MAT237-2020-0111 PDF

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MAT237 Multivariable Calculus with Proofs Term Test 2 Released Friday January 22, 2020 at 13:00 ET Submit Saturday January 23, 2020 by 13:00 ET

Instructions Please read the Term Test 2 FAQ for details on submission policies, authorized resources, rules of conduct, how to ask a question, test announcements, and more. You were expected to read them in detail in advance of the test. • Submissions are only accepted by Gradescope. Do not send anything by email. Late submissions are not accepted under any circumstance. Remember you can resubmit anytime before the deadline. • Submit your solutions using only this template PDF. You will submit a single PDF with your full written solutions. If your solution is not written using this template PDF (scanned print or digital) then you will receive zero. Organize your work neatly in the space provided. • Show your work and justify your steps on every question, unless otherwise indicated. Put your final answer in the box provided, if necessary. You must fill out and sign the academic integrity statement below; otherwise, you will receive zero.

Academic integrity statement Full Name: Student number: I confirm that: • I have not communicated with any person about the test other than a MAT237 teaching team member. • I have not used any unauthorized aids at any point during the test. • I have not viewed, participated in, or enabled any MAT237 group chat during the test. • I have not viewed the answers, solutions, term work, or notes of anyone. • I have read and followed all of the rules described in the Term Test 2 FAQ. • I have read and understand the rules of conduct. I have not violated these rules while writing this term test. • I understand the consequences of violating the University’s academic integrity policies as outlined in the Code of Behaviour on Academic Matters. I have not violated any of them while writing this assessment. By signing this document, I agree that all of the statements above are true.

Signature: Question:

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48

Score: Created and edited by B. Abuelnasr, T. Janisse, and A. Zaman. Last updated 11:31 on Thursday 21st January, 2021.

1. Let f : R3 → R be a C 4 function and suppose its 3rd Taylor polynomial at (0, 0, 0) is given by P3 (x, y, z) = 1 + 2x + 2 y + 3z + x 2 − 5x y + 2 y 2 − 7 yz + z3 . Let P0 , P1 , P2 , and P4 be the 0th , 1st , 2nd and 4th Taylor polynomials of f at (0, 0, 0). No justification is necessary. (1a) (1 point) Estimate the value f (0, 0, 0.2) with a quadratic approximation of f at (0, 0, 0).

f (0, 0, 0.2) ≈

(1b) (4 points) If possible, determine each of the following quantities. If it is not possible, write “N/A".

f (0, 0, 0) =

f (0, 0, 1) =

∂f (0, 0, 0) = ∂x

∂3f (0, 0, 0) = ∂ x∂ y∂ z

H f (0, 0, 0) =

f (x, y, z) − P3 (x, y, z) = (x , y,z)→(0,0,0) ||(x, y, z)||2 lim

lim

(x , y,z)→(0,0,0)

MAT237

f (x, y, z) − P3 (x, y, z) ||(x, y, z)||3

=

Term Test 2 - Page 2 of 11

January 23, 2020

2. (5 points) For each part below, select the BEST answer. Fill in EXACTLY ONE circle. (unfilled

filled

)

No justification is necessary. (2a) For a C 3 function f : R2 → R, you plot the quadratic form at each of its critical points a, b, c , d ∈ R2 . quadratic form of f at a quadratic form of f at b quadratic form of f at c quadratic form of f at d

If you apply the second derivative test to each critical point, at most one of them gives an inconclusive answer. Identify which one (if any) gives the inconclusive answer.  Point a

 Point b

 Point c

 Point d

 None of them will be inconclusive.

(2b) Let U ⊆ Rn be an open non-empty set and let a, b ∈ U. Let L be the line segment between a and b. Which of the following is TRUE?  If f : U → R237 is differentiable and L lies in U then ∃c ∈ L s.t. f (b) − f (a) = ∇ f (c) · (b − a).  If f : U → R237 is differentiable and U is convex then ∃c ∈ L s.t. f (b) − f (a) = f ′ (c)( b − a).  If f : U → R237 is C 1 and L lies in U then ∃c ∈ L s.t. f (b) − f (a) = ∇ f (c) · (b − a).  If f : U → R237 is C 1 and U is convex then ∃c ∈ L s.t. f (b) − f (a) = f ′ (c)(b − a)  None of the above are true.

(2c) Let f : R3 → R be C ∞ . What is the difference between ∂ (3,4) f and ∂ (3,0,4) f ?  The order of the derivatives is different.  There is no difference between the two derivatives.  Both derivatives exist but they are not necessarily equal.  One of the derivatives does not make sense.  There is not enough information to decide.

(2d) Let f : R3 → R be C 3 . Which of the following is NOT equal to ∂1 ∂2 ∂3 f ?  ∂1 ∂3 ∂2 f  ∂3 ∂1 ∂2 f  ∂2 ∂3 ∂1 f  ∂3 ∂2 ∂1 f  None of the above. They are all equal to ∂1 ∂2 ∂3 f .

(2e) Let f : Rn → R be C N +1 at a ∈ Rn . Let PN be the N th Taylor polynomial of f at a. Which statement best describes Taylor’s theorem from an analytic viewpoint?     

MAT237

f (x) is well approximated by PN (x) for N large. f (x) is well approximated by PN (x) for x near a. f (x) is equal to PN (x) with error that decays faster than || x − a|| N as N grows. f (x) is equal to PN (x) with error that decays faster than || x − a|| N as x approaches a. PN defines a polynomial surface of degree ≤ N which best fits the graph of f near a.

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January 23, 2020

3. (3 points) Below is a contour plot of the C 2 function f : R2 → R and the origin a = (0, 0) ∈ R2 . y

−2 −1 0 a

x

1

2

Determine whether each quantity is positive, negative, or zero. Select the most plausible answer. No justification is necessary. Fill in EXACTLY ONE circle. (unfilled filled ) (3a)

∂f (a)  positive ∂x

 negative

 zero

(3b)

∂f (a)  positive ∂y

 negative

 zero

(3c)

∂2f (a)  positive ∂ x2

 negative

 zero

(3d)

∂2f (a)  positive ∂ y2

 negative

 zero

(3e)

∂2f (a)  positive ∂ y∂ x

 negative

 zero

(3f)

∂2f (a)  positive ∂ x∂ y

 negative

 zero

MAT237

Term Test 2 - Page 4 of 11

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4. The parts of this question are unrelated. (4a) (3 points) Let f (x, y) = (x 2 − y 2 , 2x y). Identify the set S of points where f has a local C 1 inverse.

(4b) (2 points) Let g : R2 → R. The curve below is defined by the equation g(x, y) = 237.

Five points A, B , C , D, E ∈ R2 are labelled on the curve. No justification is necessary. Fill in ALL boxes that apply. (unfilled  and filled ) At which of these points can y be described locally as a function of x ?  A  B  C  D  E At which of these points can y be described locally as a C 1 function of x ?  A  B  C  D  E

MAT237

Term Test 2 - Page 5 of 11

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5. (5 points) For each part below, select the BEST answer. Fill in EXACTLY ONE circle. (unfilled

filled

)

No justification is necessary. (5a) Let D ⊆ Rn be a set. Let f : D → R and let a ∈ D . Which of the following statements are equivalent to " f has a local minimum at a"?  ∀x ∈ D, f (x) ≥ f (a)  ∃ϵ > 0 s.t. ∀x ∈ B ϵ (a), f (x) ≥ f (a)  ∃ϵ > 0 s.t. ∀x ∈ B ϵ (a) ∩ D, f (x) ≥ f (a)  None of the above are equivalent. (5b) Let q : Rn → R be a quadratic form. Which of the following is FALSE?  There exists a symmetric matrix A such that q(v) = v T Av for all v ∈ Rn .  q(v) = 21 v T Hq(0)v for all v ∈ Rn .  q is exactly one of definite, semidefinite, or indefinite.  q is at least one of positive definite, negative definite, semidefinite, or indefinite.  None of the above are false. (5c) The range of a quadratic form q is all of R. Which of the following is TRUE?  The determinant of q is not positive.  The matrix associated to q has at least one positive eigenvalue and one negative eigenvalue.  q is both positive definite and negative definite.  q is semidefinite but not indefinite.  None of the above are true. (5d) Let D ⊆ Rn be open and f : D → R be C 3 . Fix a ∈ D. Consider the four statements: I. II. III. IV.

If ∇ f (a) = 0 then a is a local extremum of f . If a is a local extremum of f then ∇ f (a) = 0. If ∇ f (a) = 0 and H f (a) is definite then a is a local extremum of f . If a is a local extremum of f then ∇ f (a) = 0 and H f (a) is definite.

Which of these four statements are TRUE?  I, II, III, and IV

 III and IV

 II, III, and IV

 Only II

 I and III

 Only IV

 II and III

 None of them

(5e) You have set aside 20 hours to work on two class projects. You want to maximize your grade (measured in points), which depends on how you divide your time between the two projects. Suppose that you use the method of Lagrange multipliers to solve this optimization problem and obtain the multiplier λ. What is the practical meaning of the statement λ = 3?  If you set aside about 3 more hours, you could gain about one more point on your two projects combined.  If you set aside about 1 more hour, you could gain about 3 more points on your two projects combined.  If you shift 3 hours from one project to the other, you could gain about 1 more point.  If you shift 1 hour from one project to the other, you could gain about 3 more points.  None of the above.

MAT237

Term Test 2 - Page 6 of 11

January 23, 2020

6. The parts of this question are unrelated. (6a) (3 points) Let f : R2 → R2 and g : R2 → R be differentiable functions. You have the following data: (x, y) (0, 0) (1, 2)

f ( x, y) (1, 2) (4, −2)

f x1 (x, y) 1 4

g ( x, y) 1 2

f y1 (x, y) 2 −12

f x2 (x, y) 3 3

f y2(x, y) −1 0

g x (x, y) 2 −2

g y (x, y) 1 1

Let h = g ◦ f . Use the chain rule to estimate the value of h(0.1, −0.2).

(6b) (1 point) The graph below shows a gradient vector field ∇F(x) and a constraint curve G(x) = 0. 2

1

0

−1

−2 −2

−1

0

1

2

By labelling on the graph, approximately identify where the local extrema of F on the curve are located. No justification is necessary.

MAT237

Term Test 2 - Page 7 of 11

January 23, 2020

7. For (x 1 , x 2 , y1 , y2 ) ∈ R4 , define   F(x 1 , x 2 , y1 , y2 ) = x 21 − x22 − y13 + y 22 + 4, 2x 1 x 2 + x 22 − 2 y12 + 3 y 24 + 8 . The point a = (2, −1, 2, 1) satisfies F(a) = (0, 0). You may use WolframAlpha to do any basic matrix computations without justification. (7a) (3 points) Show the equation F(x 1 , x 2 , y1 , y2 ) = 0 defines ( y1 , y2 ) locally as a C 1 function ϕ of (x 1 , x 2 ) near a.

(7b) (2 points) Compute the Jacobian of ϕ at a.

MAT237

Term Test 2 - Page 8 of 11

January 23, 2020

8. You are classifying the critical points of a C 3 real-valued function f on the closed ball S = B 237 (0) ⊆ R3 . You find it has exactly two critical points a, b ∈ S and compute the Hessian matrices at each point to be     −5 0 0 0 0 0 0 , H f (b) = 0 2 0 . H f (a) =  0 −9 0 0 −3 0 0 −16 Fill in the circle to select your answer. (unfilled

filled

)

(8a) (1 point) Classify the critical point a. No justification necessary.  local max

 local min

 saddle point

 not enough information

 none of these

(8b) (1 point) Classify the critical point b. No justification necessary.  local max

 local min

 saddle point

 not enough information

 none of these

(8c) (2 points) If possible, identify the global extrema of f on S .     

Both a and b are global extrema. One of a or b is a global extremum. There are no global extrema but there are local extrema. There is not enough information to determine any global extrema. None of the above statements are true.

Briefly justify your answer to (8c) in the box below.

MAT237

Term Test 2 - Page 9 of 11

January 23, 2020

9. (6 points) A power station company, Team Rocket, has two generators named Pikachu and Zapdos. Pikachu outputs x megawatts of power and Zapdos outputs y megawatts of power. The monthly operating costs are C P (x) = 140, 000 +

x2 , 100

C Z ( y) = 90, 000 +

y3 , 1200

measured in dollars for Pikachu and Zapdos respectively. If Team Rocket is contracted to output power at a rate of 880 megawatts, how should they balance the generating loads to minimize their costs? Explain your setup and summarize your conclusion in a full sentence. Use the method of Lagrange multipliers.

MAT237

Term Test 2 - Page 10 of 11

January 23, 2020

10. (6 points) Let m < n and let F : Rn → Rm be differentiable. Let S = {x ∈ Rn : F(x) = 0} and p ∈ S . Prove that Tp S ⊆ ker d F p .

MAT237

Term Test 2 - Page 11 of 11

January 23, 2020...