MAT137 - Winter 2020 - Test 1 PDF

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University of Toronto Department of Mathematics

MAT137 - Calculus! - Midterm 1 February 13, 2020 Examiner: T. Janisse Start time: 14:10 Duration 110 minutes. No aids permitted.

Last Name: First Name: Student Signature: Email:

@mail.utoronto.ca

Student number: UTORId: Directions: • This test is 1 hour and 50 minutes long. You are not permitted the use of any aids, such as calculators, phones, textbooks, etc. • This examination booklet contains a total of 12 pages. It is your responsibility to ensure that no pages are missing from your examination. Do not detach any pages from your examination. • Do not write on the QR code at the top of any of the pages. • You may write in either pen or pencil. • Have your student card ready for inspection. • Answer the questions on the question pages themselves. You may use the two blank pages at the back of the test paper for rough work. The extra pages WILL NOT BE MARKED unless you clearly indicate otherwise on the question pages. • There are a total of 40 marks to be earned on this test. 1

1. a) (3 points) Let a ∈ R and f a function. State the complete formal definition for “limf (x) does not x→a exist”. Make sure to specify all assumptions you make about f .

b) (3 points) Evaluate the following limits. If the limit does not exist, is ∞, or is −∞, specify this clearly. Write your final answers in the boxes provided. No justification is necessary. Only your final answer will be graded. Final Answer t

lim 2 t→2

lim

a→0



1 2+a

2 + t − t2 t−2

− a



Final Answer

1 2

Final Answer

√ 7r + r − 1 lim √ r→−∞ r2 − 2 + 1

2

2. a) (2 points) State the Intermediate Value Theorem.

b) (2 points) Let f be a function and a ∈ R. State the definition of “f differentiable at a”. Make sure to specify all assumptions about f .

c) (2 points) State L’Hopital’s rule for “00” indeterminate forms.

3. (4 points) For each of the following statements, determine if they are true or false. Circle your answer. No justification is required. a) ∀x ∈ R, ∃y ∈ R s.t. x = y 3 .

TRUE

FALSE

b) ∀x ∈ ∅, ∃y ∈ N s.t. x > y and y > x.

TRUE

FALSE

c) Let f and g be defined on R and a, b, L ∈ R. If:

TRUE

FALSE

TRUE

FALSE

• lim f (x) = b x→a

• and lim g(x) = L, x→b

then lim g(f (x)) = L. x→a

d) Let f and g be defined on R. If g ◦ f is one-to-one, then g is one-to-one. 3

4. (4 points) Consider the following graph of y = k(t) for t ∈ [−6, 6]. 2 1

-5

-4

-3

-2

-1

1

2

3

4

5

-1

Evaluate each limit. If a limit does not exist, or is ∞ or −∞, clearly state this. Write your final answer in the box provided. No justification is necessary. Only your final answer will be graded.   q √ 2 1 k(t) k ( −t) b) lim k(−k(t) ) c) lim a) lim k( t) d) lim k t→0 t→3+ t→25 k(t) t→−4+ Final Answer

a)

b)

c)

d)

5. For each question below, write your final answers in the boxes provided. No justification is necessary. Only your final answer will be graded. a) (2 points) Let f (x) = (ln(x))(e ) . Find f ′ (x). x

FINAL ANSWER

f ′ (x) = b) (2 points) Let g(x) = 2 cos2 (5x) − 1. Find g (2020) (π )

FINAL ANSWER

g (2020)(π) =

4

6. Two boats, the Calypso and the Borealis, are travelling across the Pacific ocean. When they first come into contact, the Calypso is 10km immediately south of the Borealis. The Calypso is heading north at 15km/h and the Borealis is heading southeast at 12km/h. a) (4 points) How quickly are the Calypso and Borealis moving apart 10 minutes after they first make contact?

b) (4 points) How long after the Calypso and Borealis first make contact are they closest together?

5

7. (8 points) Let f be a function defined on R so that lim f (x) = 0. Define g(x) = xf (x) for x ∈ R. Using x→0 the formal ε − δ definition of the limit, prove that: g ′ (0) = 0

6

Do not tear this page off. This page is for additional work and will not be marked, unless you clearly indicate it on the original question page.

7

Do not tear this page off. This page is for additional work and will not be marked, unless you clearly indicate it on the original question page.

8

Do not tear this page off. This page is for additional work and will not be marked, unless you clearly indicate it on the original question page.

9

Do not tear this page off. This page is for additional work and will not be marked, unless you clearly indicate it on the original question page.

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