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AP Calculus AB/BC Formula and Concept Cheat Sheet
Limit of a Continuous Function
If f(x) is a continuous function for all real numbers, then lim
𝑥→𝑐
Limits of Rational Functions
A. If f(x) is a rational function given by 𝑓(𝑥) =
𝑝
( 𝑥
)
𝑞
( 𝑥
)
,such that 𝑝(𝑥) and 𝑞(𝑥) have no common factors, and c is a real
number such that 𝑞
= 0 , then
I. lim
𝑥→𝑐
𝑓(𝑥) does not exist
II. lim
𝑥→𝑐
= ±∞ x = c is a vertical asymptote
B. If f(x) is a rational function given by 𝑓(𝑥) =
𝑝
( 𝑥
)
𝑞
( 𝑥
)
, such that reducing a common factor between 𝑝(𝑥) and 𝑞(𝑥) results
in the agreeable function k(x) , then
lim
𝑥→𝑐
= lim
𝑥→𝑐
𝑝(𝑥)
𝑞(𝑥)
= lim
𝑥→𝑐
= 𝑘(𝑐) Hole at the point (𝑐, 𝑘
Limits of a Function as x Approaches Infinity
If f(x) is a rational function given by (𝑥) =
𝑝
( 𝑥
)
𝑞
( 𝑥
)
, such that 𝑝(𝑥) and 𝑞(𝑥) are both polynomial functions, then
A. If the degree of p(x) > q(x) , lim
𝑥→∞
B. If the degree of p(x) < q(x) , lim
𝑥→∞
𝑓(𝑥) = 0 y = 0 is a horizontal asymptote
C. If the degree of p(x) = q(x) , lim
𝑥→∞
𝑓(𝑥) = 𝑐, where c is the ratio of the leading coefficients.
y = c is a horizontal asymptote
Special Trig Limits
A. lim
𝑥→ 0
sin 𝑎𝑥
𝑎𝑥
= 1 B. lim
𝑥→ 0
𝑎𝑥
sin 𝑎𝑥
= 1 C. lim
𝑥→ 0
1 −cos 𝑎𝑥
𝑎𝑥
L’Hospital’s Rule
If results lim
𝑥→𝑐
or lim
𝑥→∞
results in an indeterminate form (
0
0
∞
∞
0
∞
0
) , and
𝑝(𝑥)
𝑞(𝑥)
, then
lim
𝑥→𝑐
= lim
𝑥→𝑐
𝑝(𝑥)
𝑞(𝑥)
= lim
𝑥→𝑐
𝑝
′
(𝑥)
𝑞
′
(𝑥)
and lim
𝑥→∞
= lim
𝑥→∞
𝑝(𝑥)
𝑞(𝑥)
= lim
𝑥→∞
𝑝
′
(𝑥)
𝑞
′
(𝑥)
The Definition of Continuity
A function 𝑓
is continuous at c if
I. lim
𝑥→𝑐
𝑓(𝑥) exists
II. 𝑓
exists
III. lim
𝑥→𝑐
Types of Discontinuities
Removable Discontinuities (Holes)
I. lim
𝑥→𝑐
= 𝐿 (the limit exists)
II. 𝑓(𝑐) is undefined
Non-Removable Discontinuities (Jumps and Asymptotes)
A. Jumps
lim
𝑥→𝑐
𝑓(𝑥) = 𝐷𝑁𝐸 because lim
𝑥→𝑐
−
𝑓(𝑥) ≠ lim
𝑥→𝑐
B. Asymptotes (Infinite Discontinuities)
lim
𝑥→𝑐
Differentiability and Continuity Properties
A. If f(x) is differentiable at x = c , then f(x) is continuous at x = c.
B. If f(x) is not continuous at x = c , then f(x) is not differentiable at x = c.
C. The graph of f is continuous, but not differentiable at x = c if:
I. The graph has a cusp or sharp point at x = c
II. The graph has a vertical tangent line at x = c
III. The graph has an endpoint at x = c
Basic Derivative Rules
Given c is a constant,
Derivatives of Trig Functions
Derivatives of Inverse Trig Functions
Derivatives of Exponential and Logarithmic Functions
Explicit and Implicit Differentiation
A. Explicit Functions: Function y is written only in terms of the variable x (𝑦 = 𝑓
). Apply derivatives rules normally.
B. Implicit Differentiation: An expression representing the graph of a curve in terms of both variables x and y.
I. Differentiate both sides of the equation with respect to x. (terms with x
differentiate normally, terms with y are multiplied by
𝑑𝑦
𝑑𝑥
per the chain rule)
II. Group all terms with
𝑑𝑦
𝑑𝑥
on one side of the equation and all other terms on
the other side of the equation.
III. Factor
𝑑𝑦
𝑑𝑥
and express
𝑑𝑦
𝑑𝑥
in terms of x and y.
Tangent Lines and Normal Lines
A. The equation of the tangent line at a point (𝑎, 𝑓(𝑎)): 𝑦 − 𝑓(𝑎) = 𝑓′(𝑎)(𝑥 − 𝑎)
B. The equation of the normal line at a point (𝑎, 𝑓(𝑎)): 𝑦 − 𝑓(𝑎) = −
1
𝑓′(𝑎)
Mean Value Theorem for Derivatives
If the function f is continuous on the close interval [a, b] and differentiable on the open interval (a, b), then there exists
at least one number c between a and b such that
′
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
The slope of the tangent line is equal to the slope of the secant line.
Extrema of a Function
A. Absolute Extrema: An absolute maximum is the highest y – value of a function on a given interval or across the entire
domain. An absolute minimum is the lowest y – value of a function on a given interval or across the entire domain.
B. Relative Extrema
I. Relative Maximum: The y-value of a function where the graph of the function changes from increasing
to decreasing. Another way to define a relative maximum is the y-value where derivative of a function
changes from positive to negative.
II. Relative Minimum: The y-value of a function where the graph of the function changes from
decreasing to increasing. Another way to define a relative maximum is the y-value where derivative of a
function changes from negative to positive.
Critical Value
When f(c) is defined, if f ‘ (c) = 0 or f ‘ is undefined at x = c , the values of the x – coordinate at those points are called
critical values.
*If f(x) has a relative extrema at x = c , then c is a critical value of f.
Extreme Value Theorem
If the function f continuous on the closed interval [a, b], then the absolute extrema of the function f on the closed
interval will occur at the endpoints or critical values of f.
*After identifying critical values, create a table with endpoints and critical values. Calculate the y – value at each
of these x values to identify the extrema.
Increasing and Decreasing Functions
For a differentiable function f
A. If 𝑓
′
(𝑥) > 0 in (a, b), then f is increasing on (a, b) Tangent line has a positive slope
B. If 𝑓
′
(𝑥) < 0 in (a, b), then f is decreasing on (a, b) Tangent line has a negative slope
C. If 𝑓
′
(𝑥) = 0 in (a, b), then f is constant on (a, b) Tangent line has a zero slope (horizontal)
First Derivative Test
After calculating any discontinuities of a function f and calculating the critical values of a function f , create a sign chart
for f ‘ , reflecting the domain, discontinuities, and critical values of a function f.
A. If 𝑓
′
(𝑥) changes sign from negative to positive at 𝑥 = 𝑐, then 𝑓(𝑐) is a relative minimum of f.
B. If 𝑓′(𝑥) changes sign from positive to negative at 𝑥 = 𝑐, then 𝑓(𝑐) is a relative maximum of f.
*If there is no sign change of 𝑓′(𝑥), there exists a shelf point
Concavity
For a differentiable function f(x) ,
A. If 𝑓′′
0 , the graph of 𝑓(𝑥) is concave up
This means 𝑓′(𝑥) is increasing
B. If 𝑓
′′
(𝑥) < 0 , the graph of 𝑓(𝑥) is concave down
This means 𝑓′(𝑥) is decreasing
Second Derivative Test
For a function f(x) that is continuous at x = c
A. If 𝑓′(𝑐) = 0 and 𝑓′′(𝑐) > 0 , then 𝑓(𝑐) is a relative minimum.
B. If 𝑓′
= 0 and 𝑓
′′
( 𝑐
)
< 0 , then 𝑓
is a relative maximum.
- If 𝑓′(𝑐) = 0 and 𝑓′′(𝑐) = 0 , you must use the first derivative test to determine extrema
Antiderivatives
If 𝐹
′
= 𝑓(𝑥) for all x , 𝐹
is an antiderivative of f.
- The antiderivative is also called the Indefinite Integral
Basic Integration Rules
Let k be a constant.
Definite Integrals (The Fundamental Theorem of Calculus)
A definite integral is an integral with upper and lower limits, a and b , respectively, that define a specific interval on the
graph. A definite integral is used to find the area bounded by the curve and an axis on the specified interval (a, b).
If 𝐹
is the antiderivative of a continuous function
𝑓(𝑥), the evaluation of the definite integral to calculate
the area on the specified interval (a, b) is the First
Fundamental Theorem of Calculus:
𝑏
𝑎
Integration Rules for Definite Integrals
***** This means that c is a value of x , lying between a and b
Riemann Sum (Approximations)
A Riemann Sum is the use of geometric shapes (rectangles and trapezoids) to approximate the area under a curve,
therefore approximating the value of a definite integral.
If the interval [a, b] is partitioned into n subintervals, then each subinterval, Δx, has a width: ∆𝑥 =
𝑏−𝑎
𝑛
Therefore, you find the sum of the geometric shapes, which approximates the area by the following formulas:
A. Right Riemann Sum
[
0
1
2
𝑛− 1
)]
B. Left Riemann Sum
𝐴𝑟𝑒𝑎 ≈ ∆𝑥 [𝑓(𝑥
1
2
3
𝑛
)]
C. Midpoint Riemann Sum
𝐴𝑟𝑒𝑎 ≈ ∆𝑥 [𝑓(𝑥
1 / 2
3 / 2
5 / 2
( 2 𝑛− 1 )/ 2
)]
D. Trapezoidal Sum
∆𝑥 [𝑓(𝑥
0
1
2
𝑛− 1
𝑛
)]
Properties of Riemann Sums
A. The area under the curve is under approximated when
I. A Left Riemann sum is used on an increasing function.
II. A Right Riemann sum is used on a decreasing function.
III. A Trapezoidal sum is used on a concave down function.
B. The area under the curve is over approximated when
I. A Left Riemann sum is used on a decreasing function.
II. A Right Riemann sum is used on an increasing function.
III. A Trapezoidal sum is used on a concave up function.
BC Only: Integration by Parts
If u and v are differentiable functions of x , then
Tips: For your choice of the function u , make the selection following:
A. LIPET: Logarithmic, Inverse Trig, Polynomial, Exponential, Trig
B. LIATE: Logarithmic, Inverse Trig, Algebraic, Trig, Exponential
∗ Comes from Integration by Parts. MEMORIZE ∫ ln 𝑥 𝑑𝑥 = 𝑥 ln 𝑥 − 𝑥 + 𝐶
Integration of Trig and Inverse Trig
BC Only: Partial Fractions
Let R(x) represent a rational function of the form 𝑅
𝑁(𝑥)
𝐷(𝑥)
. If D(x) is a factorable polynomial, Partial Fractions can
be used to rewrite R(x) as the sum or difference of simpler rational functions. Then, integration using natural log.
A. Constant Numerator
B. Polynomial Numerator
BC Only: Logistic Growth
A population, P , that experiences a limit factor in the growth of the population based upon the available resources to
support the population is said to experience logistic growth.
A. Differential Equation:
𝑑𝑃
𝑑𝑡
𝑃
𝐿
B. General Solution: 𝑃(𝑡) =
𝐿
1 +𝑏𝑒
−𝑘𝑡
𝑃 = population 𝑘 = constant growth factor 𝐿 = carrying capacity 𝑡 = time,
𝑏 = constant (found with intital condition)
Graph
Exponential Growth and Decay
When the rate of change of a variable y is directly proportional to the value of y , the function y = f(x) is said to
grow/decay exponentially.
A. Differential Equation for rate of change:
𝑑𝑦
𝑑𝑡
B. General Solution: 𝑦 = 𝐶𝑒
𝑘𝑡
I. If k > 0 , then exponential growth occurs.
II. If k < 0 , then exponential decay occurs.
Characteristics of Logistics
I. The population is growing the fastest where 𝑃 =
𝐿
2
II. The point where 𝑃 =
𝐿
2
represents a point of inflection
III. lim
𝑡→∞
Area Between Two Curves
A. Let 𝑦 = 𝑓
and 𝑦 = 𝑔
represent two functions such that 𝑓
(meaning the function f is always above
the function g on the graph) for every x on the interval [a, b].
Area Between Curves = ∫
[
)]
𝑏
𝑎
B. Let 𝑥 = 𝑓
and 𝑥 = 𝑔
represent two functions such that 𝑓
(meaning the function f is always to the
right of the function g on the graph) for every y on the interval [a, b].
Area Between Curves = ∫
[
)]
𝑏
𝑎
Volumes of a Solid of Revolution: Disk Method
If a defined region, bounded by a differentiable function f , on a graph is rotated about a line, the resulting solid is called
a solid of revolution and the line is called the axis of revolution. The disk method is used when the defined region
boarders the axis of revolution over the entire interval [a, b]
A. Revolving around the x – axis
Volume = 𝜋 ∫ (𝑓
2
𝑏
𝑎
B. Revolving around the y – axis
Volume = 𝜋 ∫ (𝑓(𝑦))
2
𝑏
𝑎
C. Revolving around a horizontal line y = k
Volume = 𝜋 ∫ (𝑓(𝑥) − 𝑘)
2
𝑏
𝑎
D. Revolving around a vertical line x = m
Volume = 𝜋 ∫ (𝑓(𝑦) − 𝑚)
2
𝑏
𝑎
Volumes of Known Cross Sections
If a defined region, bounded by a differentiable function f , is used at the base of a solid, then the volume of the solid can
be found by integrated using known area formulas.
For the cross sections perpendicular to the x – axis and a region bounded by a function f , on the interval [a, b], and the
axis.
I. Cross sections are squares
Volume = ∫
[
)]
2
𝑏
𝑎
II. Cross sections are equilateral triangles
Volume =
[
]
2
𝑏
𝑎
III. Cross sections are isosceles right triangles with a leg in the base
Volume =
∫ [𝑓(𝑥)]
2
𝑏
𝑎
IV. Cross sections are isosceles right triangles with the hypotenuse in the base
Volume =
[
]
2
𝑏
𝑎
V. Cross sections are semicircles (with diameter in base)
Volume =
∫ [𝑓(𝑥)]
2
𝑏
𝑎
VI. Cross sections are semicircles (with radius in base)
Volume =
[
)]
2
𝑏
𝑎
Differential Equations
A differential equation is an equation involving an unknown function and one or more of its derivatives
𝑑𝑦
𝑑𝑥
= 𝑓(𝑥, 𝑦) Usually expressed as a derivative equal to an expression in terms of x and/or y.
To solve differential equations, use the technique of separation of variables.
Given the differential equation
𝑑𝑦
𝑑𝑥
𝑥𝑦
(𝑥
2
Step 1: Separate the variables, putting all y’s on one side, with dy in the numerator, and all x’s on the other side,
with dx in the numerator.
2
Step 2: Integrate both sides of the equation.
ln|𝑦| =
ln
2
Step 3: Solve the equation for y.
2
Given the differential equation
𝑑𝑦
𝑑𝑥
2
with the initial condition 𝑦( 3 ) = 10.
A. The general solution to a differential equation is left with the constant of integration, C , undefined.
2
2
3
B. The particular solution uses the given initial condition to calculate the value of C.
3
3
BC Only: Euler’s Method for Approximating the Solution of a Differential Equation
Euler’s method uses a linear approximation with increments (steps), h , for approximating the solution to a given
differential equation,
𝑑𝑦
𝑑𝑥
= 𝐹(𝑥, 𝑦), with a given initial value.
Process: Initial value (𝑥
0
0
1
0
1
0
0
0
2
1
2
1
1
1
3
2
3
2
2
2
***** This process repeats until the desired y – value is given.
