Calculus Exam

Differential Calculus 03-62-140 Final Exam

May 16, 2007

TIME: 3 Hours, This Exam has 9 Questions.

FAMILY NAME: GIVEN NAME: STUDENT NUMBER: SECTION:

Do Not Write Below This Line Question Question Question Question Question Question Question Question Question 1: 2: 3: 4: 5: 6: 7: 8: 9: /10 /10 /15 /10 /10 /10 /15 /10 /10

TOTAL:

/100

Question 1 (10 Marks)

Identify the following statements as either true or false: If f is an invertible function, then f ◦ f −1 ◦ f = f. The function f (x) = sin x, on the domain (−π, 0), is invertible. If f (x) = ex then f −1 (x) = e−x . If a function is continuous then it is differentiable. If lim f (x) = 2 then lim f (x) = 2. −

x→1 x→1

a) b) c) d) e) f)

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g) h) i) j)

If f is continuous and differentiable, and f (0) = 0 and f (2) = 2 then there is a number c, between 0 and 2, such that f (c) = 1. d f (x) −f (x) = dx x x2 d (f (x2 )) = 2xf (x). dx If f (c) = 0 then f has either a local maximum or a local minimum at c. If f (x) < 0 and f (x) > 0 for all x in the interval (−2, 2) then f is decreasing and concave upward, in (−2, 2).

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2

Question 2 (10 Marks)

√ The function g is obtained from the function f (x) = x, by a horizontal shift to the right by 3 units, followed by a vertical stretch by a factor of 4. Find g −1 (x).

3

Question 3 (15 Marks)

Determine the limit, or explain why it does not exist: a) b) c) |x − 1| √+ x − 2 x3 + x lim x→∞ 3x3 + 1 x2 − 5x + 6 lim x→3 x3 − 9x

x→1

lim

x2

4

Question 4 (10 Marks)

Use the limit definition of the derivative to find f (x), where f (x) = √ x2 + 1.

5

Question 5 (10 Marks)

Use a linear approximation to estimate the value of ln

1 2.70

. [Hint: e = 2.71828... ]

6

Question 6 (10 Marks)

A video camera tracks the launch of a space craft. The camera is placed 3km from the launch site. At what...