when you see…

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).