Post on 11-Dec-2015
To find the zeros...To find the zeros...
Set 0)( xf Factor or use quadratic formula if quadratic. Graph to find zeros on calculator.
Equation of the tangent lineEquation of the tangent line
Take derivative of f(x) Set maf )( Use y- y1 = m ( x – x1 )
Equation of the normal lineEquation of the normal line
Take )(xf
Set )(
1
afm
Use y – y1 = m ( x – x1)
Slope of Slope of f f (x) is increasing(x) is increasing
Find the derivative of )()( xfxf Set numerator and denominator = 0 to find critical points Make sign chart of )(xf
Determine where 0)( xf is positive
Minimum value of a functionMinimum value of a function
Find where 0)( xf To find critical numbers Plug those values and a and b
into f (x) Choose the smallest
Find critical numbersFind critical numbers
Express )(xf as a fraction Set both numerator and denominator = 0
Find inflection pointsFind inflection points
Express )(xf as a fraction Set numerator and denominator = 0 Make a sign chart of )(xf Find where )(xf changes sign ( + to - ) or ( - to + )
..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
Find vertical asymptotes of f(x)Find vertical asymptotes of f(x)
Factor/cancel f(x), if possible
Set denominator = 0
Find the average value of the derivative of f(x) :
Average rate of change of f(x)Average rate of change of f(x)
ab
afbfdxxf
ab
b
a
)()()(
1
Find the absolute minimum of f(x)Find the absolute minimum of f(x)
a) Find the derivative of f(x) b) Find all critical numbers c) Plug those values into f (x) d) Find f (a) and f (b) e) Choose the largest of c) and d)
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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
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Given v(t), find how far a particle travels (total distance traveled)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 where the particle changes direction, that is where v(t) changes sign.
Find b
c
c
a
dttvdttv )()( where t = c
is where the particle changes direction.
Or on your calculator: dttv
b
a )(
Find the average rate of change on Find the average rate of change on [a,b][a,b]
Find
vta
b
dt
basbsaba
Given v(t), determine if the particle is Given v(t), determine if the particle is speeding up at t=aspeeding up at t=a
Find v (k) and a (k).
Multiply their signs.
If positive, the particle is speeding up.
If negative, the particle is slowing down
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
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.
Find the domain of f(x)Find the domain of f(x)
Assume domain is
, .
Domain restrictions: non-zero denominators,Square root of non negative numbers,Log or ln of positive numbers
Find the range of f(x) on [a,b]Find the range of f(x) on [a,b]
Use max/min techniques to find relativemax/mins.
Then examine
f a , f b
Use max/min techniques to find relativemax/mins.
Then examine
limx
f x .
Find the range of f(x) onFind the range of f(x) on ,
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
.y is increasing proportionally to yy is increasing proportionally to y
kydt
dy
translating to
ktCey
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Find the line x = c that divides the area under
f(x) on [a, b] into two equal areas
Find the x=c so the area under f(x) is Find the x=c so the area under f(x) is
divided equallydivided equally
dxxfdxxfb
c
c
a
OR
dxxfdxxfb
a
c
a
2
1
.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 same slope at (a, b) ( afm ) and share the same y value at (a, b).
Area using left Riemann sumsArea using left Riemann sums
If subintervals have equal width:
)(...)()()( 1210
nxfxfxfxfn
abA
Area using right Riemann sumsArea using right Riemann sums
If subintervals have equal width:
)(...)()()( 321 nxfxfxfxfn
abA
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.
Area using trapezoidsArea using trapezoids
)()(2...)(2)(2)(2 1210 nn xfxfxfxfxf
n
abA
This formula only works when the width of each subinterval is the same.
If not, you have to do individual trapezoids
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. Then integrate both sides, and solve for y if possible
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
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
Horizontal tangent lineHorizontal tangent line
Write xf as a fraction.
Set the numerator equal to zero
Vertical tangent line to f(x)Vertical tangent line to f(x)
Write xf as a fraction.
Set the denominator equal to zero.
Given v(t), find minimum accelerationGiven v(t), find minimum acceleration
First find the acceleration tvta
Then minimize the acceleration by examining ta .
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.
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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 aFbFdxxfb
a
.
Solve for bF using the calculator to find the definite integral
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
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
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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
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.
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)
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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
assuming f (x) > g(x).
Minimum slope of a functionMinimum slope of a function
Make a sign chart of f ’(x) = f ” (x) Find all the relative minimums Plug those back into f ‘ (x ) Choose the smallest
Derivative of the inverse of f(x) at x=aDerivative of the inverse of f(x) at x=a
Interchange x with y.
Solve for dx
dy implicitly (in terms of y).
Plug your x value into the inverse relation and solve for y.
Finally, plug that y into your dx
dy.
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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 give you the distance from your starting point.
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
The time when the water is at a minimumThe time when the water is at a minimum
mEmF = 0,
testing the endpoints as well.
33rdrd degree Taylor polynomial... degree Taylor polynomial...
!3
))((
!2
))(())(()()(
32
3
axafaxafaxafafxP
Never include a term that has a power of x greater than the degree (3 in this case). If you use your polynomial to approximate a value of the function, don’t use an equal sign.
Radius of convergence:Radius of convergence:
Use the ratio test: 1lim 1
n
n
n b
b
Write result as: rax r is the radius of convergence
Interval of convergence:Interval of convergence:
Find the radius of convergence (previous slide) Add and subtract r from a to create a
preliminary interval Substitute each endpoint into the power series
(for x) and determine if the resulting series converges or diverges.
If the series converges include the endpoint in the interval. If it diverges, don’t include the endpoint.
Error boundError bound
1 nn aRerror
. The error is less than or equal to the first non-included term of the
series
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Use the Lagrange form of the remainder of a
Taylor polynomial centered at x=a.
Lagrange form of the remainder:Lagrange form of the remainder:
)(max!)1(
)( )1(
1
cfn
axxRerror n
n
n
Where c is some value between a and x.
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When you see…
Find the velocity and acceleration vector of a particle at t=3 when
given x(t) and y(t)
Velocity and Acceleration VectorVelocity and Acceleration Vector
Velocity vector at t = 3:
(x’(3), y’(3))
Acceleration Vector at t = 3:
(x”(3), y”(3))
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Find the total distance of a particle moving in a plane on the interval
(t1, t2).