When you see…
Find the zeros
You think…
To find the zeros...
Find roots. Set function = 0
Factor or use quadratic equation if
quadratic.
Graph to find zeros on calculator.
When you see…
Show that f(x) is even
You think…
Even function
f (-x) = f ( x)
y-axis symmetry
When you see…
Show that f(x) is odd
You think…
Odd function
.
f (-x) = -f (x)
OR
-f(-x) = f(x)
origin symmetry
When you see…
Show that lim f ( x ) exists
x a
You think…
Show lim f ( x ) exists
x a
lim
xa
Show that
f  x   lim f  x 
xa
When you see…
f x 
Find lim
, calculator allowed
x a
You think…
Find Limit with calculator
.
Use TABLE [ASK],
find y-values for x-values
close to a
from left and right
When you see…
Find
lim f  x 
x a
, no calculator
You think…
Find limits, no calculator
Substitute x = a
b
0
1) limit is value if c , incl. c  0; c  0
b
2) DNE for 0
0
3) 0 DO MORE WORK!
a) rationalize radicals
b) simplify complex fractions
c) factor/reduce
d) known trig limits
sin x
lim
1
1. x 0 x
1  cos x
0
x 0
x
lim
2.
e) piece-wise fcn: check if RH = LH at break
.
When you see…
lim f  x 
Find
,
calculator allowed
x 
You think…
Find limit with calculator
.
Use TABLE [ASK],
find y-values for large values
of x, i.e. 999999999999
When you see…
Find lim f  x  ,
no calculator
x 
You think…
Find limit, no calculator
Ratios of rates of changes
fast
 DNE
1) slow
2)
3)
.
slow
0
fast
same
 ratio
of
same
coefficients
When you see…
Find horizontal
asymptotes of f(x)
You think…
Find horizontal asymptotes of f(x)
Find
lim f  x 
x
and
lim f  x 
x  
When you see…
Find vertical
asymptotes of f(x)
You think…
Find vertical asymptotes of f(x)
lim f  x   
Find where xa
1) Factor/reduce f x  and set
denominator = 0
2) ln x has VA at x = 0
When you see…
Find the domain
of f(x)
You think…
Find the domain of f(x)
Assume domain is , .
Domain restrictions: non-zero denominators,
 of non-negative numbers,
Square root
Log or ln of only positive numbers
Real-world constraints
When you see…
Show that f(x) is
continuous
You think…
.
f(x) is continuous
Show that
1)
lim f  x  exists
x a
2)
f a  exists
3)
lim f x   f a 
xa
When you see…
Find the slope of the
tangent line to f(x) at x = a
You think…
Slope of tangent line
.
Find derivative f a   m
When you see…
Find equation of the line
tangent to f(x) at (a, b)
You think…
Equation of the tangent line
f a   m and use
y  b  m x  a
b

f
a


sometimes need to find
When you see…
Find equation of the line
normal to f(x) at (a, b)
You think…
Equation of the normal line
Same as previous but
1
m
f a 
When you see…
Find the average rate of
change of f(x) at [a, b]
You think…
Average rate of change of f(x)
Find
f (b) - f ( a)
b- a
When you see…
Show that there exists a c in
a, b such that f  c   n
You think…
Intermediate Value Theorem (IVT)
Confirm that f x  is continuous
on a, b, then show that
f (a)  n  f (b)
.
When you see…
Find the interval where
f(x) is increasing
You think…
f(x) 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. Closed interval, unless
f(x) DNE.
When you see…
Find the interval where the
slope of f (x) is increasing
You think…
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 f “ (x) is positive
Also, think f(x) is CONCAVE UP
When you see…
Find the instantaneous
rate of change of f(x)
on [a, b]
You think…
Instantaneous rate of change of f(x)
Find f ‘ ( a)
When you see…
Given s(t) (position
function), find v(t)
You think…
Given position s(t), find v(t)
Find v t   s  t 
When you see…
Find f ’(x) by the limit
definition
You think…
Find f ‘( x) by definition
frequently asked backwards
f x  h   f x 
f  x   lim
or
h 0
h
f  x   f a 
f a   lim
xa
xa
When you see…
Find the average
velocity of a particle
on [a, b]
You think…
Find the average rate of change on
[a,b]
b
 vt dt
a


sb sa
Find b  a
b a
Depending on if you know
s(t) or v(t)
When you see…
Given v(t), determine if a
particle is speeding up at
t=k
You think…
Given v(t), determine if the particle is
speeding 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
When you see…
Given a graph of f '( x )
find where f(x) is
increasing
You think…
Given a graph of f ‘(x) , find where f(x) is
increasing
Make a sign chart of f x 
Determine where f x  is positive
(above x-axis)
When you see…
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.
You think…
Straddle c, using a value, k, greater
than c and a value, h, less than c.
so
f k   f h 
f c  
k h
When you see…
Given a graph of f x ,
find where f x has a
relative maximum
You think…
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
When you see…
Given a graph of f x  ,
find where f x  is
concave down
You think…
Concave Down
.
Identify where
f x 
is decreasing
When you see…
Given a graph of f x  ,
find where f x  has
point(s) of inflection
You think…
Points of Inflection
Identify where f x changes from
increasing to decreasing or vice
versa
.
When you see…
Show that a piecewise
function is differentiable
at the point a where the
function rule splits
You think…
Show a piecewise function is
differentiable at x=a
First, be sure that the function is continuous at
x  a by evaluating a in both parts
Take the derivative of each piece and show that
lim f x   lim f x 
x a
xa 
When you see…
Given a graph of f x 
and h  x   f  x  , find h  a 
1
You think…
'
Derivative of Inverse Function
.
Find the point where a is the yvalue on f x  , sketch a tangent
line and estimate f ' b  at the point,
1
h 'a 
f ' b
then
When you see…
Given the equation for
f x  and h  x   f  x  , find
1
h a
'
You think…
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 'a 
f ' b
When you see…
Given the equation
for f x , find its derivative
algebraically
You think…
Derivative Rules
.
1) know product/quotient/chain rules
2) know derivatives of basic functions
a. Power Rule: polynomials, radicals, rationals
x
x
e
;
b
b.
c. ln x;log b x
d.
sin x;cos x; tan x
e. arcsin x;arccos x;arctan x;sin
1
x; etc
When you see…
Given a relation of x and
y, find
dy
dx
algebraically
You think…
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
dy
on one side; factor out dx and solve.
When you see…
Find the derivative of
f(g(x))
You think…
Chain Rule
f  g  x   g  x 
When you see…
Find the minimum value
of a function
You think…
Minimum value of a function
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.
When you see…
Find the minimum slope
of a function
You think…
Minimum slope of a function
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.
When you see…
Find critical numbers
You think…
Find critical numbers
Express f x  as a fraction and solve for
numerator and denominator each equal to
zero.
When you see…
Find the absolute
maximum of f(x)
You think…
Find the absolute minimum 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 f x  and lim f x  ; choose the
largest.
x
x  
When you see…
Show that there exists a c
f
'
c

0




a,
b
in
such that
You think…
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.

When you see…
Show that there exists a
c in a, b such that f '  c   m
You think…
Mean Value Theorem
Confirm that f is continuous and
differentiable on the interval. Find k
f k   f  j 
and j in a, b such that m  k  j . ,
then there is some c in k , j  such that
f   c   m.
When you see…
Find the range
of f(x) on [a, b]
You think…
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)
When you see…
Find the range
of f(x) on ( , )
You think…
Find the range of f(x) on    ,  
Use max/min techniques to find relative
max/mins.
f
Then examine xlim
 

x  .
When you see…
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 a sign chart would be
difficult.
You think…
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 minimum.
When you see…
Find inflection points of
f x  algebraically.
You think…
Find inflection points
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.
When you see…
Show that the line
y  mx  b is tangent to
f x  at  x , y 
1
1
You think…
.
y = mx+b is tangent to f(x) at a point
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.
When you see…
Find any horizontal
tangent line(s) to f x or
a relation of x and y.
You think…
Horizontal tangent line
dy
Write 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.
When you see…
Find any vertical tangent
line(s) to f x  or a
relation of x and y.
You think…
Vertical tangent line to f(x)
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.
Write
When you see…
Approximate the value
f(0.1) of by using the
tangent line to f at x = 0
You think…
Approximate f(0.1) using tangent line
to f(x) 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 
When you see…
Find rates of change for
volume problems
You think…
Rates of Change of Volumes
dV
dt
.
Write the volume formula. Find
.
Careful about product/ chain rules. Watch
positive (increasing measure)/negative
(decreasing measure) signs for rates.
When you see…
Find rates of change for
Pythagorean Theorem
problems
You think…
Pythagorean Rates of Change
x y z
2
.
2
dx
dy
dz
2x  2 y
 2z
dt
dt
dt
2
; can reduce 2’s
Watch positive (increasing distance)/negative
(decreasing distance) signs for rates.
When you see…
Find the average value
of f  x  on  a , b 
You think…
Average value of the function
b
Find
1
f
x
dx



ba a
When you see…
Find the average rate of
change of f x  on a, b
You think…
Average Rate of Change
.
f b  f  a 
ba
When you see…
Given v(t), find the total distance
a particle travels on [a, b]
You think…
Given v(t), find the total distance a
particle travels on [a,b]
b
v  t  dt

Find
a
When you see…
Given v(t), find the change in
position of a particle on [a, b]
You think…
Given v(t), find the change in position
of a particle on [a,b]
b
Find
 v  t  dt
a
When you see…
Given vt  and initial position of a
particle, find the position at t = a.
You think…
Given v(t) and the initial position of a
particle, find the position at t = a.
a
0
s

dt
t
v





Find 0
Read carefully: “starts at rest at
the origin” means s  0  0 and
v  0  0
When you see…
x
d
f t dt 

dx a
You think…
Fundamental Theorem
f x 
When you see…
d
dx
g ( x)
 f t  dt 
a
You think…
Fundamental Theorem, again
Chain Rule, too
f
 g ( x)  g '( x)
When you see…
Find area using left
Riemann sums
You think…
Area using left Riemann sums
A  basex0  x1  x2  ...  xn1 
Note: sketch a number line to visualize
When you see…
Find area using right
Riemann sums
You think…
Area using right Riemann sums
A  base  x1  x2  x3  ...  xn 
Note: sketch a number line to visualize
When you see…
Find area using
midpoint rectangles
You think…
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
When you see…
Find area using
trapezoids
You think…
Area using trapezoids
base
x0  2 x1  2 x2  ...  2 xn1  xn 
A
2
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
A  h  b1  b2 
2
When you see…
Describe how you can tell if rectangle or
trapezoid approximations over- or underestimate area.
You think…
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.
.
When you see…
b
b
a
a
Given  f x dx , find  f x  k dx
You think…
Given area under a curve and vertical
shift, find the new area under the curve
b
b
b
a
a
a






f
x

k
dx

f
x
dx

k
dx



When you see…
dy
Given dx , draw a
slope field
You think…
Draw a slope field of dy/dx
Use the given points
dy
dx
Plug them into
,
drawing little lines with the
indicated slopes at the points.
When you see…
y is increasing
proportionally to y
You think…
.
y is increasing proportionally to y
dy
 ky
dt
integrating to
y  Ae
kt
When you see…
Solve the differential
equation …
You think…
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.
When you see…
Given a base, cross
sections perpendicular to
the x-axis that are
squares
You think…
Semi-circular cross sections
perpendicular to the x-axis
The distance between the curves is the
base of your square.
b
So the volume is
  f  x   g  x   dx
a
2
When you see…
Given the value of F(a)
and the fact that the
anti-derivative of f is F,
find F(b)
You think…
Given F(a) and the that the
anti-derivative of f is F, find F(b)
Usually, this problem contains an antiderivative
you cannot do. Utilize the fact that if F x 
is the antiderivative of f,
b
 f  x dx  F  b   F  a  .
Solve for F b using the calculator
then
a
to find the definite integral, then
b
F  b    f  x dx  F (a )
a
When you see…
Meaning of
b
 f  t  dt
a
You think…
Meaning of the integral of f(t) from a to x
The accumulation function –
Net (positive f(x) => total) amount of
y –units for function f x 
from x = a to x = b.
When you see…
Given v(t) and s(0), find the
greatest distance from the
origin of a particle on [a, b]
You think…
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!)
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 0,b , find
a) the amount of water in
the tank at m minutes
You think…
Amount of water in the tank at t minutes
m
g    F  t   E  t  dt
0
b) the rate the water
amount is changing
at m
You think…
Rate the amount of water is
changing at t = m
m
d
F  t   E  t  dt  F  m   E  m 


dt 0
c) the time when the
water is at a minimum
You think…
The time when the water is at a minimum
Solve F t   E t   0 to find
candidates, evaluate
candidates and endpoints as
x  a in g    F t   E t dt , choose
the minimum value
a
0
When you see…
Find the area between f x  and
g  x  with f x  > g  x  on a, b
You think…
Area between f(x) and g(x) on [a,b]
b
A
  f x   g x dx
a
When you see…
Find the volume of the area between f x 
and g  x  with f  x   g  x  , rotated about the
x-axis.
You think…
Volume generated by rotating area between
f(x) and g(x) about the x-axis
b
2
2

A     f  x     g  x   dx


a
When you see…
Given v(t) and s(0),
find s(t)
You think…
Given v(t) and s(0), find s(t)
t
s  t    v  x  dx  s  0 
0
When you see…
Find the line x = c that
divides the area under
f(x) on [a, b] into two
equal areas
You think…
Find the x=c so the area under f(x) is
divided equally
c

a
b
1
f  x dx   f  x dx
2a
OR
c

a
f  x dx 
b

c
f  x dx
When you see…
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
You think…
Semi-circular cross sections
perpendicular to the x-axis
The distance between the curves is the
diameter of your circle. So the volume
1  f  x  g  x 
 
 dx
2 a
2

b
is
2