Download - When you see…
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When you see…
Find the zeros
You think…
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To find the zeros...To find the zeros...
Set function = 0
Factor or use quadratic equation if quadratic.
Graph to find zeros on calculator.
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When you see…
Find equation of the line tangent to f(x) at (a, b)
You think…
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Equation of the tangent lineEquation of the tangent line
Take derivative of f(x)
Set f ’(a) = m
Use y- y1 = m ( x – x1 )
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When you see…
Find equation of the line normal to f(x) at (a, b)
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Equation of the normal lineEquation of the normal line
Take f ’(x)
Set m = 1_ f ’(x)
Use y – y1 = m ( x – x1)
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When you see…
Show that f(x) is even
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Even functionEven function
f (-x) = f ( x)
y-axis symmetry
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Show that f(x) is odd
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Odd functionOdd function
. f ( -x) = - f ( x )
origin symmetry
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Find the interval where f(x) is increasing
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ff(x) increasing(x) increasing
Find f ’ (x) > 0
Answer: ( a, b ) or a < x < b
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Find the interval where the slope of f (x) is increasing
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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
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You think…
When you see…
Find the minimum value of a function
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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
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You think…
When you see…
Find the minimum slope of a function
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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
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You think…
When you see…
Find critical numbers
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Find critical numbersFind critical numbers
Express f ‘ (x ) as a fraction
Set both numerator and denominator = 0
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You think…
When you see…
Find inflection points
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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 + )
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You think…
When you see…
Show that exists limx→ a
f (x)
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Show existsShow exists limx→ a
f (x)
Show that
€
limx→ a−
f x( )=limx→ a+
f x( )
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You think…
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Show that f(x) is continuous
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..f(x) is continuousf(x) is continuous
Show that
1) ()xfax→lim exists (previous slide)
2) ()af exists
3) ()()afxfax =→lim
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Find vertical
asymptotes of f(x)
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Find vertical asymptotes of f(x)Find vertical asymptotes of f(x)
Factor/cancel f(x)
Set denominator = 0
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Find horizontal asymptotes of f(x)
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Find horizontal asymptotes of f(x)Find horizontal asymptotes of f(x)
Show
()xfx∞→lim and
()xfx−∞→lim
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You think…
When you see…
Find the average rate of change of f(x) at [a, b]
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Average rate of change of f(x)Average rate of change of f(x)
Find
f (b) - f ( a)
b - a
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You think…
When you see…
Find the instantaneous rate of change of f(x)
on [a, b]
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Instantaneous rate of change of f(x)Instantaneous rate of change of f(x)
Find f ‘ ( a)
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You think…
When you see…
Find the average valueof ()xf on [ ]ba,
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Average value of the functionAverage value of the function
Find ( )
-b a
dxxfb
a∫
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You think…
When you see…
Find the absolute minimum of f(x) on [a, b]
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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)
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You think…
When you see…
Show that a piecewise function is differentiable at the point a where the function rule splits
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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
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You think…
When you see…
Given s(t) (position function), find v(t)
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Given position s(t), find v(t)Given position s(t), find v(t)
Find ( ) ( )tstv ′=
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You think…
When you see…
Given v(t), find how far a particle travels on [a, b]
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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
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You think…
When you see…
Find the average velocity of a particle
on [a, b]
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Find the average rate of change on Find the average rate of change on [a,b][a,b]
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You think…
When you see…
Given v(t), determine if a particle is speeding up at
t = a
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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
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You think…
When you see…
Given v(t) and s(0),
find s(t)
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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
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You think…
When you see…
Show that Rolle’s Theorem holds on [a, b]
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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.
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You think…
When you see…
Show that the Mean Value Theorem holds
on [a, b]
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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.
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You think…
When you see…
Find the domain
of f(x)
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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
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You think…
When you see…
Find the range
of f(x) on [a, b]
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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)
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You think…
When you see…
Find the range
of f(x) on (−∞,∞)
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Find the range of f(x) onFind the range of f(x) on ( )∞∞− ,
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You think…
When you see…
Find f ’(x) by definition
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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
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You think…
When you see…
Find the derivative of the inverse of f(x) at x = a
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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.
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You think…
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y is increasing proportionally to y
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.y is increasing proportionally to yy is increasing proportionally to y
kydtdy=translati ngto
ktCey=
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You think…
When you see…
Find the line x = c that divides the area under
f(x) on [a, b] into two equal areas
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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∫∫ =
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You think…
When you see…
( ) =∫ dttfdx
d x
a
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Fundamental TheoremFundamental Theorem
2nd FTC: Answer is ( )xf
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You think…
When you see…
( ) =∫ dtuf
dx
d u
a
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Fundamental Theorem, againFundamental Theorem, again
2nd FTC: Answer is ( )dxdu
uf
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You think…
When you see…
The rate of change of population is …
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Rate of change of a population Rate of change of a population
...=dtdP
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You think…
When you see…
The line y = mx + b is tangent to f(x) at (a, b)
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.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.
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You think…
When you see…
Find area using left Riemann sums
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Area using left Riemann sumsArea using left Riemann sums
[ ]1210 ... −++++= nxxxxbaseA
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You think…
When you see…
Find area using right Riemann sums
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Area using right Riemann sumsArea using right Riemann sums
A =base x
1+ x
2+ x
3+ ... + x
n⎡⎣ ⎤⎦
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You think…
When you see…
Find area using midpoint rectangles
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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.
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You think…
When you see…
Find area using trapezoids
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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
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You think…
When you see…
Solve the differential equation …
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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..
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When you see…
Meaning of
( )dttfx
a∫
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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
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When you see…
Given a base, cross sections perpendicular to
the x-axis that are squares
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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
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When you see…
Find where the tangent line to f(x) is horizontal
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Horizontal tangent lineHorizontal tangent line
Write ()xf′ a s a frac .tion
Se t th enumerato r equa l tozero
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When you see…
Find where the tangent line to f(x) is vertical
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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
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When you see…
Find the minimum
acceleration given v(t)
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Given v(t), find minimum accelerationGiven v(t), find minimum acceleration
First find the acceleration ()()tvta ′= Thenminimiz et heaccelerati on by
examini ng ()ta′ .
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When you see…
Approximate the value f(0.1) of by using the
tangent line to f at x = 0
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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.
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When you see…
Given the value of F(a) and the fact that the
anti-derivative of f is F, find F(b)
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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
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When you see…
Find the derivative off(g(x))
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Find the derivative of f(g(x))Find the derivative of f(g(x))
( )( ) ( )xgxgf ′⋅′
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When you see…
Given , find ( )dxxfb
a
∫ ( )[ ]dxkxfb
a
∫ +
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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
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When you see…
Given a graph of
find where f(x) is
increasing
f '(x)
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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
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When you see…
Given v(t) and s(0), find the greatest distance from the origin of a particle on [a, b]
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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
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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
]
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You think…
a) the amount of water in
the tank at m minutes
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Amount of water in the tank at t minutesAmount of water in the tank at t minutes
( ) ( )( )dttEtFgt
t∫ −+2
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b) the rate the water
amount is changing
at m
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Rate the amount of water is changing at t = m
ddt
F t( ) −E t( )( )t
m
∫ dt =F m( ) −E m( )
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You think…
c) the time when the
water is at a minimum
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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.
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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)
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Straddle c, using a value k greater than c and a value h less than c. so
( ) ( ) ( )hk
hfkfcf
−−
≈′
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You think…
When you see…
Given , draw a
slope field dx
dy
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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.
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When you see…
Find the area between curves f(x) and g(x) on
[a,b]
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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)
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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
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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).