EXAM II REVIEW

Apps. of Derivatives

100

Find the critical point. 

f(x)=x+9/x +9

critical point= x=-3,3

200

Find the relative min and max. for the function. (1st derivative test only) 

f(x)= x+ 9/x +9

relative min= (-3,3)

relative max= (3,15)

300

What is the inflection point? 

f(x)= 4x³-3x²

inflection point= 1/4

400

What is the relative min and max? Use the second derivative test. f(x)= 4x³-3x²

relative min=(1/2,1/8)

relative max= (0,0)

500

Complete the second derivative test for the following. f(x)=4x/1+x² SOLVE FOR ALL PARTS!

a. domain= all real numbers 

b. f'(x)= -4x²+4/(1+x²)²

c. critical point= -1,1

d. increasing (-1,1) decreasing (-inf.,-1),(1,infin.) 

e. f''(x)= 4x(-6x²-6)/(1+x²)³

f. inflection point x=0 

d. concave down (-inf. , 0) concave up (0,infin.) 

Exponential Functions

100

Find the derivative of the function. f(x)= (x²+5)(e^-7x+4).

f'(x)= e^-7x+4(2x=7x²-35)

200

Solve. 

f(x)= (200/1+3e^-0.3x)=50

x=0

*you cannot take the log of a negative number*

300

Find f'(x). f(x)= 4^x/3-e^2x

f'(x)= 12^x/(3-e^2x)²

400

Find the derivative of the function. f(x)= e^{x^2+5x}/1-7e^4x+1

f'(x)=e^{x^2+5x}(2x+5)(1-7e^4x+1)-e^{x^2+5x}(-7e^4x+1)(4)/(1-7e^4x+1)²

500

Find the equation of the tangent line. f'(x)e^-x x=1

y=-1/e +2/e

Word Probs

100

A ball is thrown straight up from roof of an 80ft math building. The distance of the ball from the group at any time is given by the function. f(x)= -16x²+64x+80

When does the ball hit the ground?

t= 5 seconds 

200

You are a beneficiary of a trust fund established for your son 21 years ago. If the original amount placed in the trust fund was $10,000, how much money will your son receive if the money has earned interest at the rate of 6% per year compounded quarterly? 

A=10,000(1+(0.06/4)81

300

The quantity demanded of McDonald's big macs is related to the unit price in dollars by the equation. p= 100000/250+x (0<x<750). 

Find the level of sales win which the corresponding unit price of big macs that will result in a maximum revenue per day for the company. What is the revenue?

x=250

400

Find all the absolute extrema. f(x)=-x²+4x+6 [0,5]

absolute min (5,1) 

absolute max (-2,10)

500

A rectangular box with a square base has a volume of 30ft³. The material for the base costs 0.40c/ft². The material for the side costs 0.10c/ft². The material for the top costs 0.20c/ft². Determine the dimensions of the box to be constructed at the min. cost.

20^1/4 x 20^1/4 x 20^1/2

Log Functions

100

Solve. 

y=2^3x+2=12

x=5/6

200

Rewrite as a log function. 

ln(x³)+1/2ln(3x²-x)-2x

ln(x3sqrt(3x²-x)/e^2x)

300

Find the derivative of the function. 

f(x)=log₇(x²sqrt(x+3))

f'(x)= 3ln(x)/ln(7) +(ln x+3)/2(ln(7)

400

Find the equation of the tangent line. f(x)=xln(x) x=1

y=x-1

500

Find where the function is increasing or decreasing. f(x)= ln(x²)

increasing (0,infin.) 

decreasing (-infin. ,0) 

Word Probs. Pt.2

100

How long will it take for an investment of $6000 to grow to $7000 if the investment earns interest at a rate of 7.5% compounded continuously? 

t=ln(7/6)/0.075

200

The rate at which Dunkin Donuts gets robbed each month doubles every 8 months. Let t be the number of months, and find the time it takes for the number of thefts to triple. 

f(t)=Ae^kt

t=8ln(3)/ln(2)

300

A farmer has 4000 of fencing with a river on one side that does not require fencing. What are the max. dimension of the fence? 

L=2000 w=1000

400

Who is your most favorite SI PASS leader? 

me :)

500

Expand and simplify. 

f(x)= ln[e⁵(x+3)²/4throot(y)(x²-5x-6)

5+2ln(x+3)-1/4ln(4)-ln(x-6)-ln(x+1)