代写MATH-404-604 Advanced Calculus

Advanced Calculus Exam One

MATH-404-604 Advanced Calculus

First Examination

Spring 2024

Instructions: This is a take-home exam. It is due at the beginning of class Tuesday March 5 - please hand

a hard copy to me. You may use the written notes provided to you, the notes you take in class and the

homework sets - nothing else. You are not allowed to talk to anyone regarding this exam. Please show all

your work. Write your name on the top left of each page that you submit. Thank you and good luck!

Question1

Show that the function f(x, y) =

2x2y

x2 + y2

has a removable discontinuity at (0, 0).

Question 2

Let g : R→ R2 and f : R2 → R3 given by

g(t) =

[

t

(4/π) t

]

f

([

u

v

])

=

cosusinu

v



Let a = g(π/4) and b = f(a) = (f ◦ g)(π/4).

1. Describe either verbally or in equations the images S1 of g, S2 of f and S3 of f ◦ g : R→ R3.

2. Evaluate the Jacobian matrices Jg(π/4),Jf(a),J(f ◦ g)(π/4). Show that they are of maximal rank.

What do the the column spaces of these matrices represent?

3. State the domain and the range and write the formulae for dπ/4g,daf , and dπ/4(f ◦ g).

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Advanced Calculus Exam One

Question 3

Let f : R3 → R3, with

f(0) = p =

13

5

 Jf(0) =

−1 2 1∗2 4 ∗1

2 −3 −1



Denote its image by S.

1. Find by inspection

∂f1

∂x

,∇f3 and Djf(0).

2. Determine first ∗1 and then ∗2 if we know (a) that

Dfk/2(0) =

 0.53

−0.5



and (b) the rate of change of the function f2 at 0 is the fastest along the direction

w =

1√

14

(i+ 2j+ 3k).

3. A curve C passes through 0 with velocity u = 2i− j+2k. Compute the velocity Tu at p of the image

curve f(C).

4. Write down the formulae for the linearization

L0

xy

z

 =

 L1(x, y, z)L2(x, y, z)

L3(x, y, z)||



5. Estimate f(b) if b is obtained by moving .1 from 0 in the direction of the curve C in part (3) of this

problem.

6. Estimate

f

 0.10.3

−0.01



7. Find a function f that satisfied all the conditions mentioned above.

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Advanced Calculus Exam One

Question 4

Consider the function f : R2 → R2 given by f(x, y) = [ex cos y, ex sin y]T, the point P(1, 0). Let x > 0, y > 0

be small numbers and the rectangle R as below.

1. Compute the Jacobian Jf(P).

2. Write down the linearization LPf(x, y) : R

2 → R2 of f at P.

3. Find and describe/draw the image RL = LPf(R) of the rectangle R via the linearization LPf of f at

P.

4. Find and describe/draw the image Rf = f(R) of the rectangle R via the map f .

5. Evaluate the area of Rf and RL

6. Compute lim

(x,y)→(0,0)

area of RL

area of R

and lim

(x,y)→(0,0)

area of Rf

area of R

. Compare these limits to the determinant

of Jf(P) and comment on your findings.

The following might be useful:

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Advanced Calculus Exam One

Question 5 - For MATH-604 students only

Let f : R→ Rn and g : R→ Rn be two differentiable curves whose velocities are always nonzero for all t.

Suppose that the two points P = f(t0) and Q = g(s0) are closer than any two other points on each curve.

Show that the vector

−−→

QP is perpendicular to both velocity vectors f ′(t0) and g′(s0), i.e. to the respective

tangent lines.

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