When you see…

Find the zeros

You think…

To find the zeros...To find the zeros...

Find roots. Set function = 0 Factor or use quadratic equation if quadratic. Graph to find zeros on calculator.

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)

OR -f(-x) = f(x)

origin symmetry

You think…

When you see…

Show that exists lim ( )x a

f x

Show existsShow exists lim ( )x a

f x

Show that

xfxfaxax

limlim

You think…

When you see…

Find xfax

lim , calculator allowed

Find Limit with calculator

.

Use TABLE [ASK], find y-values for x-values

close to a from left and right

You think…

When you see…

Find xfax

lim , no calculator

Find limits, no calculatorSubstitute x = a

1) limit is value if b

c , incl. 0

0; 0cc

2) DNE for 0

b

3) 0

0 DO MORE WORK!

a) rationalize radicals b) simplify complex fractions c) factor/reduce d) known trig limits

1. 0

sinlim 1x

x

x

2. 0

1 coslim 0x

x

x

e) piece-wise fcn: check if RH = LH at break .

You think…

When you see…

Find limx

f x ,

calculator allowed

Find limit with calculator

.

Use TABLE [ASK], find y-values for large values

of x, i.e. 999999999999

You think…

When you see…

Find limx

f x ,

no calculator

Find limit, no calculator

Ratios of rates of changes

1)fast

DNEslow

2) 0slow

fast

3) same

ratiosame

of coefficients .

You think…

When you see…

Find horizontal asymptotes of f(x)

Find horizontal asymptotes of f(x)Find horizontal asymptotes of f(x)

Find

xfx lim

and

xf

x lim

You think…

When you see…

Find vertical

asymptotes of f(x)

Find vertical asymptotes of f(x)Find vertical asymptotes of f(x)

Find where limx a

f x

1) Factor/reduce xf and set denominator = 0

2) ln x has VA at x = 0

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

, . Domain restrictions: non-zero denominators, Square root of non-negative numbers, Log or ln of only positive numbers Real-world constraints

You think…

When you see…

Show that f(x) is continuous

..f(x) is continuousf(x) is continuous

Show that

1) xfax

lim exists

2) af exists

3) afxfax

lim

You think…

When you see…

Find the slope of the tangent line to f(x) at x = a

Slope of tangent line

. Find derivative maf

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

maf and use y b m x a

sometimes need to find b f a

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

Same as previous but

afm

1

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…

Show that there exists a c in ba, such that f c n

Intermediate Value Theorem (IVT)

Confirm that xf is continuous on ba, , then show that

( ) ( )f a n f b .

You think…

When you see…

Find the interval where f(x) is increasing

ff(x) increasing(x) increasing

Find xf , set both numerator and denominator to zero to find critical points, make sign chart of

xf and determine where xf is positive. Closed interval, unless f(x) DNE.

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 f “ (x) is positive

Also, think f(x) is CONCAVE UP

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…

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…

Find f ’(x) by the limit definition

Find f Find f ‘‘( x) by definition( x) by definitionfrequently asked backwardsfrequently asked backwards

ax

afxfaf

h

xfhxfxf

ax

h

lim

or lim0

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]

Find

v ta

b

dt

b as b s a b a

Depending on if you know s(t) or v(t)

You think…

When you see…

Given v(t), determine if a particle is speeding up at

t = k

Given v(t), determine if the particle is Given v(t), determine if the particle is speeding up at t = kspeeding up at t = k

Find v (k) and a (k). If signs match, the particle is speeding up. If signs opposite, the particle is slowing down

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

Determine where xf is positive (above x-axis)

You think…

When you see…

Given a table of x and xf on selected values between a and b, estimate cf

where c is between a and b.

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 a graph of xf , find where xf has a relative maximum

Relative maximum

. Identify where 0f x crosses the x-axis from above to below OR where xf is discontinuous and jumps from above to below the

x-axis

You think…

When you see…Given a graph of xf ,

find where xf is concave down

Concave Down

. Identify where xf is decreasing

You think…

When you see…Given a graph of xf ,

find where xf has point(s) of inflection

Points of Inflection

. Identify where xf changes from increasing to decreasing or vice

versa

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 at

ax by evaluating a in both parts

Take the derivative of each piece and show that

xfxfaxax

limlim

You think…

When you see…

Given a graph of xf and 1h x f x , find 'h a

Derivative of Inverse Function

.

Find the point where a is the y-value on xf , sketch a tangent

line and estimate 'f b at the point,

then

1'

'h a

f b

You think…

When you see…

Given the equation for xf and 1h x f x , find

'h a

Derivative of the inverse of f(x)Derivative of the inverse of f(x)

Understand that the point ,a b is on h x so the point ,b a is on f x . So find b where f b a

1

''

h af b

You think…

When you see…Given the equation

for xf , find its derivative algebraically

Derivative Rules. 1) know product/quotient/chain rules 2) know derivatives of basic functions

a. Power Rule: polynomials, radicals, rationals

b. ;x xe b

c. ln ; logbx x

d. sin ;cos ; tanx x x

e. 1arcsin ;arccos ;arctan ;sin ;x x x x etc

You think…

When you see…Given a relation of x and

y, find dy

dx algebraically

Implicit Differentiation

Find the derivative of each term, using product/quotient/chain appropriately,

especially, chain rule: every derivative of y

is multiplied by dy

dx ; then group all dy

dx terms

on one side; factor out dy

dx and solve.

You think…

When you see…

Find the derivative off(g(x))

Chain RuleChain Rule

xgxgf

You think…

When you see…

Find the minimum value of a function

Minimum value of a functionMinimum value of a function

Solve 0f x or DNE, make a sign chart, find sign change from negative to positive for relative minimums and evaluate those candidates along with endpoints back into

xf and choose the smallest. NOTE: be careful to confirm that xf exists for any x-values that make 'f x

DNE.

You think…

When you see…

Find the minimum slope of a function

Minimum slope of a functionMinimum slope of a function

Solve " 0f x or DNE, make a sign chart, find sign change from negative to positive for

relative minimums and evaluate those candidates along with endpoints back into

'f x and choose the smallest. NOTE: be careful to confirm that xf exists

for any x-values that make "f x DNE.

You think…

When you see…

Find critical numbers

Find critical numbersFind critical numbers

Express xf as a fraction and solve for

numerator and denominator each equal to zero.

You think…

When you see…

Find the absolute maximum of f(x)

Find the absolute minimum of f(x)Find the absolute minimum of f(x)

Solve 0f x or DNE, make a sign chart, find sign change from positive to

negative for relative maximums and evaluate those candidates into xf , also

find xfx lim and xf

x lim ; choose the

largest.

You think…

When you see…

Show that there exists a c in ba, such that ' 0f c

Confirm that f is continuous and differentiable on the interval. Find k and j in ba, such that f k f j , then

there is some c in ,k j such that

f c 0.

Rolle’s Theorem

You think…

When you see…

Show that there exists a c in ba, such that 'f c m

Mean Value Theorem

Confirm that f is continuous and differentiable on the interval. Find k

and j in ba, such that

.f k f j

mk j

,

then there is some c in ,k j such that .f c m

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 values at relative max/mins. Also compare f a and f b

(endpoints)

You think…

When you see…

Find the range

of f(x) on ( , )

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 ,

You think…

When you see…Find the locations of relative extrema of

xf given both 'f x and "f x .

Particularly useful for relations of x and y where a sign chart would be difficult.

Second Derivative Test

.

Find where ' 0 OR DNEf x then check the

value of "f x there. If "f x is positive, xf

has a relative minimum. If "f x is negative, xf has a relative minimum.

You think…

When you see…

Find inflection points of xf algebraically.

Find inflection pointsFind inflection points

Express xf as a fraction and set both numerator and denominator equal to zero. Make sign chart of xf to find where

xf changes sign. (+ to – or – to +) NOTE: be careful to confirm that xf exists for any x-values that make "f x DNE.

You think…

When you see…

Show that the line bmxy is tangent to

xf at 1 1,x y

.y = mx+b is tangent to f(x) at a pointy = mx+b is tangent to f(x) at a point

Two relationships are required: same slope and point of intersection. Check that

1'm f x and that 1 1,x y is on both xf and the tangent line.

You think…

When you see…Find any horizontal

tangent line(s) to xf or a relation of x and y.

Horizontal tangent lineHorizontal tangent line

Write dy

dx as a fraction. Set the numerator equal to zero. NOTE: be careful to confirm that any values are on the curve. Equation of tangent line is y = b. May have to find b.

You think…

When you see…Find any vertical tangent

line(s) to xf or a relation of x and y.

Vertical tangent line to f(x)Vertical tangent line to f(x)

Write

dy

dx as a fraction. Set the denominator equal to zero.

NOTE: be careful to confirm that any values are on the curve.

Equation of tangent line is x = a. May have to find a.

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 where 0fm and the point is 0,0 f . Then plug in 0.1 into this

line; be sure to use an approximate sign.

Alternative linearization formula: 'y f a x a f a

You think…

When you see…

Find rates of change for volume problems

Rates of Change of Volumes

.

Write the volume formula. Find dV

dt . Careful about product/ chain rules. Watch

positive (increasing measure)/negative (decreasing measure) signs for rates.

You think…

When you see…Find rates of change for Pythagorean Theorem

problems

Pythagorean Rates of Change

.

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

; can reduce 2’s

Watch positive (increasing distance)/negative (decreasing distance) signs for rates.

You think…

When you see…

Find the average valueof xf on ba,

Average value of the functionAverage value of the function

Find 1

b

a

f x dxb a

You think…

When you see…

Find the average rate of change of xf on ba,

Average Rate of Change

.

f b f a

b a

You think…

When you see…

Given v(t), find the total distance a particle travels on [a, b]

Given v(t), find the total distance a Given v(t), find the total distance a particle travels on [a,b]particle travels on [a,b]

Find b

a

dttv

You think…

When you see…

Given v(t), find the change in position of a particle on [a, b]

Given v(t), find the change in position Given v(t), find the change in position of a particle on [a,b]of a particle on [a,b]

Find b

a

v t dt

You think…

When you see…

Given tv and initial position of a particle, find the position at t = a.

Given Given v(t)v(t) and the initial position of a and the initial position of a particle, find the position at particle, find the position at tt = = aa..

Find

0

0a

v t dt s

Read carefully: “starts at rest at the origin” means 0 0s and 0 0v

You think…

When you see…

dttfdx

d x

a

Fundamental TheoremFundamental Theorem

xf

You think…

When you see…

( )

g x

a

df t dt

dx

Fundamental Theorem, againFundamental Theorem, againChain Rule, tooChain Rule, too

( ) '( )f g x g x

You think…

When you see…

Find area using left Riemann sums

Area using left Riemann sumsArea using left Riemann sums

1210 ... nxxxxbaseANote: sketch a number line to visualize

You think…

When you see…

Find area using right Riemann sums

Area using right Riemann sumsArea using right Riemann sums

1 2 3 ... nA base x x x x

Note: sketch a number line to visualize

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. Note: sketch a number line to visualize

You think…

When you see…

Find area using trapezoids

Area using trapezoidsArea using trapezoids

nn xxxxxbase

A 1210 2...222

This formula only works when the base (width) is the same. Also trapezoid area is the average of LH and RH. If different widths, you have to do individual

trapezoids, 1 2

1

2A h b b

You think…

When you see…Describe how you can tell if rectangle or trapezoid approximations over- or under-

estimate area.

Over- or Under-estimates

Overestimate area: LH for decreasing; RH for increasing; and trapezoids for concave up

Underestimate area: LH for increasing; RH for decreasing and trapezoids for concave down

DRAW A PICTURE with 2 shapes.

.

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

dxkdxxfdxkxfb

a

b

a

b

a

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 dx

dy,

drawing little lines with theindicated slopes at the points.

You think…

When you see…

y is increasing proportionally to y

.y is increasing proportionally to yy is increasing proportionally to y

kydt

dy

integrating to

kty Ae

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. Integrate each side, add C. Find C before solving for y,[unless ln y , then solve for y first and find A]. When solving for y, choose + or – (not both), solution will be a continuous function passing through the initial value.

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 distance between the curves is the base of your square.

So the volume is 2b

a

f x g x dx

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 antiderivative you cannot do. Utilize the fact that if xF is the antiderivative of f,

then b

a

f x dx F b F a .

Solve for bF using the calculator to find the definite integral, then

( )b

a

F b f x dx F a

You think…

When you see…

Meaning of

b

a

f t dt

Meaning of the integral of f(t) from a to xMeaning of the integral of f(t) from a to x

The accumulation function –

Net (positive f(x) => total) amount of y –units for function xf

from x = a to x = b.

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]]

Solve 0 OR DNEv t . Then integrate tv adding 0s to find ts . Finally, compare s(each candidate) and s(each endpoint). Choose greatest distance (it might be negative!)

When you see…

Given a water tank with g gallons initially being filled at

the rate of tF gallons/min and

emptied at the rate of tE gallons/min on 0,b , find

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

0

m

g F t E t dt

You think…

b) the rate the water

amount is changing

at m

Rate the amount of water is changing at t = m

0

mdF t E t dt F m E m

dt

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

Solve 0F t E t to find candidates, evaluate

candidates and endpoints as x a in

0

a

g F t E t dt , choose the minimum value

You think…

When you see…

Find the area between xf and g x with xf > g x on ba,

Area between f(x) and g(x) on [a,b]Area between f(x) and g(x) on [a,b]

dxxgxfAb

a

You think…

When you see…

Find the volume of the area between xf and g x with f x g x , 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

2 2b

a

A f x g x dx

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)

0

0t

s t v x dx s

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

1

2

c b

a a

f x dx f x dx

OR

You think…

When you see…

Find the volume given a base bounded by xf and g x with xf > g x and cross

sections perpendicular to the x-axis are semi-circles

Semi-circular cross sections Semi-circular cross sections perpendicular to the x-axisperpendicular to the x-axis

The distance between the curves is the diameter of your circle. So the volume

is 2

1

2 2

b

a

f x g xdx