Name:

Math 6: Test 1

February 20, 2012Please write neatly and clearly. For problems 1-4, you do not need to show your work.

Problems 5-9 are partial credit; you should show your work and identify your final answer.You may use a calculator. However, no cell phones or books may be used.

Question: 1 2 3 4 5 6 7 8 Total

Points: 25 5 15 5 10 10 15 15 100

Score:

1. (25 points) Compute the following.

(a) If f(x) = 2x− 4, then f ′(4) = A. 0 B. 2 C. 3 D. 4

(b) If f(x) = 9x3 − 4x2 + 2, then f ′(x) =

A. 12x3 − 6x+ 2 B. 9x3(3x2)− 4x2(2x) C. 27x2 − 8x D. 9 + 3x2 − 4− 2x+ 2

(c) If g(x) = (4π + π2)10, then g′(x) =

A. 0 B. 10(4π + π2)9(4 + 2π) C. 111 (4π + π2)11 D. Undefined

(d) If h(x) =1

4x+ x2, then h′(x) =

A.1

4 + 2xB. −4x−2 − 2x−3 C.

(4x+ x2)− (4 + 2x)

(4x+ x2)2D.−(4 + 2x)

(4x+ x2)2

(e) If f(t) =√

1 + t2, then f ′(3) = A.3√10

B.1

2√

10C.√

10 D. −505

2. (5 points) Find the equation for the tangent line to the curve y = x2 at x = 4.

A. y − 8 = x− 4 B. y − 16 = 8(x− 4) C. y − x2 = 2x(x− 4) D. y − 16 = 2x(x− 4)

3. (15 points) Use the graph of g(x) to answer to following questions.

-2 2 4

-2

2

4

6

8y=gHxL

(a) g(1) =

(b) g′(1) ≈

(c) For what values of x will g′(x) = 0?

4. (5 points) Suppose the temperature varies on a wire. Let x be the distance in cm to the right of thebeginning of the wire, and let T (x) be the temperature, in degrees Celsius, at the point x. Supposethat T (50) = 42, and T ′(50) = −7. As you move slightly to the right (as x increases slightly), then thetemperature

A. increases B. stays the same C. decreases D. There is not enough information.

5. (10 points) Let f(x) = x4 − x2 + 1.

(a) Find the average rate of change of f(x) over the interval [1, 2].

(b) Find the average rate of change of f(x) over the interval [1, 1.01].

(c) Find the instantaneous rate of change of f(x) at x = 1.

6. (10 points) Find the equation of the tangent line at x = 4 to the function

f(x) =(1 +

√x

x2

)3

7. (15 points) A baseball is thrown into the air off the top of a building. Its height in feet after t secondsis given by y(t) = −16t2 − 32t+ 128.

(a) How fast is the ball traveling after .5 seconds?

(b) How fast is the ball traveling when it hits the ground?

8. (15 points) A company manufactures and sells widgets. The monthly cost (in thousands of dollars) ofmanufacturing x (thousand) widgets is given by C(x), and the revenue is given by R(x).

C(x) = 50√x+ 10

R(x) = 40x− x2

The company currently produces 10 thousand widgets a month.

(a) Find the marginal revenue function. If production increases, will revenue increase or decrease?

(b) What is the current profit?

(c) What is the current marginal profit? Should the company increase or decrease production?