math calculus

NAME:_______________________________ MAT 135 BUSINESS CALCULUS I

DUE: 6/5/2017

TAKE-HOME FINAL Indicate the derivative for the given general function using the derivative toolbox. (2 pts each)

1) 𝑑

𝑑𝑥 𝑙𝑛 𝑥𝑎 =

2) 𝑑

𝑑𝑥 𝑎𝑥 =

3) 𝑑

𝑑𝑥 𝑎𝑥𝑛 =

4) 𝑑

𝑑𝑥 𝑙𝑛[𝑓(𝑥)] =

5) 𝑑

𝑑𝑥 𝑒𝑥 =

6) 𝑑

𝑑𝑥 𝑒𝑔(𝑥) =

7) 𝑑

𝑑𝑥

𝑔(𝑥)

ℎ(𝑥) =

8) 𝑑

𝑑𝑥 𝑔(𝑥)ℎ(𝑥) =

9) 𝑑

𝑑𝑥 [𝑓(𝑥)]𝑛 =

10) 𝑑

𝑑𝑥 log

𝑏 𝑥𝑎 =

TAKE HOME FINAL

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Find the derivatives for the following functions. SHOW ALL WORK. (2 pts each)

11) 𝑓(𝑥) = 6𝑥 ln𝑥

12) 𝑓(𝑥) = ln(2𝑥 − 9)

13) 𝑓(𝑥) = 𝑒3𝑥 2+5

14) 𝑓(𝑥) = 3𝑥 − 𝑒𝑥 + 𝑒2 − 25𝑥

15) 𝑓(𝑥) = 𝑒𝑥

5𝑥3−3𝑥

16) 𝑓(𝑥) = 2𝑥

𝑥2−25

TAKE HOME FINAL

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Given that a particular company who makes four-slice toasters have determined that the market

bears the following estimate sales: a weekly demand of 300 toasters at a price of $25 per toaster,

and a weekly demand of 400 toasters at a price of $20 per toaster. The financial department

estimates that weekly fixed costs will be $5000 and variable costs per unit will be $5. Answer the

following questions:

17) (2 pts) Assuming that it is linear, find the price-demand function p(x) where x is the amount of

toasters sold.

18) (2 pts) Find the revenue function R(x).

19) (2 pts) What is the domain for the price-demand function and the revenue function?

20) (2 pts) Find the cost function C(x).

21) (2 pts) Find the break-even points.

TAKE HOME FINAL

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Using the prompts provided, graph the function 𝑓(𝑥) = 3𝑥2+2

𝑥2−9

22) (2 pts) What is the domain of f?

23) (2 pts) Find the intercepts of f :

y-intercepts: f (0) =

x-intercepts: f (x) = 0 (If there are no solutions to this equation, then there are no x-intercepts.)

24) (2 pts) What is the equation for the horizontal asymptote of f? (This indicates the end behavior of the

function. Remember the rules of degrees from section 3.2 …)

25) (2 pts) What is(are) the equation(s) for the vertical asymptote(s) of f ?

26) (4 pts) What is the first derivative of f?

TAKE HOME FINAL

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27) (2 pts) What are the critical points of f?

28) (4 pts) Use the number line provided to show where f is increasing or decreasing, and any local max

or mins.

f is increasing on the interval(s) _________________________________________________

f is decreasing on the interval(s) _________________________________________________

f has a local max at the point(s) x = ______________________________________________

f has a local min at the point(s) x = ______________________________________________

TAKE HOME FINAL

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29) (4 pts) What is the second derivative of f?

30) (4 pts) Use the number line provided to show where f is concave up and concave down:

f is concave up on the interval(s) _________________________________________________

f is concave down on the interval(s) _______________________________________________

TAKE HOME FINAL

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31) (1 pt) Given the information in this problem, choose the graph of f from the choices below:

Graph A

Graph B

Graph C

Graph D

The correct graph for the function f is: (circle one) A B C D

TAKE HOME FINAL

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32) (5 pts) Find the absolute maximum and the absolute minimum for the function 𝑓(𝑥) = −8𝑥

𝑥2+4 on the

interval [-9,9].

f has an absolute maximum of ______________ at the point x = ______________.

f has an absolute minimum of ______________ at the point x = ______________.

TAKE HOME FINAL

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33) (5 pts) Find the maximum profit for a company that has the following price-demand and cost

functions:

𝑝(𝑥) = 300 − 𝑥

30 , 𝐶(𝑥) = 90,000 + 30𝑥

Then find the price per item that results in the maximum profit. Record your answers at the bottom of the

page.

The maximum profit to be gained is: ________________________________________________

The price per item that is required to achieve the maximum profit is: ______________________