note: exam 2 will cover more material than what is listed ......note: exam 2 will cover more...

MATH%151%SI% Exam%2%Review% 10/20/15%

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Note: Exam 2 will cover more material than what is listed here and on the other sample test! Study your notes, homework, and textbook to cover all material from Sections 4.1-4.5 and 5.1.

1. Find the Derivative of the function f(x) (4.1-4.5) a) ! ! = !!! + 4!

b) ! ! = (2! + 3)(3!!)

c) ! ! = ( ! + 3)(!! − 5!)

d) ! ! = !!!!!

e) ! ! = !!!!!/!


f) ! ! = !!2−4

g) ! ! = log!(3!)+ 5! + 23

h) ! ! = ln!(4!! + 3! + 2)

i) ! ! = 15!

j) ! ! = 6!!! − 5!"#!!

k) ! ! = 7!! − 8 ln !


2. Find the equation of the line tangent to the curve ! ! = ! 3− !! (4.2) at the point (2,-2). Use the product rule for the derivative of f.

3. The total number bacteria (in millions) t hours after the present time is given by the function N(t)=52t-4. What is the function for the rate of growth of the bacterial population (dN/dt)? (4.3,4.4)

4. Find all critical numbers for the following (5.1-5.2) a) ! ! = !! − 12!

b) ! ! = !! − 4


5. Find the intervals for which the following functions are increasing, decreasing, and find

the points for all relative maxima and minima (5.1-5.2) a) ! ! = !! − 3!+4

6. Below is the graph of F’(x). Indicate where the f(x) is increasing and decreasing.%%

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Things%to%Know%

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Logarithms:%

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• log![!(!) ∗ ! ! ] = log! ! ! + log! ! ! %

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• log![!(!) /! ! ] = log! ! ! − log! ! ! %

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• log![!(!)!] = !log! ! ! %

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• log! ! = ! → !! = !%%

Interest:%

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• Compound:!! = ! 1 + !!

!"%

• Continuous:!! = !!!"%• %

Definition%of%Derivative:%

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• !! ! = lim!→!! !!! !! !

! %

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Properties%of%derivatives:%

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• Increasing%function:%dy/dx>0%

• Decreasing%function:%dy/dx<0%

• Critical%numbers%dy/dx=0%or%DNE%(but%function%does%exist)%

• Relative%max:%increasing→decreasing%

• Relative%min:%decreasing→increasing%

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Formulas%(continued):%

• Revenue=Unit%Price%x%Units%sold%

• Profits=RevenueUTotal%Cost%

• Average%Revenue=Revenue/Units%

• Average%Cost=Total%Cost/Units%

• Average%Profit=Profit/Units%

• !!"!"#"$%" = !"#$%&"'!!"#"$%"%

• !!" !"#$ = !"#$%&"'!!"#$%

• !!" !"#$%& = !"#$%&"'!!"#$%

• !!" !"#$%$"& = !"#$%&'(%

• !!" !"#$%&'( = !""#$#!"#$%&%%

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Differentiation%Rules:%

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• !"#$%&#%:! !!" ! = 0!%• !"#$%!!"#$: !!" !

! = !!!!!%• !"#$%&#%!!"#$%&#'!!"#$: !!" !" ! = ! !

!" ! ! %

• !"# !"## !"#$%: !!" ! ! ± ! ! = !!" ! ! ± !

!" ! ! %

• !"#$%&'!!"#$: !!" ! ! ∙ ! ! = !! ! ! ! + !! ! ! ! %

• !"#$%&'$!!"#$: !!"! !! ! = !! ! ! ! !!! ! !(!)

! ! ! %

• !"#$%&%'()*!!"#$: !!" !! = !!!"#%

• !"#$%&'ℎ!"#!!"#$: !!" log! ! =!

!"#$%

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• !"#$%!!"#$: !!" ! ! ! = !! ! ! !! ! %

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• !"#$%!!"#$ + !"#$%!!"#$:%!

!!" [! ! ]! = ![!(!)]!!!!′(!)%%

• !"#$!"!#$%&!!"#$ + !"#$%!!"#$:!!

!!" !

!(!) = !! ! !"# ∙ !′(!)%%

• !"#$%&'()&*!!"#$ + !"#$%!!"#$:!%

!!" log! !(!) =

!′(!)!(!)!"#%

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How!to!determine!increasing/decreasing!intervals!and!Relative!Extrema!

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1. Find%the%derivative%of%the%function%in%

question%

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2. Find%the%undefined%values%of%both%derivative%

and%original%find%and%critical%numbers.%%%

What%are%critical%numbers?%

Where!the!derivative!is!equal!to!zero!or!undefined!where!the!function!is!defined.%%

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3. Plot%the%critical%numbers%and%undefined%

numbers%on%a%number%line%

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4. Pick%intermediate%values%on%each%interval%

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5. Test%intermediate%values%with%the%%

First%Derivative%test%

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6. Draw%a%“rough%sketch”%of%the%function%using%

the%results%

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7. Write%down%increasing%and%decreasing%

intervals%using%%(%%open%%%/%closed%%)%brackets.%%

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8. Label%extrema%and%determine%y%values%for%

points%

Example%%%%%%! ! = 4 − x!%!!" ! ! = 1

2 4 − !! !!/! ∙ −2!%

!!(!) = − !! − !!

!

!!

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! ! is!undefined!where!4 − !! < 0%%(negative%root)%

! ! !!"!!"#$%&"$#!!"#$#!! > !!!"#!! < −!!!! ! !is!undefined!where!4 − !! = 0!!

(bottom=0)%

!! ! !!"!!"#$%&"$#!!"#$#!! = !!!"#!! = −!!!!! ! = 0!when!top = 0%

x=0!!!!!!!!

!! −! = !!!,!positive!

!! ! = − !!,!negative!

!!!!!!!

Increasing:!(D2,0)!Decreasing:!(0,2)!

!!!!!!!

Increasing%to%decreasing%creates%a%maximum%%

Relative!maximum!(0,2)!Relative!minima!(D2,0)!&!(2,0)!

U2% %0% %2%

%X% %%%%%%%%%%%X%

U1% %%%%%%%%%%%1%U2% %0% %2%

%X% %%%%%%%%%%%X%

U1% %%%%%%%%%%%1%U2% %0% %2%++++++++%%%%%U%U%U%U%U%U%U%U%U%

note: exam 2 will cover more material than what is listed ......note: exam 2 will cover more...

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