When you see…

Find the zeros

You think…

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 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 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 function

f (-x) = f ( x)

y-axis symmetry

You think…

When you see…

Show that f(x) is odd

Odd function

. f ( -x) = - f ( x )

origin symmetry

You think…

When you see…

Find the interval where

f(x) is increasing

f(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 f (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 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 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 numbers

Express f ‘ (x ) as a fraction

Set both numerator and denominator = 0

You think…

When you see…

Find inflection points

Find 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 lim ( )x a

f x

Show existslim ( )x a

f x

Show that

limxa

f x limxa

f x

You think…

When you see…

Show that f(x) is

continuous

.f(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)

Factor/cancel f(x)

Set denominator = 0

You think…

When you see…

Find horizontal

asymptotes of f(x)

Find horizontal asymptotes of f(x)

Show

xf

x 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)

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)

Find f ‘ ( a)

You think…

When you see…

Find the average value

of xf on ba ,

Average value of the function

Find

b-a

dxxf

b

a

You think…

When you see…

Find the absolute

minimum(or maximum) of

f(x) on [a, b]

Find the absolute minimum of f(x)

a) Make a sign chart of f ’(x)

b) Find all relative minima (or maxima)

c) Plug those values into f (x)

d) Find f (a) and f (b) -----endpoints!

e) Choose the smallest (largest for max)

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

differentiable at x=a

First, be sure that the function is continuous at

ax .

Take the derivative of each piece and show that

xfxfaxax

limlim

You think…

When you see…

Given s(t) (position

function), 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

travels on [a,b]

Find b

a

dttv

You think…

When you see…

Find the average

velocity of a particle

on [a, b]

Find the average rate of change on

[a,b]

Find

v t a

b

dt

b a

s b s a b a

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

speeding 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)

s t v t dt C

Plug in t = 0 to find C

You think…

When you see…

Show that Rolle’s

Theorem holds on [a, b]

Show that Rolle’s Theorem holds on

[a,b]

Show that f is continuous and differentiable

on the interval

If

f a f b , then find some c in

a , b

such that

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 f is continuous and differentiable

on 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)

Assume domain is

, .

Domain restrictions: non-zero denominators,

Square root of non negative numbers,

Log or ln of positive numbers

You think…

When you see…

Find the range

of f(x) on [a, b]

Find the range of f(x) on [a,b]

Use max/min techniques to find relative

max/mins.

Then examine

f a , f b

You think…

When you see…

Find the range

of f(x) on ( , )

Use max/min techniques to find relative

max/mins.

Then examine

limx

f x .

Find the range of f(x) on ,

You think…

When you see…

Find f ’(x) by definition

Find f ‘( 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=a

If x=a and y=b on the inverse, then

find the derivative of f(x) at y=b and

find its reciprocal

You think…

When you see…

y is increasing

proportionally to y

.y is increasing proportionally to y

kydt

dy

translating to

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

divided equally

dxxfdxxf

b

c

c

a

You think…

When you see…

dttfdx

dx

a

Fundamental Theorem

2nd FTC: Answer is xf

You think…

When you see…

dtufdx

du

a

Fundamental 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

...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)

Two relationships are true.

The two functions share the same

slope ( xfm )

and share the same y value at 1x.

You think…

When you see…

Find area using left

Riemann sums

Area using left Riemann sums

1210 ... nxxxxbaseA

You think…

When you see…

Find area using right

Riemann sums

Area using right Riemann sums

nxxxxbaseA ...321

You think…

When you see…

Find area using

midpoint rectangles

Area 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 trapezoids

nn xxxxxbase

A 1210 2...222

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...

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

dttf

x

a

Meaning 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…

Find where the tangent

line to f(x) is horizontal

Horizontal 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)

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 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

to f(x) at x = 0

Find the equation of the tangent line to f

using 11 xxmyy

where 0fm and the point is 0,0 f .

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 F(a) and the that the

anti-derivative of f is F, find F(b)

Usually, this problem contains an antiderivative

you cannot take. Utilize the fact that if xF

is the antiderivative of f,

then aFbFdxxF

b

a

.

Solve for bF using the calculator

to find the definite integral

You think…

When you see…

Find the derivative of

f(g(x))

Find the derivative of f(g(x))

xgxgf

You think…

When you see…

Given , find dxxf

b

a

dxkxf

b

a

Given area under a curve and vertical

shift, find the new area under the curve

dxkdxxfdxkxf

b

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

increasing

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 v(t) and s(0), find the greatest distance from

the origin of a particle on [a, b]

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.

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 minutes

dttEtFg

t

t

2

1

You think…

b) the rate the water

amount is changing

at m

Rate the amount of water is

changing at t = m

mEmFdttEtFdt

dm

t

You think…

c) the time when the

water is at a minimum

The 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/dx

Use the given points

Plug them into dx

dy,

drawing little lines with the

indicated 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]

dxxgxfA

b

a

,

assuming f (x) > g(x)

You think…

When you see…

Find the volume if the

region between the

curve f(x) and a given

horizontal line (y=k) is

rotated about that line

Volume generated by rotating region

between f(x) and y=k about the line y=k

creating a disk

kxfand

horizontalisrotationofaxisthethatgiven

dxkxfV

b

a

)(

))(( 2

You think…

When you see…

Find the volume if the

region between the

curves f(x) and g(x) is

rotated about a

horizontal axis of rotation

Volume generated by rotating region

between f(x) and g(x) about a given

horizontal axis of rotation creating a washer

)()(

]))(())([( 22

xgxfand

horizontalisrotationofaxisthethatgiven

dxaxisxgaxisxfV

b

a

You think…

When you see…

Find the volume of a

known cross-section cut

perpendicular to the x-

axis with a given side on

the region specified by

f(x) and g(x)

Volume of known cross-sections cut

perpendicular to the x-axis on region

between f(x) and g(x) where typically

f(x)-g(x) is the base of the figure

b

adxbasebasetheondiametersemicircle

b

adxbasebasetheonhyprtisosceles

b

adxbasebasetheonlegrtisosceles

b

adxbasebasetheonlequilatera

dxb

abasebasetheonsquares

2)(8

:,

2)(4

1:,

2)(2

1:,

2)(4

3:

2)(: