Find the zeros

Find roots. Set function = 0, factor or use quadratic

equation if quadratic, graph to find zeros on calculator

Show that f (x) is even

Show that f (−x) = f ( x )

symmetric to y-axis

Show that f (x) is odd

Show that f (−x) = − f (x) OR f (x) = − f (−x)

symmetric around the origin

Show that lim f (x) exists

x-> a

Show that lim f (x )= lim f (x ); exists and are equal

x->a- x->a+

Find lim f ( x ) , calculator allowed

x->a

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

from left and right

Find lim f ( x ) , no calculator

x->a

Substitute x = a

1) limit is value if b/c, incl. 0/c=0; c cannot equal 0.

2) DNE for b/0

3) 0/0 DO MORE WORK!

a) rationalize radicals

b) simplify complex fractions

c) factor/reduce

d) know trig limits

1. lim sinx/x= 1

x-\>0 2. lim 1-cosx/x= 0 x-\>0 e) piece-wise function: check if RH = LH at break

Find lim f ( x ) , calculator allowed

x →∞

Use TABLE [ASK], find y values for large values of x,

i.e. 999999999999

Find lim f ( x ) , no calculator

x →∞

Ratios of rates of changes

1) fast/slow= DNE

2) slow/fast= 0

3) same/same= ratio of coefficients

Find horizontal asymptotes of f (x)

Find lim f ( x ) and lim f ( x )

x →∞ x → −∞

Find vertical asymptotes of f (x)

Find where lim f ( x ) = ±∞

x-\>a±

1) Factor/reduce f (x ) and set denominator = 0

2) ln x has VA at x = 0

Find domain of f (x)

Assume domain is (−∞, ∞). Restrictable domains:

denominators ≠ 0, square roots of only non-negative

numbers, log or ln of only positive numbers, real-world

constraints

Show that f (x) is continuous

Show that…

1) lim f(x) exists (limf(x)=limf(x))

x-\>a x-\>a- x-\>a+

2) f (a) exists

3) lim f ( x ) = f (a )

x→a

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

x = a.

Find derivative f ′(a ) = m

Find equation of the line tangent to f ( x ) at

( a, b )

f ′(a ) = m and use y − b = m ( x − a )

sometimes need to find b = f ( a )

Find equation of the line normal

(perpendicular) to f (x ) at ( a, b )

Same as above but m =

−1/f ′(a )

Find the average rate of change of f ( x ) on

[a, b]

Find (f (b ) − f (a ))/(b-a)

Show that there exists a c in [a, b] such that

f (c) = n

Intermediate Value Theorem (IVT)

Confirm that f ( x ) is continuous on [a, b] , then show that

f (a) ≤ n ≤ f (b) .

Find the interval where f ( x ) is increasing

Find f ′(x ) , set both numerator and denominator to zero

to find critical points, make sign chart of f ′( x ) and

determine where f ′( x ) is positive.

Find interval where the slope of f (x ) is

increasing

Find the derivative of f ′( x ) = f ′′( x ) , set both numerator

and denominator to zero to find critical points, make

sign chart of f ′′( x ) and determine where f ′′( x ) is

positive.

Find instantaneous rate of change of f (x ) at

a

Find f ′(a )

Given s (t ) (position function), find v(t )

Find v(t ) = s ′(t )

Find f ′( x ) by the limit definition

Frequently asked backwards

f ‘(x)= lim f ( x + h) − f ( x)/h or

h-\>0

f ‘(a)= lim f(x)-f(a)/x-a

x-\>a

Find the average velocity of a particle on

[a, b]

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

Find v ( k ) and a ( k ) . If signs match, the particle is

speeding up; if different signs, then the particle is

slowing down.

Given a graph of f ′( x ) , find where f (x ) is

increasing

Determine where f ′( x ) is positive (above the x-axis.)

Given a table of x and f ( x ) on selected

values between a and b, estimate f ′(c )

where c is between a and b.

Straddle c, using a value, k, greater than c and a value, h, less than c. So f ‘(c) ≈ f(k)-f(h)/ k-h

Given a graph of f ′( x ) , find where f (x ) has

a relative maximum.

Identify where f ′ ( x ) = 0 crosses the x-axis from above to below OR where f ′(x ) is discontinuous and jumps from above to below the x-axis.

Given a graph of f ′( x ) , find where f (x ) is

concave down.

Identify where f ′(x ) is decreasing.

Given a graph of f ′( x ) , find where f (x ) has

point(s) of inflection.

Identify where f ′(x ) changes from increasing to

decreasing or vice versa.

Show that a piecewise function is

differentiable

at the point a where the function rule

splits

First, be sure that the function is continuous at x = a by

evaluating each function at x = a. Then take the

derivative of each piece and show that

lim f ‘(x)=lim f ‘(x)

x->a- x->a+

Given a graph of f ( x ) and h ( x ) = f^-1(x)

find h ‘ ( a )

Find the point where a is the y-value on f ( x ) , sketch a

tangent line and estimate f ‘ ( b ) at the point, then

h ‘(a)= 1/f ‘(b)

Given the equation for f ( x ) and

h ( x ) = f ^-1( x ) , find h ‘ ( a )

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

h ‘(a) = 1/f ‘(b)

Given the equation for f ( x ) , find its

derivative algebraically.

1) know product/quotient/chain rules

2) know derivatives of basic functions

a. Power Rule: polynomials, radicals, rationals

b. e^x ; b^x

c. ln x;logb x

d. sin x;cos x; tan x

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

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.

Find the derivative of f ( g ( x ))

Chain Rule

f ′( g ( x )) ⋅ g ′( x )

Find the minimum value of a function on

[a, b]

Solve f ′ ( x ) = 0 or DNE, make a sign chart, find sign

change from negative to positive for relative minimums

and evaluate those candidates along with endpoints backi nto f (x ) and choose the smallest. NOTE: be careful to confirm that f ( x ) exists for any x-values that make f ‘ ( x ) DNE.

Find the minimum slope of a function on

[a, b]

Solve f “ ( x ) = 0 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 f ( x ) exists for any x-values that make

f “ ( x ) DNE.

Find critical values

Express f ′( x ) as a fraction and solve for numerator and

denominator each equal to zero.

Find the absolute maximum of f ( x )

Solve f ′ ( x ) = 0 or DNE, make a sign chart, find sign

change from positive to negative for relative maximums

and evaluate those candidates into f (x ) , also find

lim x->∞ f(x) and lim x->-∞; choose the largest.

Show that there exists a c in [a, b] such that

f ‘(c) = 0

Rolle’s Theorem

Confirm that f is continuous and differentiable on the

interval. Find k and j in [a, b] such that f ( k ) = f ( j ) ,

then there is some c in [k , j] such that f ′(c)= 0.

Show that there exists a c in [a, b] such that

f ‘(c) = m

Mean Value Theorem

Confirm that f is continuous and differentiable on the

interval. Find k and j in [a, b] such that

m= f(k)-f(j)/(k-j), then there is some c in [k,j] such that f ‘(c)=m

Find 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)

Find range of f (x ) on (− ∞, ∞ )

Use max/min techniques to find values at relative

max/mins. Also compare lim f (x ).

x →±∞

Find the locations of relative extrema of

f ( x ) given both f ‘ ( x ) and f “ ( x ) .

Particularly useful for relations of x and y

where finding a change in sign would be

difficult.

Second Derivative Test

Find where f ‘ ( x ) = 0 OR DNE then check the value of

f “ ( x ) there. If f “ ( x ) is positive, f ( x ) has a relative

minimum. If f “ ( x ) is negative, f ( x ) has a relative

maximum.

Find inflection points of f (x ) algebraically.

Express f ′′( x ) as a fraction and set both numerator and

denominator equal to zero. Make sign chart of f ′′( x ) to

find where f ′′( x ) changes sign. (+ to – or – to +)

NOTE: be careful to confirm that f (x ) exists for any x-

values that make f “ ( x ) DNE.

Show that the line y = mx + b is tangent to

f ( x ) at ( x1 , y1 )

Two relationships are required: same slope and point of

intersection. Check that m = f ‘ ( x1 ) and that ( x1 , y1 ) is on both f (x ) and the tangent line.

Find any horizontal tangent line(s) to f ( x )

or a relation of x and y.

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.

Find any vertical tangent line(s) to f ( x ) or a

relation of x and y.

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.

Approximate the value of f (0.1) by using

the tangent line to f at x = 0

Find the equation of the tangent line to f using

y − y1 = m( x − x1 ) where m = f ′(0 ) and the point

is (0, f (0 )) . 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 )

Find rates of change for volume problems.

Write the volume formula. Find dV/dt. Careful about product/chain rules. Watch positive (increasing measure)/negative (decreasing measure) signs for rates.

Find rates of change for Pythagorean

Theorem problems.

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

Find the average rate of change of f ( x ) on

[a, b]

f (b) − f ( a )/ b-a

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

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

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

f (x )

f ( g ( x) ) g ‘( x)

Find area using left Riemann sums

A = base[x0 + x1 + x 2 + … + x n −1]

Note: sketch a number line to visualize

Find area using right Riemann sums

A = base[x1 + x 2 + x3 + … + x n]

Note: sketch a number line to visualize

Find 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.

Note: sketch a number line to visualize

Find area using trapezoids

Describe how you can tell if rectangle or

trapezoid approximations over- or under-

estimate area.

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.

Given dy/dx, draw a slope field

Use the given points and plug them into dy/dx, drawing little lines with the indicated slopes at the points.

y is increasing proportionally to y

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.

Find the volume given a base bounded by

f ( x ) and g ( x ) with f (x ) > g ( x ) and

cross sections perpendicular to the x-axis are

squares

Given the value of F (a ) and F ‘ ( x ) = f ( x ) ,

find F (b )

The accumulation function: net (total if f ( x ) is positive)

amount of y-units for the function f ( x ) beginning at

x = a and ending at x = b.

Given v(t ) and s (0 ) , find the greatest

distance

from the origin of a particle on [a, b]

Solve v ( t ) = 0 OR DNE . Then integrate v(t ) adding

s (0 ) to find s (t ) . Finally, compare s(each candidate) and

s(each endpoint). Choose greatest distance (it might be

negative!)

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 [0, b] , find

a) the amount of water in the tank at m

minutes

b) the rate the water amount is changing

at m

c) the time when the water is at a

minimum

Find the volume of the area between f ( x ) and g ( x ) with f ( x ) \> g ( x ) , rotated about

the x-axis.

Given v(t ) and s (0 ) , find s (t )

Find the line x = c that divides the area under f ( x ) on [a, b] to two equal areas

Find the volume given a base bounded by

f ( x ) and g ( x ) with f (x ) > g ( x ) and

cross sections perpendicular to the x-axis are

semi-circles