University of Toronto Scarborough Department of Computer ...
University of Toronto Scarborough Department of Computer & Mathematical Sciences MATA30: Calculus I - Midterm Test
Examiner: Sophie Chrysostomou
Date: Friday, November 23, 2012 Duration: 110 minutes
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO. FAMILY NAME: GIVEN NAMES: STUDENT NUMBER: DAY AND TIME OF YOUR TUTORIAL: SIGNATURE: CIRCLE THE NAME OF YOUR TEACHING ASSISTANT: Fazle Chowdhury
Kathleen Smith
Jianhao Yang
NOTES: • No calculators, cell phones or any electronic aids are permitted at the test room. • No books, notebooks or scrap paper are permitted near your examination table. • There are 12 numbered pages in the test. It is your responsibility to ensure that, at the start of the test, this booklet has all its pages. The last two pages (11 and 12) are empty. • Please leave all the pages of this booklet stapled. Do not remove any pages. • Answer all questions in the space provided. Show your work and justify your answers for full credit.
FOR MARKERS ONLY 1
/ 10
2
/10
3a
/5
3b
/5
3c
/5
3d
/5
3e
/5
4
/6
5
/7
6
/ 10
7
/7
8a
/5
8b
/5
8c
/5
8d
/5
8e
/5
TOTAL
/100
MATA30Y
page 1
1. (a) [ 4 marks] Give the definition of g(x) = arctan x and give the domain and its range.
(b) [ 2 marks] Give the graph of g(x) = arctan x.
(c) [ 4 marks] Find the exact value of the expression: √ 1 cos arctan 3 + arcsin √ 2
page 2
MATA30Y
2. (a) [ 4 marks] Find all x satisfying: 4− log4 (x)+log9 (3) = 3
(b) [ 6 marks] Prove the identity:
sin x + sin(2x) = tan x . 1 + + cos(x) cos(2x)
page 3
MATA30Y
3. [5 marks each; 25 marks total] Evaluate each of the following limits, else explain why the limit does not exist. Justify your answer. Do not use L’Hˆospital’s rule. x2 − 9 (a) lim √ √ x→3 3− x
1 1 (b) lim x sin cos x→0 x x 2
|x + 5| x→−5 2x + 10
(c) lim
page 4
MATA30Y
1 − cos x x→0 x tan x
(d) lim
(e) lim x→2
√
x2 + 5 − (x + 1) 2−x
page 5
MATA30Y
4. [6 marks] Let f (x) =
√
x2 + 4 . Find all the horizontal and vertical asymptotes of f . x−6
MATA30Y
5. [7 marks] Given some numbers a, b and c, define f : R → R by a+x if x ≤ 1 , 2bx if 1 < x < 2 , f (x) = x3 + c if 2 ≤ x < 3 , 9x if x ≥ 3 , Find all possible a, b and c so that f is continuous on R.
page 6
MATA30Y
page 7
x3 + 5x − 7 6. [10 marks] Fully justify that f (x) = has a root. Give the statement of x(x − 4) any theorem used.
MATA30Y
7. [7 marks] Let f (x) =
page 8
1 . Use the definition of the derivative to find f ′ (x). 1 + 2x
page 9
MATA30Y
8. [5 marks each; 25 marks total] For the following given functions use differentiation rules to find the required derivatives. Do not simplify. 2
4
3
(a) Let f (x) = (ax 3 + x 3 + b) 5 where a and b are constants. Compute f ′ (x).
(b) If g(x) =
2 + x2 ln x , find g ′ (x). x + ln 2x
2
(c) Let h(x) = (sin x + 2)x . Find h′ (x).
page 10
MATA30Y
(d) If f (x) = arcsin
√
x2
− 1 , find f ′ (x)
(e) If cos(xy) + 1 = yex + x2 + y 2, find
dy . dx
page 11
MATA30Y
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page 12
MATA30Y
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Examiner: Sophie Chrysostomou
Date: Friday, November 23, 2012 Duration: 110 minutes
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO. FAMILY NAME: GIVEN NAMES: STUDENT NUMBER: DAY AND TIME OF YOUR TUTORIAL: SIGNATURE: CIRCLE THE NAME OF YOUR TEACHING ASSISTANT: Fazle Chowdhury
Kathleen Smith
Jianhao Yang
NOTES: • No calculators, cell phones or any electronic aids are permitted at the test room. • No books, notebooks or scrap paper are permitted near your examination table. • There are 12 numbered pages in the test. It is your responsibility to ensure that, at the start of the test, this booklet has all its pages. The last two pages (11 and 12) are empty. • Please leave all the pages of this booklet stapled. Do not remove any pages. • Answer all questions in the space provided. Show your work and justify your answers for full credit.
FOR MARKERS ONLY 1
/ 10
2
/10
3a
/5
3b
/5
3c
/5
3d
/5
3e
/5
4
/6
5
/7
6
/ 10
7
/7
8a
/5
8b
/5
8c
/5
8d
/5
8e
/5
TOTAL
/100
MATA30Y
page 1
1. (a) [ 4 marks] Give the definition of g(x) = arctan x and give the domain and its range.
(b) [ 2 marks] Give the graph of g(x) = arctan x.
(c) [ 4 marks] Find the exact value of the expression: √ 1 cos arctan 3 + arcsin √ 2
page 2
MATA30Y
2. (a) [ 4 marks] Find all x satisfying: 4− log4 (x)+log9 (3) = 3
(b) [ 6 marks] Prove the identity:
sin x + sin(2x) = tan x . 1 + + cos(x) cos(2x)
page 3
MATA30Y
3. [5 marks each; 25 marks total] Evaluate each of the following limits, else explain why the limit does not exist. Justify your answer. Do not use L’Hˆospital’s rule. x2 − 9 (a) lim √ √ x→3 3− x
1 1 (b) lim x sin cos x→0 x x 2
|x + 5| x→−5 2x + 10
(c) lim
page 4
MATA30Y
1 − cos x x→0 x tan x
(d) lim
(e) lim x→2
√
x2 + 5 − (x + 1) 2−x
page 5
MATA30Y
4. [6 marks] Let f (x) =
√
x2 + 4 . Find all the horizontal and vertical asymptotes of f . x−6
MATA30Y
5. [7 marks] Given some numbers a, b and c, define f : R → R by a+x if x ≤ 1 , 2bx if 1 < x < 2 , f (x) = x3 + c if 2 ≤ x < 3 , 9x if x ≥ 3 , Find all possible a, b and c so that f is continuous on R.
page 6
MATA30Y
page 7
x3 + 5x − 7 6. [10 marks] Fully justify that f (x) = has a root. Give the statement of x(x − 4) any theorem used.
MATA30Y
7. [7 marks] Let f (x) =
page 8
1 . Use the definition of the derivative to find f ′ (x). 1 + 2x
page 9
MATA30Y
8. [5 marks each; 25 marks total] For the following given functions use differentiation rules to find the required derivatives. Do not simplify. 2
4
3
(a) Let f (x) = (ax 3 + x 3 + b) 5 where a and b are constants. Compute f ′ (x).
(b) If g(x) =
2 + x2 ln x , find g ′ (x). x + ln 2x
2
(c) Let h(x) = (sin x + 2)x . Find h′ (x).
page 10
MATA30Y
(d) If f (x) = arcsin
√
x2
− 1 , find f ′ (x)
(e) If cos(xy) + 1 = yex + x2 + y 2, find
dy . dx
page 11
MATA30Y
This page is intentionally blank.
page 12
MATA30Y
This page is intentionally blank.
