Note: All work must be shown and explained to receive full credit.
1. Find the derivative of the following functions. (Do not simplify.)
a. ???? = 2????2 5???? + 3????2 + 4????3
b. ????(????) = (5????3 + ????2)(2???? 4)
c. ????(????) = ????2 2???? + 5
d. ???? = (7 2????)3
2. Let????(????) = ????3 ???? 1.
a. Find the equation of the tangent line to the graph of the function at x = 0.
b. What does the derivative of the function at x = 0 tell you about the direction at that point?
Is it increasing, decreasing, or neither?
3. Let????(????) = ????3 27????.
a. Find all of the relative extrema of g(x) algebraically.
b. Sketch the graph of g(x), and label all of the relative extrema on the graph.
4. The height, h (in feet), of an object shot vertically upward from the ground with an initial velocity
of v0 ft/sec is modeled by the following function:
?(????) = 16????2 + ????0????
where t is the time (in seconds).
a. Choose one value for v0 between 120 and 480 to be used in your height function equation
b. Important: By Wednesday night at midnight, submit a Word document stating only
your name and your chosen value for v0 above in Part a. Submit this in the Unit 2 IP
submissions area. This submitted Word document will be used to determine the
Last Day of Attendance for government reporting purposes.
c. How fast is the object moving after 5 seconds?
d. How long will it take the object to reach its maximum height?
e. How long will it take the object to hit the ground?
f. What is the velocity of the object when it hits the ground?
5. The profit (in millions of dollars) derived from selling x units of a certain software is modeled by
the following function:Page 2 of 2
????(????) = 0.003????3 + 100????
a. If the rate of change in profit, called the marginal profit, is modeled by the derivative of
P(x), Find P(x).
b. Find the marginal profit for a production level of 100 units.
c. Find the actual gain in profit by increasing the production from 100 units to 101 units.
d. Based on your calculations from b and c, what can you say about the actual increase in
profit and the marginal profit?
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