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. Simplygiving the derivative of the function will not receive muchcredit.

2. For the function f (x) = x² + 2x

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

b. Find the slope of the tangent line at (1, f (1)) . For fullcredit show steps with proper notation. Show all work and clearlylabel 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 labelanswer below.

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

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

a. Find the average rate of change in profit if production goesfrom 800 car seats to 850 car seats. Show all work and clearlylabel the answer below.

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

c. Find P?(800). Interpret the meaning of this result in acomplete sentence with correct units.

4. Use derivative rules to find d/dt ( 5/t³ ? 4 ? t ). Show allwork and clearly label answer below.

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

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

a. Find N?(x) .  Students may use derivative rules tofind this derivative. Show all work and clearly label the answerbelow.

b. Find N?(10) and N?(25) . Interpret the meaning of theseresults with correct units.