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When you see…. You think…. Find the zeros. To find the zeros. When you see…. Find equation of the line tangent to f ( x ) at ( a , b ). You think…. Equation of the tangent line. When you see…. Find equation of the line normal to f ( x ) at ( a , b ). You think…. - PowerPoint PPT Presentation
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When you see…
Find the zeros
You think…
To find the zeros...To find the zeros...
Set function = 0
Factor or use quadratic equation if quadratic.
Graph to find zeros on calculator.
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 equation of the line normal to f(x) at (a, b)
Equation of the normal lineEquation of the normal line
Take f ’(x)
Set m = 1_ f ’(x)
Use y – y1 = m ( x – x1)
You think…
When you see…
Show that f(x) is even
Even functionEven function
f (-x) = f ( x)
y-axis symmetry
You think…
When you see…
Show that f(x) is odd
Odd functionOdd function
. f ( -x) = - f ( x )
origin symmetry
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 it is positive
You think…
When you see…
Find the minimum value of a function
Minimum value of a functionMinimum value of a function
Make a sign chart of f ‘( x)
Find all relative minimums
Plug those values 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
Make a sign chart of f ’(x) = f ” (x)
Find all the relative minimums
Plug those back 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 exists limx→ a
f (x)
Show existsShow exists limx→ a
f (x)
Show that
€
limx→ a−
f x( )=limx→ a+
f x( )
You think…
When you see…
Show that f(x) is continuous
..f(x) is continuousf(x) is continuous
Show that
1) ()xfax→lim exists (previous slide)
2) ()af exists
3) ()()afxfax =→lim
You think…
When you see…
Find vertical
asymptotes of f(x)
Find vertical asymptotes of f(x)Find vertical asymptotes of f(x)
Factor/cancel f(x)
Set denominator = 0
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
()xfx−∞→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…
Find the absolute minimum of f(x) on [a, b]
Find the absolute minimum of f(x)Find the absolute minimum of f(x)
a) Make a sign chart of f ’( x)
b) Find all relative maxima
c) Plug those values into f (x)
d) Find f (a) and f (b)
e) Choose the largest of c) and d)
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
First, be sure that the function is continuous atax=.
Tak eth ederivativ eo f eac hpiec e an d sho w that() ()xfxf axax ′=′ +→→− 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 ()∫ba dttv
You think…
When you see…
Find the average velocity of a particle
on [a, b]
Find the average rate of change on Find the average rate of change on [a,b][a,b]
You think…
When you see…
Given v(t), determine if a particle is speeding up at
t = a
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
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 lug int = 0 t o fi ndC
You think…
When you see…
Show that Rolle’s Theorem holds on [a, b]
Show that Rolle’s Theorem holds on Show that Rolle’s Theorem holds on [a,b][a,b]
Show that f is continuous and differentiableon the interval
If
€
f a( )=f b( ) , t hen fi ndsom ec i n
€
a,b[ ]s uchtha t
€
′ f c( )=0.
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]
Show that f is continuous and differentiableon the interval.
Then find some c such that
€
′ f c( ) =f b( ) − f a( )
b−a.
You think…
When you see…
Find the domain
of f(x)
Find the domain of f(x)Find the domain of f(x)
Assume domain is
€
−∞,∞( ) .
Domai n restrictions: non-zer o denominators,Squar eroo t of non negativ enumbers,
Log orl n of positi ve numbers
You think…
When you see…
Find the range
of f(x) on [a, b]
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)
You think…
When you see…
Find the range
of f(x) on (−∞,∞)
Find the range of f(x) onFind the range of f(x) on ( )∞∞− ,
You think…
When you see…
Find f ’(x) by definition
Find f Find f ‘‘( x) by definition( x) by definition
€
′ f x( )= limh→ 0
f x + h( )− f x( )h
or
′ f x( )= limx→ a
f x( )− f a( )x −a
You think…
When you see…
Find the derivative of the inverse of f(x) at x = a
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 dxdy implicitly (in terms of y).
Plug your x value into the inverse relation and solve for y.
Finally, plug that y into your dxdy.
You think…
When you see…
y is increasing proportionally to y
.y is increasing proportionally to yy is increasing proportionally to y
kydtdy=translati ngto
ktCey=
You think…
When you see…
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∫∫ =
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 ( )dxdu
uf
You think…
When you see…
The rate of change of population is …
Rate of change of a population Rate of change of a population
...=dtdP
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′=)
andshar e th esa mey val uea t 1x.
You think…
When you see…
Find area using left Riemann sums
Area using left Riemann sumsArea using left Riemann sums
[ ]1210 ... −++++= nxxxxbaseA
You think…
When you see…
Find area using right Riemann sums
Area using right Riemann sumsArea using right Riemann sums
A =base x
1+ x
2+ x
3+ ... + x
n⎡⎣ ⎤⎦
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
[ ]nn xxxxxbaseA +++++= −1210 2...222This 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 ()xfstar tinga t som e consta nta a nde ndinga tx
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 base2( )ab∫ dx
You think…
When you see…
Find where the tangent line to f(x) is horizontal
Horizontal tangent lineHorizontal tangent line
Write ()xf′ a s a frac .tion
Se t th enumerato r equa l tozero
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′ a s a frac .tion
Se t th edenominato r equa l tozer .o
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 ′= Thenminimiz et heaccelerati on by
examini ng ()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 −=− wher e ()0fm′= an d th e poin t is ()()0,0f.
Th enplu g in 0.1 in to this lin .eB e su re t ous e an approximatio n()≈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 antiderivativeyou cannot take. Utilize the fact that if ()xF is the antiderivativ eo f f,
then Fx()ab∫dx=Fb()−Fa().
Solve for ()bF using the calculator to find the definite integral
You think…
When you see…
Find the derivative off(g(x))
Find the derivative of f(g(x))Find the derivative of f(g(x))
( )( ) ( )xgxgf ′⋅′
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
fx()+k⎡⎣ ⎤⎦ab∫ dx=fx()a
b∫dx+kab∫dx
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′Determin e 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 t o find turni ng points.Integrat e ()tv usi ng ()0s t o fi ndthe constan t t o fi nd ()ts.F inds(al l turni ng points) whi chwi ll give
yout he distanc efro m yourstart ing point.
Adjust fort he ori .gin
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
[t1,t2
]
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
ddt
F t( ) −E t( )( )t
m
∫ dt =F m( ) −E m( )
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
Fm( )−Em( )=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 dxdy,
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]
A=fx()−gx()⎡⎣ ⎤⎦ab∫ dx,
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
A=fx()( )2−gx()( )2⎡⎣⎢ ⎤⎦⎥ab∫ dx
assuming f (x) > g(x).
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