · · · 5 01 0 0 · · · 0 5

.

(i) Find detA5.

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(ii) Find rankA5.

(iii) Find detAn.

(iv) Find rankAn.

MAS212 and MATH1B Mid-Term Test


Consider the curve give by parametric equations

x = t, y = t2, z = t3.

(a) At which points is the tangent line parallel to the plane x + 2y + z = 0?

(b) At which points is the tangent line parallel to the plane x + 2y + z = 4?


Consider the PDE (xy + y2)fx + (xy + x2)fy = 0. Substitute u = x + y, v = x2 − y2 andfind the general solution to this PDE. You may assume for simplicity that x + y 6= 0.


In the land of Fantasia, the king ordered a swimming pool with sides and bottom coveredwith gold. The shape of the pool must be a rectangular box (see Figure 15) of volume100 m3. Thus the amount of gold needed is proportional to the total area of the bottomface and four side faces of the parallelepiped. Find the dimensions of the swimmingpool (length, width, and depth) minimizing the expense of gold.

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0

1

2

3

4

01

23

4

0

1

2

3

4

Width

Lengh

Depth

Figure 15: A rectangular box.


Change the order of integration and evaluate

∫ 1

0dx

∫ 1−x2

0

e1−y√1 − y

dy.



V

x cos(√

1 − x2z)

dxdydz,

where V is the region in the first octant (so x, y, z ≥ 0) common to the cylinders x2+y2 ≤1 and x2 + z2 ≤ 1. The region of integration is shown in Figure 16


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0

0.5

1

1.5 0

0.5

1

1.5

0

0.5

1

1.5

OyOx

Oz

Figure 16: Region of integration for Question 5, MAS212 and MATH1B Mid-Term Test2009

Re-write the integral∫∫∫

V

z3dxdydz

in spherical coordinates and evaluate it. Here, V is the region in R3 given by x2+y2 +z2 ≤

z.

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Documents

Exam Papers in Calculus IV and Calculus for CN Yang Scholars 2006-2009