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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) ! ! = !!!!!/!
MATH%151%SI% Exam%2%Review% 10/20/15%
f) ! ! = !!2−4
g) ! ! = log!(3!)+ 5! + 23
h) ! ! = ln!(4!! + 3! + 2)
i) ! ! = 15!
j) ! ! = 6!!! − 5!"#!!
k) ! ! = 7!! − 8 ln !
MATH%151%SI% Exam%2%Review% 10/20/15%
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
MATH%151%SI% Exam%2%Review% 10/20/15%
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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MATH%151%SI% Exam%2%Review% 10/20/15%
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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MATH%151%SI% Exam%2%Review% 10/20/14%
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%
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