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+