AP Exam
The Final Hours of Test Prep…
Tonight
• Don’t cram – you’ve spent a month studying for this exam!
• Spend a little time reviewing– Practice tests– FRQ’s– Stand and Deliver
• Relax • Calculator check – good batteries, radians• Go to bed early – get a good nights rest
Tomorrow Morning
• Get up early
• Eat a good, healthy breakfast– Avoid sugar or caffeine (it’s a long test… your
blood sugar will spike and drop!)
• Be confident!
During the Test
• Focus on the problems you know• Star and come back to other questions
later (esp. on FRQ’s!)• READ carefully!• MC – eliminate answers, guessing is ok!• FRQ’s
– NEAT and ORGANIZED work!– Use proper notation, ( ), dx, units– Start with the questions you know!
GOOD LUCK!!
• YOU CAN DO IT!
• You’ve studied and worked soooo hard
• Be confident!
When you see…
Find equation of the line tangent to f(x) at (a, b)
You think…
Equation of the tangent lineEquation of the tangent line
Take derivative of f(x)
Set f ’(a) = m
Use y- y1 = m ( x – x1 )
You think…
When you see…
Find the interval where f(x) is increasing
ff(x) increasing(x) increasing
Find f ’ (x) > 0
Answer: ( a, b ) or a < x < b
You think…
When you see…
Find the interval where the slope of f (x) is increasing
Slope of Slope of f f (x) is increasing(x) is increasing
Find the derivative of f ''(x) = f ''(x) Set numerator and denominator = 0 to find critical points Make sign chart of f '' (x)
Determine where f''(x) is positive
You think…
When you see…
Find the minimum value of a function
Minimum value of a functionMinimum value of a function
Find all critical numbers (where f '( x) = 0 or undefined) Plug those values into f (x) Plug endpoints into f(x)
Choose the smallest
You think…
When you see…
Find the minimum slope of a function
Minimum slope of a functionMinimum slope of a function
Find critical numbers of f ’(x) (Where f ” (x) = 0 or undefined) Plug crit. numbers into f ’(x) Plug endpoints into f ’(x) Choose the smallest
You think…
When you see…
Find critical numbers
Find critical numbersFind critical numbers
Express f ‘ (x ) as a fraction
Set both numerator and denominator = 0
You think…
When you see…
Find inflection points
Find inflection pointsFind inflection points
Express f “ (x) as a fraction
Set numerator and denominator = 0
Make a sign chart of f “ (x)
Find where it changes sign ( + to - ) or ( - to + )
You think…
When you see…
Show that f(x) is continuous
..f(x) is continuousf(x) is continuous
S h o w t h a t
1 ) xfax
lim e x i s t s ( p r e v i o u s s l i d e )
2 ) af e x i s t s
3 ) afxfax
lim
You think…
When you see…
Find horizontal asymptotes of f(x)
Find horizontal asymptotes of f(x)Find horizontal asymptotes of f(x)
Show
xfx lim
and
xf
x lim
You think…
When you see…
Find the average rate of change of f(x) at [a, b]
Average rate of change of f(x)Average rate of change of f(x)
Find
f (b) - f ( a)
b - a
You think…
When you see…
Find the instantaneous rate of change of f(x)
on [a, b]
Instantaneous rate of change of f(x)Instantaneous rate of change of f(x)
Find f ‘ ( a)
You think…
When you see…
Find the average valueof xf on ba,
Average value of the functionAverage value of the function
Find
b-a
dxxfb
a
You think…
When you see…
Show that a piecewise function is differentiable at the point a where the function rule splits
Show a piecewise function is Show a piecewise function is
differentiable at x=adifferentiable at x=a
F i r s t , b e s u r e t h a t t h e f u n c t i o n i s c o n t i n u o u s a t
ax .
T a k e t h e d e r i v a t i v e o f e a c h p i e c e a n d s h o w t h a t
xfxfaxax
limlim
You think…
When you see…
Given s(t) (position function), find v(t)
Given position s(t), find v(t)Given position s(t), find v(t)
Find tstv
You think…
When you see…
Given v(t), find how far a particle travels on [a, b]
Given v(t), find how far a particle Given v(t), find how far a particle travels on [a,b]travels on [a,b]
Find b
a
dttv
You think…
When you see…
Given v(t) and s(0),
find s(t)
Given v(t) and s(0), find s(t)Given v(t) and s(0), find s(t)
s t v t dt C
P lu g in t = 0 to fin d C
You think…
When you see…
Show that the Mean Value Theorem holds
on [a, b]
Show that the MVT holds on [a,b]Show that the MVT holds on [a,b]
S h o w th a t f is c o n tin u o u s a n d d if fe re n tia b leo n th e in te rv a l .
T h e n f in d s o m e c s u c h th a t
f c f b f a
b a.
You think…
When you see…
Find f ’(x) by definition
Find f Find f ‘‘( x) by definition( x) by definition
f x limh0
f x h f x h
or
f x limx a
f x f a x a
You think…
When you see…
y is increasing proportionally to y
.y is increasing proportionally to yy is increasing proportionally to y
kydt
dy
translating to
ktCey
You think…
When you see…
dttfdx
d x
a
Fundamental TheoremFundamental Theorem
2nd FTC: Answer is xf
You think…
When you see…
dtuf
dx
d u
a
Fundamental Theorem, againFundamental Theorem, again
2nd FTC: Answer is dx
duuf
You think…
When you see…
The rate of change of population is …
Rate of change of a population Rate of change of a population
...dt
dP
You think…
When you see…
The line y = mx + b is tangent to f(x) at (a, b)
.y = mx+b is tangent to f(x) at (a,b)y = mx+b is tangent to f(x) at (a,b)
Two relationships are true.
The two functions share the sameslope ( xfm)
and share the same y value at 1x.
You think…
When you see…
Find area using left Riemann sums
Area using left Riemann sumsArea using left Riemann sums
0 1 2 1( ) ( ) ( ) ... ( )nA base f x f x f x f x
You think…
When you see…
Find area using right Riemann sums
Area using right Riemann sumsArea using right Riemann sums
1 2 3( ) ( ) ( ) ... ( )nA base f x f x f x f x
You think…
When you see…
Find area using midpoint rectangles
Area using midpoint rectanglesArea using midpoint rectangles
Typically done with a table of values.
Be sure to use only values that are given.
If you are given 6 sets of points, you can only do 3 midpoint rectangles.
You think…
When you see…
Find area using trapezoids
Area using trapezoidsArea using trapezoids
0 1 2 1( ) 2 ( ) 2 ( ) ... 2 ( ) ( )2 n n
baseA f x f x f x f x f x
This formula only works when the base is the same. If not, you have to do individual trapezoids
You think…
When you see…
Solve the differential equation …
Solve the differential equation...Solve the differential equation...
Separate the variables –
x on one side, y on the other.
The dx and dy must all be upstairs..
You think…
When you see…
Meaning of
dttfx
a
Meaning of the integral of f(t) from a to xMeaning of the integral of f(t) from a to x
The accumulation function –
accumulated area under the function xf
starting at some constant a and ending at x
You think…
When you see…
Given a base, cross sections perpendicular to
the x-axis that are squares
Semi-circular cross sections Semi-circular cross sections perpendicular to the x-axisperpendicular to the x-axis
The area between the curves typically is the base of your square.
So the volume is dxbase
b
a 2
You think…
When you see…
Find where the tangent line to f(x) is horizontal
Horizontal tangent lineHorizontal tangent line
Write xf as a fraction.
Set the numerator equal to zero
You think…
When you see…
Find where the tangent line to f(x) is vertical
Vertical tangent line to f(x)Vertical tangent line to f(x)
Write xf as a fraction.
Set the denominator equal to zero.
You think…
When you see…
Find the minimum
acceleration given v(t)
Given v(t), find minimum accelerationGiven v(t), find minimum acceleration
First find the acceleration tvta
Then minimize the acceleration by examining ta .
You think…
When you see…
Approximate the value f(0.1) of by using the
tangent line to f at x = 0
Approximate f(0.1) using tangent line Approximate f(0.1) using tangent line to f(x) at x = 0to f(x) at x = 0
Find the equation of the tangent line to f using 11 xxmyy
where 0fm and the point is 0,0f .
Then plug in 0.1 into this line.Be sure to use an approximation sign.
You think…
When you see…
Given the value of F(a) and the fact that the
anti-derivative of f is F, find F(b)
Given Given FF((aa)) and the that the and the that the anti-derivative of anti-derivative of ff is is FF, find , find FF((bb))
Usually, this problem contains an antiderivative you cannot take. Utilize the fact that if xF is the antiderivative of f,
then b
a
f x dx F b F a .
Solve for bF using the calculator to find the definite integral
You think…
When you see…
Given , find dxxfb
a
dxkxfb
a
Given area under a curve and vertical Given area under a curve and vertical shift, find the new area under the curveshift, find the new area under the curve
dxkdxxfdxkxfb
a
b
a
b
a
You think…
When you see…
Given a graph of
find where f(x) is
increasing
'( )f x
Given a graph of f ‘(x) , find where f(x) is Given a graph of f ‘(x) , find where f(x) is increasingincreasing
Make a sign chart of xf
Determine where xf is positive
You think…
When you see…
Given v(t) and s(0), find the greatest distance from the origin of a particle on [a, b]
Given Given vv((tt)) and and ss(0)(0), find the greatest distance from , find the greatest distance from the origin of a particle on [the origin of a particle on [aa, , bb]]
Generate a sign chart of tv to find turning points.Integrate tv using 0s to find the constant to find ts .Find s(all turning points) which will giveyou the distance from your starting point.
Adjust for the origin.
When you see…
Given a water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at the rate of E(t) gallons/min on
, find
1 2[ , ]t t
You think…
a) the amount of water in
the tank at m minutes
Amount of water in the tank at t minutesAmount of water in the tank at t minutes
dttEtFgt
t 2
You think…
b) the rate the water
amount is changing
at m
Rate the amount of water is changing at t = m
mEmFdttEtFdt
d m
t
You think…
c) the time when the
water is at a minimum
The time when the water is at a minimumThe time when the water is at a minimum
mEmF = 0,
testing the endpoints as well.
You think…
When you see…
Given a chart of x and f(x) on selected values between a and b, estimate where c is between a and b.
'( )f x
Straddle c, using a value k greater than c and a value h less than c.
so hk
hfkfcf
You think…
When you see…
Given , draw a
slope field dx
dy
Draw a slope field of dy/dxDraw a slope field of dy/dx
Use the given points
Plug them into dx
dy,
drawing little lines with theindicated slopes at the points.
You think…
When you see…
Find the area between curves f(x) and g(x) on
[a,b]
Area between f(x) and g(x) on [a,b]Area between f(x) and g(x) on [a,b]
dxxgxfAb
a ,
assuming f (x) > g(x)
You think…
When you see…
Find the volume if the area between the curves
f(x) and g(x) is rotated about the x-axis
Volume generated by rotating area between Volume generated by rotating area between f(x) and g(x) about the x-axisf(x) and g(x) about the x-axis
dxxgxfAb
a 22
assum ing f (x) > g(x).