Calculus I Midterm 1 October, 2016 2

1. (6 marks) Answer each question in the space provided. Do NOT simplify your
answers for parts a) and b).
( )( )
a) (2 marks) If f ( x) = 2 x −2 + x 5 2 x 3 + 1 , find f ′(x) .

f ′(x) = ______________________________________________________

(e x )(3 x )
b) (2 marks) If f ( x) = , find f ′(x) .
x2 + 3

f ′(x) = ______________________________________________________

c) (2 marks) Let f ( x) = g ( x)h( x) , g (10) = −4 , h(10) = 560 , g ′(10) = 0 , and

h′(10) = 35 . Find f ′(10) . Show ALL work.

f ′(10) = ______________________________________________________

(You can use this space for rough work, but anything below this line will NOT be
marked!)
Calculus I Midterm 1 October, 2016 3

2. (3 marks each; total 9 marks) Answer each question in the space provided, and
enter your final answer as indicated. You MUST show your work.

a) Simplify tan(arccos( x 5))

Answer: tan(arccos( x 5)) = __________________________________________

b) Sketch a SINGLE function f (x ) satisfying ALL of the following:

o f ′( x ) = 0 for x < −2

o f ′( 2) = 2 , f ( −2) = −1 , lim f ( x) = 3 , lim f ( x) = ∞

x → −∞ x→ ∞

o f (x ) is continuous on its domain and is not differentiable at x = −2

c) Use the Intermediate Value Theorem (IVT) to prove that x2 = x + 1 has a

root.
Calculus I Midterm 1 October, 2016 5

3. (Total 11 marks) Answer the following in the space provided.

x2 −1
a) (1 mark) lim
x→ 1− | x − 1 |

b) (3 marks) What do we mean when we say that f (x) is continuous at x = a ?

There are three properties. List all three.

1)

2)

3)

2 + cos x
c) (3 marks) lim [Hint: Squeeze Theorem]
x→ ∞ x+3

d) (4 marks) Based on the following graph of f , answer the below questions:

lim f ( x ) = __________ lim f ( x) = ___________ lim f ( x) = _________

x → −∞ x → −6 x→ 1−

At how many values of x is f discontinuous __________

Calculus I Midterm 1 October, 2016 6

4. (Total 8 marks) Answer each question in the space provided. You do NOT have to
show your work; only the final answer will be marked.
a) (2 marks) Use the given graph of f ( x ) = 1 / x to find a number δ such that if
1
| x − 1 |< δ then | − 1 |< 0 . 2
x

Answer (to 4 decimal places):__________________________________________

b) (2 marks) Find lim x − x

2
x→ ∞

Answer: _____________________________________________________
x 2 − 3x + 7
c) (1 mark) State the horizontal asymptote(s) of f ( x) = 3
x + 10 x − 4

Answer: _____________________________________________________
x 7 − 5x + 5
d) (1 mark) Evaluate lim
x→ 0 7x2 − 2
Answer: _____________________________________________________

ax 2 + 1, x <1

e) (2 marks) Given the following function f ( x) =  5 − bx, 1≤ x < 3
 − 2, x≥3

Determine the values of a and b that make the function continuous.

Answer: _____________________________________________________
Calculus I Midterm 1 October, 2016 7

5. (6 marks) Use the definition of the derivative (i.e. “First Principles”) to find f ′(x) if
f ( x) = 4 x + 3 .
Calculus I Midterm 1 October, 2016 9

6. (2 marks each; 10 marks total) For each of the following questions, select ALL of
the correct answers by clearly shading in the appropriate boxes. Each question is worth
2 marks, but there may be anywhere from 0 to 4 correct answers. You will lose 1 mark
for each mistake (i.e. selecting something that’s wrong, or missing something that’s
correct, up to a maximum of 2 marks deduction per question (i.e. no negative marks ☺)

a) Which of the following functions have a domain the set of all real numbers?
f ( x) = sin( x + 7) − e x
f ( x) = ln( x − 8) f ( x) = ( )2
x+2 −6
1
f ( x) =
5 + x2
b) Based on the following graph of f , which of the following statements are true?

There are 2 values for which the function is not differentiable

The function is continuous from the left at x = 0 .
f ′(−5) < 0
lim f ( x ) = ∞
x→ ∞

c) Which of the following limits do NOT exist?

x3 − 7 x x 4 + 5x − 3 1 3x 2 − x − 10
lim lim lim cos  lim
x→ 0 x3 x →0
2 − x2 + 4 x→ ∞
 x x→ 2 x+2

d) Let f be a continuous function on the closed interval [1,5] , where f (1) = 1 and
f (5) = −3 . Which one of the following is guaranteed by the Intermediate Value
Theorem?
f (c) = 2 for at least one c between -3 and 1
f (c) = −2 for at least one c between -3 and 1
f (c) = 2 for at least one c between 1 and 5
f (c) = −2 for at least one c between 1 and 5

e) Consider the graphs of f (x) and g (x) below. Which of the following are true?
f (x) g (x)

g (x) is the derivative of f (x) g (x) is differentiable everywhere

g (x) is an odd function f (x) is an even function