Question

1. For the function f (x) = 2x² + 8 use the limit definition (four-step process) to find f′(x) . Students must show all steps (more or less four of them) to compute the derivative. Simply giving the derivative of the function will not receive much credit.

2. For the function f (x) = x² + 2x

a. Find f′(x) . Students may use derivative rules to find this derivative. Show all work and clearly label the answer below.

b. Find the slope of the tangent line at (1, f (1)) . For full credit show steps with proper notation. Show all work and clearly label the answer below.

c. Find the equation, y = mx + b , of the tangent line at (1, f (1)) . For full credit all steps required for the final answer. Show all work and clearly label the answer below. Clearly label answer below.

3. The profit (in dollars) from the sale of x infant car seats is given by:

P (x) = 45x − 0.025x² − 5000 where 0 ≤ x ≤ 2, 400

a. Find the average rate of change in profit if production goes from 800 car seats to 850 car seats. Show all work and clearly label the answer below.

b. Find P′(x) . Students may use derivative rules to find this derivative. Clearly label the answer below.

c. Find P′(800). Interpret the meaning of this result in a complete sentence with correct units.

4. Use derivative rules to find d/dt ( 5/t³ − 4 √ t ). Show all work and clearly label answer below.

5. A manufacturer will sell N (x) speed boats after spending $x thousand on advertising, as given by

N (x) = 1200 − 3,845/x where 5 ≤ x ≤ 30

a. Find N′(x) .  Students may use derivative rules to find this derivative. Show all work and clearly label the answer below.

b. Find N′(10) and N′(25) . Interpret the meaning of these results with correct units.

Answer 1