Problem #1:

For what value of the constant c is the following function continuous at x = 5?

f(x) =

{	1 x + 1 5 x + 5	if x 5 (and x 0)
c	if x = 5

Problem #1:

1 25	Correct Answer: 1 52 Your Mark: 2/2

Problem #1	Attempt #1	Attempt #2	Attempt #3
Your Answer:	1 5	1 25
Your Mark:	0/2	2/2

Problem #2:

Evaluate the following limit

lim

h0

72 5(x + h)

72 5x

h

Problem #2:

5

2

72 5x

Correct Answer:

5

2

72 5x

Your Mark: 2/2

Problem #2

Attempt #1

Attempt #2

Attempt #3

Your Answer:

5

2

72 5x

Your Mark:

2/2

Problem #3:

Consider the following equations. In each case suppose that we apply the Intermediate Value Theorem using the interval [0,1]. (i.e., we take a = 0, b = 1 in the Intermediate Value Theorem.) (i) x2 + x 1 = 0 (ii) 2ex = x 4 (iii) ln(x+1) = 2 2x For which equations does the Intermediate Value Theorem conclude that there must be a root of the equation in the interval (0,1)?

(A) (i) and (ii)

(B) (i) only

(C) all of them

(D) (i) and (iii)

(E) none of them

(F) (ii) and (iii)

(G) (iii) only

(H) (ii) only

Problem #3:

D

Correct Answer: D Your Mark: 2/2

Problem #3	Attempt #1	Attempt #2	Attempt #3
Your Answer:	D
Your Mark:	2/2

Problem #4:

Suppose that f is continuous on [0,6] and that the only solutions of the equation f(x) = 3 are x = 1 and x = 5. If f(4) = 5, then which of the following statements must be true? (i) f(2) (ii) f(0) > 3 (iii) f(6) > 3

(A) none of them

(B) (ii) only

(C) (i) and (ii)

(D) (ii) and (iii)

(E) (iii) only

(F) (i) only

(G) (i) and (iii)

(H) all of them

Problem #4:

A

Correct Answer: A Your Mark: 2/2

Problem #4	Attempt #1	Attempt #2	Attempt #3
Your Answer:	A
Your Mark:	2/2

Problem #5:

(a)	If an equation of the tangent line to the curve y = f(x) at the point where a = 9 is y = 6x 9, find f(9) and f'(9).

(b)	If the tangent line to y = f(x) at (2,2) passes through the point (8,12), find f(2) and f'(2).

Problem #5(a):

45,6	separate your answers with a comma. (Your answers MUST be in the correct order.) Correct Answer: 45, 6 Your Mark: 2/2

Problem #5(b):

2, 1	separate your answers with a comma. (Your answers MUST be in the correct order.) Correct Answer: 2, 1 Your Mark: 2/2

Problem #5	Attempt #1	Attempt #2	Attempt #3
Your Answer:	5(a) 45,6 5(b) 2, 1	5(a) 5(b)	5(a) 5(b)
Your Mark:	5(a) 2/2 5(b) 2/2	5(a) 5(b)	5(a) 5(b)

Problem #6:

Let f(x) = cosx. Which of the following would you use to calculate f'(/2) using the definition of derivative (i.e., first principles)?

(A)

lim

h0

cos(/2 + h)

h

(B)

lim

h0

cos(/2 + h) 1

h

(C)

lim

h0

sin(/2 + h)

h

(D)

lim

h0

sin(/2 + h) 1

h

(E)

lim

h0

cos(/2 + h) + 1

h

(F)

lim

h0

sin(/2 + h)

h

(G)

lim

h0

sin(/2 + h) + 1

h

(H)

lim

h0

cos(/2 + h)

h

Problem #6:

H

Correct Answer: H Your Mark: 2/2

Problem #6	Attempt #1	Attempt #2	Attempt #3
Your Answer:	H
Your Mark:	2/2

Problem #7:

The following limit represents the derivative of some function f at some number a. Find such an f and a.

lim

h0

36 + h

6

h

(A) f(x) =

x

, a = 6 + h

(B) f(x) =

x + h

6, a = 36

(C) f(x) =

x

, a = 6

(D) f(x) =

x

, a = 36 + h

(E) f(x) =

x + h

, a = 36

(F) f(x) =

x + h

, a = 6

(G) f(x) =

x

6, a = 6

(H) f(x) =

x

, a = 36

Problem #7:

H

Correct Answer: H Your Mark: 2/2

Problem #7	Attempt #1	Attempt #2	Attempt #3
Your Answer:	H
Your Mark:	2/2

Problem #8:

Which of the below graphs is an example of a function that satisfies the following conditions? g'(0) = 1, g'(1) = 1, g'(2) = 0, g'(3) = 1, g'(4) = 0

(A)

(B)

(C)

(D)

(E)

(F)

(G)

(H)

Problem #8:

B

Correct Answer: B Your Mark: 2/2

Problem #8	Attempt #1	Attempt #2	Attempt #3
Your Answer:	A	H	B
Your Mark:	0/2	0/2	2/2