Sample Midterm Examination

Time: 3 hoursPassing grade: 55%Total points: 64

1. Give the exact value of cos(− π

12

).5 points

2. Let f(x) = 2x2 − 5 and g(x) =x+ 5

2x− 9.6 points

Find the composite functions and their corresponding domains.

a. f (g(x))

b. g (f(x))

3. Give a labeled graph of the function g(x) = 3(x+ 4)2 by starting with the graph5 pointsof a basic function, and then applying the appropriate transformations.

Explain the procedure you are using.

Note: No credit will be given if any other method is used.

4. Evaluate each of the limits below. If a limit does not exist explain why.16 points

a. limx→2

(3x2 − 2x+ 1

)b. lim

x→−2

3x2 − 2x− 16

(x+ 2)2

c. limx→3

2x2 − x+ 1

x− 3

d. limx→2

√6− x− 2√3− x− 1

e. limx→0

sin(3x)

x2 − x

5. Compute the derivatives of each of the functions below. You may not need to8 pointssimplify your answers.

a. y = 4x5 + 3x4 − 6x3 + 6

b. y =2x− 16

(x+ 3)2

c. y = sin(2x2 − x+ 1)

d. y = (−4x3 − x2 + 3x+ 7)4

6. Find the values of x such that tangent line to the curve f(x) = 3x2 + 4x− 3 is4 pointsperpendicular to the line y = 6x+ 2.

7. Gravel is being dumped from a conveyor belt at a rate of 0.5 m3 / min. It forms a5 pointspile in shape of a cone whose base diameter and height are always equal. Howfast is the height of the pile increasing when the pile is 4 m high?

8. Find y′ using implicit differentiation:5 pointsx3y + xy2 = 4xy + 7.

9. Use differentials to find the approximate value of√9.2.5 points

10. Sketch the graph of a single function f(x) that satisfies all of the conditions5 pointslisted below.

• the limit limx→0

f(x) does not exist.

• f(0) = 0.

• limx→∞

f(x) = −1.

• the function f is not differentiable at x = −2.