Often Used Trig Identities

sin 2 x  cos2 x  1

1  tan2 x  sec 2 x

1  cot 2 x  csc 2 x

sin 2x  2 sin x cos x

cos 2 x  cos2 x  sin 2 x

cos 2 x  2 cos2 x  1

cos 2 x  1  2 sin 2 x

1  cos 2 x
sin 2 x 
2

1  cos 2 x
cos2 x 
2

Inverse Trig Output Restrictions

𝜋 𝜋
arcsin 𝜃 output will be in [− 2 , 2 ]

arccos 𝜃 output will be in [0, 𝜋]

𝜋 𝜋
arctan 𝜃 output will be in (− 2 , 2 )

arccot 𝜃 output will be in (0, 𝜋)

𝜋 𝜋
arcsec 𝜃 output will be in [0, 2 ) ∪ ( 2 , 𝜋]

𝜋 𝜋
arccsc 𝜃 output will be in [− 2 , 0) ∪ (0, 2 ]
 Logarithm Properties

ln 1 = 0

ln e = 1

ln 𝑎 + ln 𝑏 = ln(𝑎𝑏)
𝑎
ln 𝑎 − ln 𝑏 = ln ( )
𝑏

𝑏 ln 𝑎 = ln(𝑎𝑏 )

𝑒 ln 𝑎 = 𝑎

Limit Existence

In order for a limit to exist at x = c, lim− 𝑓(𝑥) = lim+ 𝑓(𝑥) = 𝐿

𝑥→𝑐 𝑥→𝑐
where L is a real number.

NOTE: ∞ is not a real number

Also, if the function is continuous at x = c, lim 𝑓(𝑥) = 𝐿 = 𝑓(𝑐).

𝑥→𝑐
L can be easily found by substituting x = c into the function in this case.

Special Limits Involving Trig Functions

sin 𝑎𝑥 𝑎𝑥
lim 𝑎𝑥
=1 lim =1
𝑥→0 𝑥→0 sin 𝑎𝑥

1−cos 𝑥 cos 𝑥−1

lim 𝑥
=0 lim 𝑥
=0
𝑥→0 𝑥→0
 Vertical Asymptotes

If lim− 𝑓(𝑥) = ±∞ or lim+ 𝑓(𝑥) = ±∞, then x = c is a vertical asymptote.

𝑥→𝑐 𝑥→𝑐

NOTE: There can be an infinite number of vertical asymptotes.

Horizontal Asymptotes

If lim 𝑓(𝑥) = 𝑑, then y = d is a horizontal asymptote.

𝑥→±∞

NOTE: Horizontal asymptotes only exist at the ends of the graph. Therefore, the max number of
horizontal asymptotes is 2. There can be 2, 1, or 0 horizontal asymptotes.

NOTE: Determining the limits above for ∞ and –∞ also tell you the end behavior on each side in
addition to determining the horizontal asymptotes.

Constant Over Zero or ∞

When evaluating a limit:

constant
= ±∞ (be careful of sign!)
approaching 0

constant
=0
approaching ± ∞

Limits at Infinity: RATIONAL FUNCTIONS

polynomial
For finding limits as 𝑥 → ±∞ for a function of the form:
polynomial

If Degree Limit Equals Horizontal Asymptote

N<D lim 𝑓(𝑥) = 0 y=0
𝑥→±∞

N=D lim 𝑓(𝑥) = ratio of leading coefficients y = ratio of leading coeff.

𝑥→±∞

N>D lim 𝑓(𝑥) = ±∞ No horizontal asymptote

𝑥→±∞
 Continuity

lim 𝑓(𝑥) = 𝑓(𝑐) if and only if f(x) is continuous at x = c.

𝑥→𝑐

NOTE: To check for continuity, you must find 𝑓(𝑐) and lim 𝑓(𝑥).
𝑥→𝑐

 If f(c) is undefined, the function is not continuous at x = c.

 To find lim 𝑓(𝑥), you may have to find the left and right limits
𝑥→𝑐
and see if they match. You only have to find the left and right
limits if you can’t substitute or massage the limit.

Intermediate Value Theorem

If a function is continuous on [a, b], then it takes on every value between f(a) and f(b).
 Definitions of the Derivative

Definition of the Derivative: Derivative at any point.

Answer will usually contain a variable.

𝑓(𝑥 + ℎ) − 𝑓(𝑥)
lim
ℎ→0 ℎ

Definition of Derivative at a Point (version 1): Derivative for specific x-value, x = a.

Answer is a number.

𝑓(𝑥) − 𝑓(𝑎)
lim
𝑥→𝑎 𝑥−𝑎

Definition of Derivative at a Point (version 2): Derivative for specific x-value, x = a.

Answer is a number.

𝑓(𝑎 + ℎ) − 𝑓(𝑎)
lim
ℎ→0 ℎ

REMEMBER! The derivative is the slope of the tangent line at that point on the graph.

Tangent Line & Normal Line

Tangent Line
 The slope of the tangent line is the derivative of the function.

 To find the slope of the tangent line at a particular point, find the
derivative and then substitute the x-value into the result.

 To write the equation of the tangent line at a particular point, find the
slope and then use point-slope form to write the equation of the line.

Normal Line
 The normal line at a point is perpendicular to the tangent line at that
point.

 To find the slope of the normal line, first find the slope of the tangent
line. Then change the sign and take the reciprocal to determine the
slope of the normal line.
 Differentiability
A function f(x) is differentiable at x = c if the derivative from the left of x = c is
equal to the derivative from the right of x = c. That is:

𝑓(𝑥) − 𝑓(𝑐) 𝑓(𝑥) − 𝑓(𝑐)

lim− = lim+
𝑥→𝑐 𝑥−𝑐 𝑥→𝑐 𝑥−𝑐

Some notes about differentiability

 You don’t have to find the derivative using limits. In most cases, you can
just take the derivative of the function.

 If the original function is undefined at a certain x-value, the function has

a discontinuity at that x-value (and is not differentiable there because
there is no point there).

 If the derivative of a function is undefined at a certain x-value, the

function is not differentiable at that x-value. The function may be
continuous at this x-value though as the original function may be defined
while the derivative is undefined.

 Piecewise functions are often not differentiable where the function

changes. For these, you will have to find the left and right derivative to
see if they match.

Relationship between differentiability & continuity

 If a function is differentiable, then the function is continuous.

 If a function is not continuous at a point, it is not differentiable at that

point.

 The converse is not necessarily true: A continuous function is not always

differentiable everywhere.

Horizontal & Vertical Tangents

Horizontal tangents occur where the numerator of the derivative equals zero.

Vertical tangents occur where the denominator of the derivative equals zero.
 Critical Values & Relative Extrema
Definition of Critical Value
Let f be defined at c. If f ′(c) = 0 or f ′(c) is undefined, then c is called a critical
value of f.

1st Derivative Test

At a critical point c:
1. If f ′ changes sign from positive to negative at c, then f has a relative
maximum value at c.
2. If f ′changes sign from negative to positive at c, then f has a relative
minimum value at c.
3. If f ′ does not change sign at c, then f has no relative extreme value at c.

At an endpoint on a closed interval [a, b]:

1. Left endpoint:
 If f ′ < 0 for x > a, then f has a relative maximum at x = a. In other
words, if the function decreases from the left endpoint, then the left
endpoint is a relative maximum.
 If f ′ > 0 for x > a, then f has a relative minimum at x = a. In other
words, if the function increases from the left endpoint, then the left
endpoint is a relative minimum.
2. Right endpoint:
 If f ′ > 0 for x < b, then f has a relative maximum at x = b. In other
words, if the function increases into the right endpoint, then the right
endpoint is a relative maximum.
 If f ′ < 0 for x < b, then f has a relative minimum at x = b. In other
words, if the function decreases into the right endpoint, then the right
endpoint is a relative minimum.

Intervals of Increase & Decrease

A function f is increasing when f ′ > 0 (positive).

A function f is decreasing when f ′ < 0 (negative).

To find intervals of increase and decrease, find all points of discontinuity and
critical values. Then organize a sign chart.
 Concavity and Points of Inflection

If f ′′ > 0 on (a, b), then f ′ is increasing and f is concave up on (a, b).

If f ′′ < 0 on (a, b), then f ′ is decreasing and f is concave down on (a, b).

f has a point of inflection at x if f ′′ changes sign at x.

2nd Derivative Test for Extrema

If f ′(c) = 0 and f ′′(c) > 0, then f(c) is a relative minimum of f.

If f ′(c) = 0 and f ′′(c) < 0, then f(c) is a relative maximum of f.

If f ′(c) = 0 and f ′′(c) = 0, then no conclusion regarding relative extrema is

possible. You must use the 1st Derivative sign chart instead.

Chain Rule
𝑑
Version 1 (most used): 𝑑𝑥
[𝑓(𝑔(𝑥))] = 𝑓′(𝑔(𝑥)) ∙ 𝑔′(𝑥)

𝑑𝑦 𝑑𝑦 𝑑𝑢
Version 2: 𝑑𝑥
= 𝑑𝑢 ∙ 𝑑𝑥

Product Rule & Quotient Rule

𝑑
Product Rule: [𝑓(𝑥)𝑔(𝑥)] = 𝑔(𝑥)𝑓 ′ (𝑥) + 𝑓(𝑥)𝑔′(𝑥)
𝑑𝑥

𝑑 𝑓(𝑥) 𝑔(𝑥)𝑓′ (𝑥)−𝑓(𝑥)𝑔′(𝑥) 𝐿𝑜𝑤∙𝑑𝐻𝑖𝑔ℎ−𝐻𝑖𝑔ℎ∙𝑑𝐿𝑜𝑤

Quotient Rule: [
𝑑𝑥 𝑔(𝑥)
] = [𝑔(𝑥)]2
= 𝐿𝑜𝑤∙𝐿𝑜𝑤

NOTE: The chain rule must often be used to take the derivatives in the product & quotient rule.
Derivative Rules (Version 1 – Don’t Glue Yet!)
Note: These include the chain rule:

Power Rule: Trig Derivatives:

𝑑 𝑛 𝑑
[𝑢 ] = 𝑢′ ∙ 𝑛𝑢𝑛−1 [sin 𝑢] = 𝑢′ ∙ cos 𝑢
𝑑𝑥 𝑑𝑥
𝑑
Constant Rule: [cos 𝑢] = 𝑢′ ∙ (−sin 𝑢)
𝑑𝑥
𝑑 𝑑
[𝑐𝑜𝑠𝑡𝑎𝑛𝑡] = 0 [tan 𝑢] = 𝑢′ ∙ sec 2 𝑢
𝑑𝑥 𝑑𝑥
Scalar Rule: 𝑑
[cot 𝑢] = 𝑢′ ∙ (− csc 2 𝑢)
𝑑 𝑑 𝑑𝑥
[𝑎 ∙ 𝑓(𝑥)] = 𝑎 ∙ [𝑓(𝑥)] 𝑑
𝑑𝑥 𝑑𝑥 [sec 𝑢] = 𝑢′ ∙ sec 𝑢 tan 𝑢
Where a is a constant 𝑑𝑥
𝑑
[csc 𝑢] = 𝑢′ ∙ (− csc 𝑢 cot 𝑢)
𝑑𝑥

E
 Motion

Velocity
 Velocity is the derivative of position: v(t) = s′(t)
 When velocity > 0, object is moving in a positive direction (right or up for linear
motion)
 When velocity < 0, object is moving in a negative direction (left or down for linear
motion)
 When velocity equals 0, the object is at rest.

Acceleration
 Acceleration is the derivative of velocity and the 2nd derivative of position:
a(t) = v′(t) = s′′(t)
 When acceleration > 0, the velocity of the object is increasing.
 When acceleration < 0, the velocity of the object is decreasing.

Speed & Speeding Up or Down

 Speed = |velocity|
 Speeding up when velocity and acceleration have the same sign.
 Slowing down when velocity and acceleration have opposite signs.
Derivative Rules (Version 2 – Still Don’t Glue!)
Note: These include the chain rule:

Power Rule: Trig Derivatives:

𝑑 𝑛 𝑑
[𝑢 ] = 𝑢′ ∙ 𝑛𝑢𝑛−1 [sin 𝑢] = 𝑢′ ∙ cos 𝑢
𝑑𝑥 𝑑𝑥
𝑑
Constant Rule: [cos 𝑢] = 𝑢′ ∙ (−sin 𝑢)
𝑑𝑥
𝑑 𝑑
[𝑐𝑜𝑠𝑡𝑎𝑛𝑡] = 0 [tan 𝑢] = 𝑢′ ∙ sec 2 𝑢
𝑑𝑥 𝑑𝑥
Scalar Rule: 𝑑
[cot 𝑢] = 𝑢′ ∙ (− csc 2 𝑢)
𝑑 𝑑 𝑑𝑥
[𝑎 ∙ 𝑓(𝑥)] = 𝑎 ∙ [𝑓(𝑥)] 𝑑
𝑑𝑥 𝑑𝑥 [sec 𝑢] = 𝑢′ ∙ sec 𝑢 tan 𝑢
Where a is a constant 𝑑𝑥
𝑑
Exponential: [csc 𝑢] = 𝑢′ ∙ (− csc 𝑢 cot 𝑢)
𝑑𝑥
𝑑 𝑢
(𝑒 ) = 𝑢′ ∙ 𝑒 𝑢
𝑑𝑥

𝑑 𝑢
(𝑎 ) = 𝑢′ ∙ 𝑎𝑢 ∙ ln 𝑎
𝑑𝑥

Logarithmic:
𝑑 𝑢′
(ln 𝑢) =
𝑑𝑥 𝑢

𝑑 𝑢′
(log 𝑎 𝑢) =
𝑑𝑥 𝑢 ∙ ln 𝑎
Extreme Value Theorem (EVT) & Absolute Extrema

Extreme Value Theorem (EVT)

If f is continuous on a closed interval [a, b], then f has both an absolute minimum and
an absolute maximum.

Finding Both the Absolute Maximum & the Absolute Minimum on a Closed Interval
1. Find the critical values of f on the interval (a, b).
2. Evaluate f at the critical values AND the endpoints x = a and x = b.
3. The smallest value is the absolute minimum. The largest value is the absolute
maximum.

Mean Value Theorem (MVT) & Rolle’s Theorem

Mean Value Theorem

If f(x) is continuous on [a, b], and differentiable on (a, b), then there exists an x =c in
the interval (a, b) such that:

𝑓(𝑏) − 𝑓(𝑎)
𝑓 ′ (𝑐) =
𝑏−𝑎

Rolle’s Theorem
If f(x) is continuous on [a, b], differentiable on (a, b), AND f(a) = f(b), then there
exists an x =c in the interval (a, b) such that:

𝑓(𝑏) − 𝑓(𝑎)
𝑓 ′ (𝑐) = =0
𝑏−𝑎

NOTE: f(a) & f(b) don’t have to be 0. They just have to be the same. But when
Rolle’s applies, the derivative is set equal to 0.

L’Hôpital’s Rule
𝑓(𝑥) 𝑓(𝑥) 𝑓′(𝑥)
If lim 𝑔(𝑥) is indeterminate, then lim 𝑔(𝑥) = lim 𝑔′(𝑥) .
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎

NOTE: The limit must be 0/0 or ∞/∞ to use L’Hôpital’s Rule.

 Implicit Differentiation

Steps for Implicit Differentiation:

1. Differentiate both sides with respect to x. Don’t forget the chain rule! When
𝑑𝑦
taking the derivative a term with a y, the chain rule will create a .
𝑑𝑥
𝑑𝑦
2. Collect all terms containing 𝑑𝑥
on one side of the equation.
𝑑𝑦
3. Factor out 𝑑𝑥.
𝑑𝑦
4. Solve for using division.
𝑑𝑥

NOTE: Students often make mistakes when the term –xy is part of the equation.
Remember, you need to use the product rule for this with –x as one part and y
as the other part. Many students lose the negative sign when doing the
product rule because they don’t think about it being “attached” to the x!

NOTE: The derivative of an implicit equation usually includes both x and y. If you are
asked to find the derivative at a particular x-value, you must first substitute
that x-value into the original equation to find the corresponding y-value. Then
you will substitute the x- and y-value in simultaneously to find the derivative
at that particular ordered pair.
 Derivative Rules
Note: These include the chain rule:

Power Rule: Trig Derivatives:

𝑑 𝑛 𝑑
[𝑢 ] = 𝑢′ ∙ 𝑛𝑢𝑛−1 [sin 𝑢] = 𝑢′ ∙ cos 𝑢
𝑑𝑥 𝑑𝑥
𝑑
[cos 𝑢] = 𝑢′ ∙ (−sin 𝑢)
Constant Rule: 𝑑𝑥
𝑑
𝑑 [tan 𝑢] = 𝑢′ ∙ sec 2 𝑢
[𝑐𝑜𝑠𝑡𝑎𝑛𝑡] = 0 𝑑𝑥
𝑑𝑥
𝑑
[cot 𝑢] = 𝑢′ ∙ (− csc 2 𝑢)
𝑑𝑥
Scalar Rule: 𝑑
[sec 𝑢] = 𝑢′ ∙ sec 𝑢 tan 𝑢
𝑑 𝑑 𝑑𝑥
[𝑎 ∙ 𝑓(𝑥)] = 𝑎 ∙ [𝑓(𝑥)]
𝑑𝑥 𝑑𝑥 𝑑
Where a is a constant [csc 𝑢] = 𝑢′ ∙ (− csc 𝑢 cot 𝑢)
𝑑𝑥

Exponential: Inverse Trig Derivatives:

𝑑 𝑢 𝑑 𝑢′
(𝑒 ) = 𝑢′ ∙ 𝑒 𝑢 [arcsin 𝑢] =
𝑑𝑥 𝑑𝑥 √1 − 𝑢2
𝑑 𝑢 𝑑 𝑢′
(𝑎 ) = 𝑢′ ∙ 𝑎𝑢 ∙ ln 𝑎 [arctan 𝑢] =
𝑑𝑥 𝑑𝑥 1 + 𝑢2
𝑑 𝑢′
[arcsec 𝑢] =
Logarithmic: 𝑑𝑥 |𝑢|√𝑢2 − 1
𝑑 𝑢′ 𝑑 −𝑢′
(ln 𝑢) = [arccos 𝑢] =
𝑑𝑥 𝑢 𝑑𝑥 √1 − 𝑢2
𝑑 −𝑢′
𝑑 𝑢′ [arccot 𝑢] =
(log 𝑎 𝑢) = 𝑑𝑥 1 + 𝑢2
𝑑𝑥 𝑢 ∙ ln 𝑎
𝑑 −𝑢′
[arccsc 𝑢] =
𝑑𝑥 |𝑢|√𝑢2 − 1

Evaluating the Derivative of an Inverse Function

Formula for Evaluating the Derivative of an Inverse Function

1
(𝑓 −1 )′ (𝑥) =
𝑓′(𝑓 −1 (𝑥))

Example: To find 𝑓 −1 (2), you need to find where the function f(x) = 2. Remember:
The input of an inverse function 𝑓 −1 is the output of the function f.
 Related Rates

Steps for Related Rates Problem:

5. Draw a picture.
6. Write down known information.
7. Write down what you are looking for.
8. Write an equation that relates the quantities.
9. Differentiate both sides with respect to t (Don’t forget the chain rule!) to
show how the rates of the quantities are related.
10. Substitute and evaluate.

NOTE: Look to see if any quantities are constant in the situation. If so, you can
substitute them if before taking the derivative, making your derivative
much easier. However, be careful not to substitute any values that are only
equal to that measurement or rate at a specific instant in time. You must
wait to substitute these values until after you have taken the derivative.

NOTE: Sometimes you have to use a second equation to relate two quantities in
order to eliminate variables in your main equation. Realizing you have to do
this often saves time and makes your derivative and work less messy.
 Linearization & Differentials
Linearization Equation
Used to estimate the value of a function that would be difficult to find without a
calculator.
𝐿(𝑥) = 𝑓 ′ (𝑎)(𝑥 − 𝑎) + 𝑓(𝑎) where a is the center of the approximation
NOTE: This is just the tangent line at x = a in a slightly different form.

Overestimation & Underestimation Using Linearization

 The tangent line (linearization equation at x = a) underestimates the value
of the function when the function is concave up at x = a.
 The tangent line (linearization equation at x = a) overestimates the value of
the function when the function is concave down at x = a.

Differentials
Often used to find how one variable changes with respect to the other. Also used
to estimate the value of a function as it is the same as the linearization equation in
a slightly different format.
𝑓(𝑥 + Δ𝑥) ≈ 𝑓 ′ (𝑥)𝑑𝑥 + 𝑓(𝑥)
NOTE: This is also just the tangent line at x = a in a slightly different form.
 Curve Sketching

Given graph of f ʹ(x), when is f (x) increasing/decreasing?

 f (x) is increasing when f ʹ(x) > 0 (f ʹ(x) graph above x-axis)
 f (x) is decreasing when f ʹ(x) < 0 (f ʹ(x) graph below x-axis)

Given graph of f ʹ(x), where are the critical values of f (x)?

 Critical values exist where f ʹ(x) = 0 (f ʹ(x) graph crosses x-axis) or where f ʹ(x)
is undefined
 NOTE: To be a critical value, the point must also exist in the f(x) function.

Given graph of f ʹ(x), where are the relative extrema of f (x)?

 How do you tell whether the critical value is a relative extremum? f ʹ(x) graph
changes sign
o from negative to positive = relative minimum
o from positive to negative = relative maximum
 At the endpoints (NOTE: only applies on CLOSED INTERVAL):
o Left endpoint
 f ʹ(x) negative after ⇒ f(x) decreasing ⇒ Left is relative max
 f ʹ(x) positive after ⇒ f(x) increasing ⇒ Left is relative min

o Right endpoint
 f ʹ(x) negative before ⇒ f(x) decreasing ⇒ Right is relative min
 f ʹ(x) positive before ⇒ f(x) increasing ⇒ Right is relative max

Based on the graph of f ʹ(x), when is f (x) concave up/concave down?

 f (x) is concave up where graph of f ʹ(x) is increasing.
 f (x) is concave down where graph of f ʹ(x) is decreasing

Given graph of f ʹ(x), where are the inflection points of f (x)?

 The inflection points of f (x) are at the x-values that are turning points on the
graph of f ʹ(x).
 RAM & Trapezoidal Rule

RAM (Rectangular Approximation Method)

 NOTE: Subintervals may not be of equal width
 LRAM (Left):
o Uses the left-hand function value for each subinterval as the height of
the rectangle.
o The distance between x-values is the width of the subinterval.
o Approximation of area (draw picture to help you)
 Underestimate if function is increasing
 Overestimate if function is decreasing
 RRAM (Right):
o Uses the right-hand function value for each subinterval as the height of
the rectangle.
o The distance between x-values is the width of the subinterval.
o Approximation of area (draw picture to help you)
 Overestimate if function is increasing
 Underestimate if function is decreasing
 MRAM (Midpoint):
o Uses the midpoint function value for each subinterval as the height of
the rectangle.
o The distance between the left and right x-values is the width of the
subinterval.

Trapezoidal Rule
For equal width subintervals, the area under the curve can be approximated by
the Trapezoidal Rule:
Δ𝑥
𝐴𝑟𝑒𝑎 ≈ (𝑓(𝑥0 ) + 2𝑓(𝑥1 ) + 2𝑓(𝑥2 ) + ⋯ + 2𝑓(𝑥𝑛−1 ) + 𝑓(𝑥𝑛 ))
2

NOTE: If subintervals are not equal width, you must find the areas of individual
1
trapezoids and add their areas together. 𝐴 = ℎ(𝑏1 − 𝑏2 )
2
 Integration Formulas

Trig:
∫ 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶 , 𝑘 is a consant
∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶
𝑛
𝑥 𝑛+1
∫ 𝑥 𝑑𝑥 = + 𝐶, if 𝑛 ≠ −1
𝑛+1
∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶
∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
∫ sec 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶
1
∫ 𝑑𝑥 = ln|𝑥| + 𝐶
𝑥
∫ csc 2 𝑥 𝑑𝑥 = −cot 𝑥 + 𝐶
𝑎𝑥
∫ 𝑎 𝑥 𝑑𝑥 = +𝐶
ln 𝑎
∫ sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝐶
1
∫ 𝑑𝑥 = arctan 𝑥 + 𝐶
1 + 𝑥2
∫ csc 𝑥 cot 𝑥 𝑑𝑥 = −csc 𝑥 + 𝐶
1
∫ 𝑑𝑥 = arcsin 𝑥 + 𝐶
√1 − 𝑥 2
 Properties of Definite Integrals
𝑏
Integral Evaluation Theorem: ∫𝑎 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
where F is an antiderivative of f.

𝑎 𝑏
Order of Integration Rule: ∫𝑏 𝑓(𝑥)𝑑𝑥 = − ∫𝑎 𝑓(𝑥)𝑑𝑥

𝑎
Zero Rule: ∫𝑎 𝑓(𝑥)𝑑𝑥 = 0

𝑏 𝑏
Scalar Rule: ∫𝑎 𝑘𝑓(𝑥)𝑑𝑥 = 𝑘 ∫𝑎 𝑓(𝑥)𝑑𝑥

𝑏 𝑏 𝑏
Sum & Difference Rule: ∫𝑎 (𝑓(𝑥) + 𝑔(𝑥)) 𝑑𝑥 = ∫𝑎 𝑓(𝑥)𝑑𝑥 + ∫𝑎 𝑔(𝑥)𝑑𝑥

𝑏 𝑐 𝑐
Additivity Rule: ∫𝑎 𝑓(𝑥)𝑑𝑥 + ∫𝑏 𝑓(𝑥)𝑑𝑥 = ∫𝑎 𝑓(𝑥)𝑑𝑥

Average Value of Function & MVT (Definite Integrals)

Average Value of a Function
If f is integrable on the closed interval [a, b], then the average value of f on the
interval is:
𝑏
1
∫ 𝑓(𝑥)𝑑𝑥
𝑏−𝑎 𝑎

Mean Value Theorem for Integrals

If f is continuous on the closed interval [a, b], then there exists a number c in the
closed interval [a, b] such that:
𝑏
1
𝑓(𝑐) = ∫ 𝑓(𝑥)𝑑𝑥
𝑏−𝑎 𝑎
In other words, the average value equals the function value at some c in [a, b]

NOTE: Another way to write the MVT for Integrals is shown below. This is the
result of taking the formula above and multiplying by (b – a)
𝑏
𝑓(𝑐)(𝑏 − 𝑎) = ∫ 𝑓(𝑥)𝑑𝑥
𝑎
 Fundamental Theorem of Calculus
If f is continuous on an open interval I containing a, then, for every x in the
interval,
𝑥
𝑑
[∫ 𝑓(𝑡)𝑑𝑡] = 𝑓(𝑥)
𝑑𝑥 𝑎

NOTES:
 The bottom limit of integration must be a constant.
 The upper limit of integration must be a function.

When the upper limit is not just x, but function of x (like x2):
𝑢
𝑑
[∫ 𝑓(𝑡)𝑑𝑡] = 𝑢′𝑓(𝑢)
𝑑𝑥 𝑎
Where u is a function of x.

When the lower and upper limit are both functions of x:

𝑣
𝑑
[∫ 𝑓(𝑡)𝑑𝑡] = 𝑣 ′ 𝑓(𝑣) − 𝑢′𝑓(𝑢)
𝑑𝑥 𝑢
Where u and v are both functions of x.
 Motion (Again)

Displacement (net change in position)

𝑏
∫ 𝑣(𝑡)𝑑𝑡
𝑎

Total Distance Traveled

𝑏
∫ |𝑣(𝑡)|𝑑𝑡
𝑎

NOTE: When finding total distance by hand, you must find where v(t) changes
sign. You must set up multiple integrals if v(t) changes sign, then add the
absolute values of each integral.

NOTE: When finding total distance on a calculator, you can just type in the
integral above with the absolute value bars around the v(t) function.

Position (Finding position s(b) given position s(a) at x = a)

𝑏
𝑠(𝑏) = 𝑠(𝑎) + ∫ 𝑣(𝑡)𝑑𝑡
𝑎

Velocity
 Velocity is the derivative of position: v(t) = s′(t)
 When velocity > 0, object is moving in a positive direction (right or up for
linear motion)
 When velocity < 0, object is moving in a negative direction (left or down for
linear motion)
 When velocity equals 0, the object is at rest.

Acceleration
 Acceleration is the derivative of velocity and the 2nd derivative of position:
a(t) = v′(t) = s′′(t)
 When acceleration > 0, the velocity of the object is increasing.
 When acceleration < 0, the velocity of the object is decreasing.

Speed & Speeding Up or Down

 Speed = |velocity|
 Speeding up when velocity and acceleration have the same sign.
 Slowing down when velocity and acceleration have opposite signs.
 Net Change

A net change problem is when you are given a rate and are asked to find things
about the rate and the total amount. This is related to motion problems about
displacement, position, and distance. However, the rate given is not velocity but is
about a situation other than motion.

Finding the total amount of something given a rate

𝑏
𝑅(𝑏) = 𝑅(𝑎) + ∫ 𝑟(𝑡)𝑑𝑡
𝑎
where r(t) is a rate and R is an antiderivative of r.

Finding change in total amount of something given a rate

𝑏
∫ 𝑟(𝑡)𝑑𝑡
𝑎
where r(t) is a rate
 Exponential Growth & Decay

Differential Equation for Exponential Growth

𝑑𝑦
= 𝑘𝑦
𝑑𝑥
In words: increase/decrease at a rate proportional to the amount present

Exponential Growth/Decay Function

If you know a function grows/decays exponentially, then 𝑦 = 𝑦0 𝑒 𝑘𝑡
NOTE: k is positive if growth, k is negative if decay

Area Between Curves

Area Between Curves: Vertical Strip
𝑏
∫ [𝑓(𝑥) − 𝑔(𝑥)]𝑑𝑥
𝑎

Where a & b are x-values and f(x) is above g(x)

Area Between Curves: Horizontal Strip

𝑑
∫ [𝑓(𝑦) − 𝑔(𝑦)]𝑑𝑦
𝑐

Where c & d are y-values and f(y) is to the right of g(y). NOTE: Functions are
x = something in terms of y.
 Volume: Disk Method

Disk Method:
• We use the disk method when the area to be revolved is touching the axis of
revolution.
• The representative rectangle (strip) must be perpendicular to the axis of
revolution.
• Disks are just flat cylinders. Therefore we are integrating the volume of a
cylinder where 𝜋𝑟 2 is the area of the base and dx (or dy)is the very small
height of the cylinder.

Rotate about a Horizontal Axis (like the x-axis, y = #)

• When revolving around the x-axis, r is the function given and the formula is:
𝑏
𝑉 = 𝜋 ∫ [𝑓(𝑥)]2 𝑑𝑥
𝑎

• When revolving around another horizontal axis y = k, the radius is not just
the function given. You must adjust the radius by subtracting the value of k.
The formula becomes:
𝑏
𝑉 = 𝜋 ∫ [𝑓(𝑥) − 𝑘]2 𝑑𝑥
𝑎

• VERTICAL STRIPS!

Rotate about a Vertical Axis (like the y-axis, x = #)

• When revolving around the y-axis, r is the function given and the formula is:
𝑑
𝑉 = 𝜋 ∫ [𝑓(𝑦)]2 𝑑𝑦
𝑐

• When revolving around another vertical axis x = k, the radius is not just the
function given. You must adjust the radius by subtracting the value of k. The
formula becomes:
𝑑
𝑉 = 𝜋 ∫ [𝑓(𝑦) − 𝑘]2 𝑑𝑦
𝑐

• HORIZONTAL STRIPS!
 Volume: Washer Method

Washer Method:
• We use the disk method when the area to be revolved is not touching the
axis of revolution.
• The representative rectangle (strip) must be perpendicular to the axis of
revolution.
• Washers are the outer flat cylinders minus the inner flat cylinder. Therefore
we are integrating the volume of a cylinder where 𝜋𝑅 2 is the area of the
base of the outer cylinder, 𝜋𝑟 2 is the area of the base of the inner cylinder
and dx (or dy)is the very small height of the cylinder.

Rotate about a Horizontal Axis (like the x-axis, y = #)

• When revolving around the x-axis, R(x) and r(x) are the functions given.
R(x) is the function further from the axis of revolution, and r(x) is the
function closer to the axis of revolution.
𝑏
𝑉 = 𝜋 ∫ ([𝑅(𝑥)]2 − [𝑟(𝑥)]2 )𝑑𝑥
𝑎

• When revolving around another horizontal axis y = k, the radius is not just
the function value. You must adjust each radius by subtracting the value of
k. The formula becomes:
𝑏
𝑉 = 𝜋 ∫ ([𝑅(𝑥) − 𝑘]2 − [𝑟(𝑥) − 𝑘]2 )𝑑𝑥
𝑎

• VERTICAL STRIPS!

Rotate about a Vertical Axis (like the y-axis, x = #)

• When revolving around the y-axis, R(y) and r(y) are the functions given.
R(y) is the function further from the axis of revolution, and r(y) is the
function closer to the axis of revolution.
𝑑
𝑉 = 𝜋 ∫ ([𝑅(𝑦)]2 − [𝑟(𝑦)]2 )𝑑𝑦
𝑐

• When revolving around another vertical axis y = k, the radius is not just the
function value. You must adjust each radius by subtracting the value of k.
The formula becomes:
𝑑
𝑉 = 𝜋 ∫ ([𝑅(𝑦) − 𝑘]2 − [𝑟(𝑦) − 𝑘]2 )𝑑𝑦
𝑐

• HORIZONTAL STRIPS!
Volume: Cross Sections & Area Formulas to Know

Volume: Cross Sections

𝑏
𝑉 = ∫ (Area formula of cross section) 𝑑𝑥
𝑎

NOTE: The area of the cross section formula that becomes the integrand will
typically contain the function given. However, it is not always just the
function. Draw a picture to help you figure out the area formula in
terms of x (or y).

NOTE: Draw a representative rectangle (strip). Make sure it is perpendicular

to the appropriate axis.

Area Formulas to Know

Square: 𝐴 = 𝑠 2
Rectangle: 𝐴 = 𝑏ℎ (base & height must be perpendicular)
1
Triangle: 𝐴 = 𝑏ℎ (base & height must be perpendicular)
2
√3 2
Equilateral Triangle: 𝐴 = 𝑠
4
1
Semicircle: 𝐴 = 2 𝜋𝑟 2
Limit of Riemann Sum vs Definite Integral
𝑛 𝑏
lim ∑ 𝑓(Δ𝑥𝑘 + 𝑎)Δ𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥
𝑥→∞
𝑘=1 𝑎

Steps for Converting from Definite Integral to Limit of Riemann Sum:

𝑏−𝑎
1. Find Δ𝑥 =
𝑛

2. Find 𝑥𝑘 = Δ𝑥𝑘 + 𝑎
3. Substitute into limit notation above.

Steps for Converting from Limit of Riemann Sum to Definite Integral:

1. Determine Δ𝑥 from limit notation.
2. Determine 𝑥𝑘 from limit notation. Then identify 𝑎.
𝑏−𝑎
3. Determine 𝑏 using Δ𝑥 =
𝑛

4. Write the integral.

Integration Using Long Division

Use long division before integrating when ALL of the following are present:

1. Polynomial over a polynomial.

2. Numerator degree ≥ Denominator degree

3. Denominator is two or more terms (Recall: If denominator is only one

term, you can divide all terms in the numerator by the term in the
denominator)
 Integration Using Completing the Square

Use completing the square when:

1. Numerator degree < Denominator degree

2. Numerator is not the derivative of the denominator (or different by

scalar)

3. Denominator does not factor

NOTE: Once you have completed the square, the resulting integrand will have an
antiderivative that is an inverse trig function.

Integration by Parts

Indefinite: ∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢

𝑏 𝑏
Definite: ∫𝑎 𝑢 𝑑𝑣 = [𝑢𝑣]𝑏𝑎 − ∫𝑎 𝑣 𝑑𝑢

To determine what is u and what is dv, use the following to determine u:

• Logarithmic
• Inverse Trig
• Polynomial
• Exponential
• Trig

Tabular Integration by Parts can be used if and only if one function differentiates
to zero eventually, and the other continually integrates.
 L’Hôpital’s Rule (Part 2)
𝑓(𝑥) 𝑓(𝑥) 𝑓′(𝑥)
If lim 𝑔(𝑥) is indeterminate, then lim 𝑔(𝑥) = lim 𝑔′(𝑥) .
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎

0 ∞
NOTE: The limit must be 0
or ∞
to use L’Hôpital’s Rule.

Other Indeterminate Forms: ∞∙0 ∞−∞ 1∞ 00 ∞0

0 ∞
• For ∞ ∙ 0, reciprocate one of the terms and put it on the bottom to create 0
or ∞

• For ∞ − ∞, combine fractions into one fraction by getting a common denominator

0 ∞
to create or
0 ∞

• For 1 , 0 , and ∞0 , take the ln of both sides to bring down the exponent. Then
∞ 0

follow the steps for ∞ ∙ 0. Don’t forget to exponentiate to reverse the ln!

Determinate Forms that do not require L’Hôpital’s Rule:

∞ 0
∞+∞=∞ −∞ − ∞ = −∞ 0∞ = 0 0−∞ = ∞ 0
=∞ ∞
=0
 Improper Integrals

Improper Integrals with Infinite Limits of Integration:

1. If f is continuous on the interval [a, ∞), then
 b
 a
f ( x)dx = lim
b → a  f ( x)dx
2. If f is continuous on the interval (–∞, b], then
b b
 −
f ( x)dx = lim
a → − a  f ( x)dx
3. If f is continuous on the interval (–∞, ∞), then
 c b
 −
f ( x)dx = lim
b → − b  f ( x)dx + lim
b → c  f ( x)dx
Where c is any real number. However, usually c = 0 is used.

Improper Integrals with Infinite Discontinuities:

1. If f is continuous on the interval [a, b) and has an infinite discontinuity at b, then
b c
a
f ( x)dx = lim−
c →b 
a
f ( x)dx

2. If f is continuous on the interval (a, b] and has an infinite discontinuity at a, then

b b
a
f ( x)dx = lim+
c →a  f ( x)dx
c

3. If f is continuous on the interval [a, b], except for some c in (a, b) at which f has an
infinite discontinuity, then
b c b
a
f ( x)dx = lim−
c →c 
a
f ( x)dx + lim+
c →c  f ( x)dx
c
 Euler’s Method

• Euler’s Method is used to estimate the value of the function from a differential
equation, particularly when the variables cannot be separated to find the particular
solution.

• The accuracy of the estimation increases as the value of dx decreases.

• The accuracy of the estimation increases as the distance between the initial
condition x-value and the desired x-value decreases.

• When f is concave up at the initial condition, Euler’s Method is an underestimate.

• When f is concave down at the initial condition, Euler’s Method is an overestimate.

Arc Length (Don’t Glue Yet!)

𝑏 𝑑𝑦 2 𝑑 𝑑𝑥2
Arc Length (Regular Functions): 𝐿 = ∫𝑎 √1 + (𝑑𝑥 ) 𝑑𝑥 or 𝐿 = ∫𝑐 √1 + (𝑑𝑦) 𝑑𝑦

Newton’s Law of Cooling

Formulas using Temperature of Liquid at any time t

𝑑𝑇
Differential Equation: 𝑑𝑡 = 𝑘(𝑇0 − 𝑇𝑆 )

Temperature Function: 𝑇 = 𝑇𝑆 + (𝑇0 − 𝑇𝑆 )𝑒 𝑘𝑡

Formulas using Temperature Difference at any time t

𝑑𝑇𝑑𝑖𝑓𝑓
Differential Equation: = 𝑘𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑑𝑡

Temperature Function: 𝑇𝑑𝑖𝑓𝑓 = (𝑇𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 )𝑒 𝑘𝑡

 Logistic Growth
𝑑𝑃 𝑀−𝑃 𝑑𝑃 𝑃
Differential Equations: = 𝑘𝑃 ( ) or = 𝑘𝑃 (1 − )
𝑑𝑡 𝑀 𝑑𝑡 𝑀

𝑀 𝑀−𝑃0
Logistic Growth Model: 𝑃 = 1+𝐴𝑒 −𝑘𝑡 where M = carrying capacity and 𝐴 = 𝑃0

1
Fastest Growth Occurs when 𝑃 = 2 𝑀

lim 𝑃(𝑡) = 𝑀
𝑡→∞
 Parametric Equations

Parametric equations can be used to define motion that is not a function:

Position:
• Horizontal position = x(t)
• Vertical position = y(t)

𝑑𝑦
𝑑𝑦 𝑑𝑡
=
𝑑𝑥 𝑑𝑥
𝑑𝑡

𝑑𝑦
𝑑2 𝑦 derivative of 𝑑𝑥
=
𝑑𝑥 2 𝑑𝑥
𝑑𝑡

𝑏 𝑑𝑥 𝑑𝑦 2 2
Length of Curve (aka distance object travels): 𝐿 = ∫𝑎 √( ) + ( ) 𝑑𝑡
𝑑𝑡 𝑑𝑡

Object is moving:
𝑑𝑥
• Left when 𝑑𝑡 < 0
𝑑𝑥
• Right when 𝑑𝑡 > 0
𝑑𝑦
• Down when <0
𝑑𝑡
𝑑𝑦
• Up when 𝑑𝑡 > 0

𝑑𝑥 𝑑𝑦
Object is at rest when 𝑑𝑡 and 𝑑𝑡 both equal zero at the same value of t.

Speed increases if the derivative of the speed function is positive.

Speed decreases if the derivative of the speed function is negative.

𝑑𝑥
Tangent line is vertical when 𝑑𝑡 is zero. Must often find the position (x, y) for the
solution for t.

𝑑𝑦 𝑑𝑥
Tangent line is horizontal when 𝑑𝑡 is zero (as long as 𝑑𝑡 is not also 0). Must often
find the position (x, y) for the solution for t.
 Vectors

Vector notation: 〈𝑥(𝑡), 𝑦(𝑡)〉

Position: 〈𝑥(𝑡), 𝑦(𝑡)〉

Velocity: 〈𝑥′(𝑡), 𝑦′(𝑡)〉

Speed (t = a): |𝑣(𝑎)| = ‖𝑣(𝑎)‖ = √(𝑥′(𝑎))2 + (𝑦′(𝑎))2 (Result is a number)

𝑣(𝑎)
Direction (t = a): |𝑣(𝑎)|
(Result is a vector)

Acceleration: 〈𝑥′′(𝑡), 𝑦′′(𝑡)〉

Object is moving:
𝑑𝑥
• Left when 𝑑𝑡 < 0
𝑑𝑥
• Right when >0
𝑑𝑡
𝑑𝑦
• Down when 𝑑𝑡 < 0
𝑑𝑦
• Up when 𝑑𝑡 > 0

𝑑𝑥 𝑑𝑦
Object is at rest when 𝑑𝑡 and 𝑑𝑡 both equal zero at the same value of t.

Speed increases if the derivative of the speed function is positive.

Speed decreases if the derivative of the speed function is negative.

𝑏 𝑑𝑥 𝑑𝑦 2 2
Distance traveled by particle: 𝐿 = ∫𝑎 √( 𝑑𝑡 ) + ( 𝑑𝑡 ) 𝑑𝑡
 Polar Equations

Helpful Facts for Moving from Polar to Cartesian:

• x = r cos θ
• y = r sin θ
• 𝑟2 = 𝑥2 + 𝑦2
𝑦 𝑦
• 𝜃 = arctan (or tan 𝜃 = )
𝑥 𝑥

𝑑𝑦 𝑑𝑟
𝑑𝑦 𝑑𝜃 𝑟 ′ sin 𝜃 + 𝑟 cos 𝜃 𝑑𝜃 sin 𝜃 + 𝑟 cos 𝜃
= = =
𝑑𝑥 𝑑𝑥 𝑟 ′ cos 𝜃 − 𝑟 sin 𝜃 𝑑𝑟 cos 𝜃 − 𝑟 sin 𝜃
𝑑𝜃 𝑑𝜃

1 𝑏
Area inside polar curve: 𝐴 = 2 ∫𝑎 𝑟 2 𝑑𝜃 (may have to find bounds for one period)

1 𝑏
Area between polar curves: 𝐴 = 2 ∫𝑎 (𝑅 2 − 𝑟 2 ) 𝑑𝜃 (may have to find intersections)

𝑏
Polar arc length: 𝐿 = ∫𝑎 √(𝑟)2 + (𝑟′)2 𝑑𝜃

Arc Length (All 3 Formulas)

𝑏 𝑑𝑦 2 𝑑 𝑑𝑥 2
Arc Length for functions of x: 𝐿 = ∫𝑎 √1 + (𝑑𝑥 ) 𝑑𝑥 or 𝐿 = ∫𝑐 √1 + (𝑑𝑦) 𝑑𝑦

𝑏 𝑑𝑥 𝑑𝑦2 2
Arc Length for parametric (equivalent to distance travelled): 𝐿 = ∫𝑎 √( ) + ( ) 𝑑𝑡
𝑑𝑡 𝑑𝑡

𝑏 𝑏 𝑑𝑟 2
Arc Length for polar curves: 𝐿 = ∫𝑎 √(𝑟)2 + (𝑟′)2 𝑑𝜃 = ∫𝑎 √𝑟 2 + (𝑑𝜃) 𝑑𝜃