Mth 241 Test 2 Extra Practice: ARC and IRC Name:

This is not a practice exam. This is a supplement to the activities v\/e do in class and the homework.

1. Use the graph to answer the following

questions.

a. Draw a line connecting the points (1,5)

and (4, 8).

b. The line you just drew is called the

' ^ C g y x - f line.

c. The slope of this line is called the

d. Calculate the slope of this line.

e. Draw the tangent line to the point (1, 5).

f. Calculate the slope of this line and find the equation of the tangent line.

g. The slope ofthe tangent line is the: "kA/v»W<iDi*5 YU^te^ of

2. Calculate the ARC over the interval [1,1+h] for the function - 6x :

h .1 .01 .001 .0001

ARC "3.^ - 5 7 9 9 ^

Use the table above to estimate the IRC atx = l : ^

3. Use shortcut rules to find the following derivatives: ^

a. / (x ) = 3 x ' - 2 x - 6 f'(2)= f Y>c - */ZX - 2.

b. / (x ) = -x^- ; r^ r(x)= -SX"" f'( ^ ^ -^x"-

4. For the function Ix^ - x +1:

a. Use <3 = 3

to find an expression for the ARC at(^=^ ^ >-i

b. What is the derivative at x = 3?

6 ^

5. For the f u n c t i o n -3x -5 : ,. f{a + h)-f(a) ,

a. Use — t o find an expression for the ARC at x a.

U = - 3

b. What is the derivative at x = a?

-3 -t3] v-^' 6. IVIore practice using the shortcut rules to find the derivative of a function:

Express answer using positive exponents. 1 3 1

a. y = - + 5 X x' x'

b. = ln{x) +

c. = 5e'

d. >' = log5 X

6 ^ " 5

dx

dx

dy

dx

dy

dx

X ><•

X

7. The Clearwater Company is deciding to get into the bottled water business. The company knows that it costs $3.00 per case of bottled water plus $20,000 in fixed costs. Their marketing team does a bit of research to determine that the demand equation for bottled water in the Corvallis area is p = - .001^ +15, where p is price ($) per case and q is the number of cases.

a) In terms of quantity, find the cost function: C(q) = _

b) In terms of quantity, find the revenue function. R(q)= (~ > ^^'^ 'h ' " -^^^^ ^ ^

c) What is the fewest that must be sold to breakeven? q =

.1

d) Find the marginal cost function. MC =

e) Find the marginal revenue function. MR = _

f) What is the MR at the breakeven point (include units). MR = _

g) What is the MC at the breakeven point (include units). MC = jCOJjA.

h) Find the quantity that must be sold such that MR=MC. q ^ L

i) Find the quantity that must be sold such that MP - 0.

j) Find the quantity that must be sold to maximize profit. q =

k) Find the quantity that must be sold such that MR = 0.

I) When MR -= 0 what does this quantity represent? Write using sentences.

m) What is the MP when MR = 0: Positive, Negative, or zero? /1/g< t<r Vc

n) In order to maximize revenue what should you do when MR > 0: Increase production? Keep production constant? Decrease production? Explain! ^

A stone is thrown upwards at an initial velocity of 120 feet per second from the top of a dam whose

height, h, is unknown. The following model represents the height of stone, s{t), as a function of time:

5(0 = -16/'+120/ + /2

a. What is the velocity function? - 3 7 - 6 4 f - ^ 0 ^ d l^w>»4-w< o-< ^

b. What is the velocity of the stone after 1 second? {^l%Cc

c. What is the velocity of the stone after 7 seconds? ~

d. Is the velocity of the stone after 1 second is (greater than, less than, or approximately equal

to) the velocity of the stone after 7 seconds? Explain. . yAW < ^

just u.r^ — w t i - • ^ f ' ' e. What is the velocity of the stone when it is at the maximum height? ^ ^ - ^ p f. Use the velocity function to determine how many seconds have elapsed when velocity is

9. Use the graph of / ( x ) to complete the table below.

fix) fXx) X P,N,Z l,D, Min, Max P,N,Z l,D, Min, Max Concave U, D, or

Inflection

-5 X u. -4

i T7 -2 X 7 r 0 •D

1.5 A/ X

3 JZ u

4 I f T

Challenge:

10. Find a function whose derivative is:

a. / ' (x ) = 6 x ' - 2 x + 3

b. f'(x) = - + e' +7r X