• Credit: Simplestudies.org
• This study guide has a lot of images so if u cant scroll to the very bottom without jittering just let it load for a bit

Units 1-8 are the same as AP Calculus AB Guide

• Here is a Cheat Sheet/Shorter and compressed version of what's described. Final Notes for AB and BC
• Limit evaluation chart: here
• Key words pdf here

What is a limit and how to find it:

Limit: If f(x) becomes close to a unique number L as x approaches c from either side, then
the limit of f(x) as x approaches c is L.

• A limit refers to the y-value of a function

* The general limit exists when the right and left limits are the same_equal each other

• DNE = does not exist.

Examples of estimating a limit numerically:

Example of using a graph to find a limit:

When limits don’t exist:
When the Left limit ≠ Right limit, then the limit is said to not exist.

• In the picture below, you can tell that the two limits don’t equal each other, thus theanswer to this limit is DNE.

Unbounded Behavior:

Evaluating Limits Analytically:

Limits Theorem:
Given:
Lim and Lim

Limits at Infinity:

* If m<n, then the limit equals 0

• If m=n, then the limit equals a_b

• If m>n, then the limit DNE

Finding Vertical Asymptotes
The only step you have to do is
set the denominator equal to zero and solve.

• Example:
  • (x+2)(x-2) = 0 → x = 2, -2
    • 2 is a removable hole while -2 is the non-removable vertical asymptote.

Finding Horizontal Asymptotes
Use the
two terms of the highest degree in the numerator and denominator

• Example
  • x and x2 are the two terms of the highest degree in the numerator and denominatorrespectively. After finding it, use the limits at infinity rule to determine the limit.

Intermediate Value Theorem
A continuous function on a
closed interval cannot skip values.
● f(x) must be continuous on the given interval [a,b]
● f(a) and f(b) cannot equal each other.
● f© must be in between f(a) and f(b)

Example #1: Apply the IVT, if possible on [0,5] so that f©=1 for the function

f(x)=x2+x+1

1. f(x) is continuous because it is a polynomial function.

1. f(a)=f(0)=1f(b)=f(5)=29

1. By the IVT, there exists a value c where f©=1 since 1 is between -1 and 29.Example #2:

Example #2:

- For 0<t<60, must there be a time t when v(t) = -5?

1. f(a) = f(0) = -20f(b) = f(60) =10

1. By the IVT, there is a time t where v(t)=-5 on the interval [0,60] since -20 < -5 < 10

The Squeeze Theorem

that means f(x) equals h(x) and g(x)

What is a derivative?

• Derivative: The slope of the tangent line at a particular point; also known as theinstantaneous rate of change.

• The derivative of f(x) is denoted as f’(x) or

Derivatives as Limits

Steps to find derivatives as limits:

1. Identify the form of the derivative first (look at the image above)… is it form a? b? c?

1. Identify f(x)

1. Derive f(x) using the corresponding equations next to each form

1. Plug in the “c” value if applicable

Rules of Differentiation

Derivatives of Trigonometric Functions:

* HINT: If the original function starts with C, then the derivative is negative!

• Example: cosx, cotx, & cscx

Derivative Rule for LN

* HINT: [Derive over copy]

• Example: h(x) = ln(2x^2 + 1)
  • First derive 2x^2 + 1. That would be 4x! And then put that over theoriginal function, which would be 2x^2 + 1.
  • Your answer would then be 4x_(2x^2 + 1)

Deriving Exponential Functions

Continuity

A function f is continuous at “c” if:

• The value exists- The value of the function is defined at “c” and f© exists

• The limit exists - The limit of the function must exist at “c”
  • The left and right limits must equal

• Function=limit. The value of the function at “c” must equal the value of the limit at “c”

Discontinuity

• Removable → discontinuity at “c” is called removable if the function can becontinuous by defining f©

• Non-removable → discontinuity at “c” is called non-removable if the function cannotbe made continuous by redefining f©

Differentiability

In order for a function to be differentiable at x = c:

• The function must be continuous at x = c

• Its left and right derivative must equal each other at x = c

Example:

The Chain Rule

The chain rule helps us find the derivative of a composite function. For the formula, g’(x)
would be the chain.

Example:

General Rule Power

We use the general rule power when finding the derivative of a function that is raised to the
nth power
. In the formula given, f’(x) is the chain.

Implicit Differentiation

Inverse Trig Functions: Differentiation

Example:

Derivatives of Inverse Functions

Higher-Order Derivatives

Particle Motion

• s(t) represents the position function, aka f(x)
  • t stands for time, s(t) is the position at a specific time.

• v(t) represents the velocity function, aka f’(x)
  • t stands for time, v(t) is the speed and direction at a specific time.
  • Velocity is the derivative of position.
    • A particle is moving to the right or up when velocity is positive.
    • A particle is moving to the left or down when velocity is negative.
    • A particle’s position is increasing when velocity is positive.
    • A particle’s position is decreasing when its velocity is negative.
    • A particle is at rest or stopped when its velocity is zero

• a(t) represents the acceleration function aka f’’(x)
  • t stands for time, a(t) is the rate at which the velocity is changed at specific times

• Example: s(t)=6t^3 -4t^2 → v(t)=18t^2 -8t → a(t)=36t-8

Particle Moving Away_Toward the Origin(x-axis)

* A particle is moving towards the origin when its position and velocity have opposite signs.

• A particle is moving away from the origin when its position and velocity have the samesigns,

Particle Speeding Up_Slowing Down

* A particle is speeding up (speed is increasing) if the velocity and acceleration have thesame signs at the point

• A particle is slowing down (speed is decreasing) if the velocity and acceleration haveopposite signs at the point.

Related Rates

What is the purpose of related rates?

• The purpose is to find the rate where a quantity changes

• The rate of change is usually with respect to time

How to solve it?

1. Identify all given quantities to be determined.

1. Make a sketch of the situation and label everything in terms of variables, even if you aregiven actual values.

1. Find an equation that ties your variables together.

1. Using chain rule, implicitly differentiate both sides of the equation with respect to time.Substitute or plug in the given values and solve for the value that is being asked for
   1. *Don’t forget to put the correct units!

The Different Types of Related Rates Problems

• Algebraic

• Circle

• Triangles

• Cube

• Right Cylinder

• Sphere

• Circumference

Related Rates: Algebraic

Example: A point moves along the curve y = 2x^2 - 1 in which y decreases at the rate of 2 units
per second. What rate is x changing when x = -3_2?

Related Rates: Circle

Example: The radius of a circle is increasing at a rate of 3cm_sec. How fast is the circumference
of the circle changing?

Related Rates: Triangle

Example: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a
rate of 2 ft_s, how fast will the ladder be moving away from the wall when the top is 5ft above
the ground?

Related Rates: Cube

Example: The volume of a cube is increasing at a rate of 10cm^3_min. How fast is the surface area
increasing when the length of an edge is 30cm?

Related Rates: Right Cylinder

Example: The radius of a right circular cylinder increases at the rate of 0.1cm_min and the height
decreases at the rate of 0.2 cm_mm. What is the rate of change of the volume of the cylinder, in
cm3_min, when the radius is 2cm and the height is 3cm?

Related Rates: Sphere

Example: As a balloon in the shape of a sphere is being blown up, the volume is increasing at a
rate of 4in3_s. At what rate is the radius increasing when the radius is 1 inch.

Related Rates: Circumference

Example: What is the value of the circumference of a circle at the instant when the radius is
increasing at 1_6 the rate the area is increasing?

L’Hopital’s Rule

Let f and g be continuous and differentiable functions on an open interval (a,b). If the limit of
f(x) and g(x) as x approaches c produces the indeterminate form 0_0 or ∞_∞ then,

Mean Value Theorem

If f(x) is a function that is continuous on the closed intervals [a,b] and differentiable on the
open interval (a,b),
then there must exist a value c between (a,b)

* Example: Confirm f(x)=x^3 on [0,3] and find a value that satisfies this theorem

Function Increasing or Decreasing

• f(x) is increasing when f’(x) is positive

• f(x) is decreasing when f’(x) is negative

Extreme Value Theorem
If f(x) is continuous on a closed interval [a,b], then f(x) has both a minimum and maximum on the interval.

(Absolute max on top and absolute min on bottom)

First Derivative Test

For first derivative tests, derive the function once and set it to 0. After that, find the zeros and
plug them into a number line. Using your derived function, plug-in numbers before and after
your constant (the zeros of the function) to see if it becomes negative or positive, as shown
below.

• If it’s positive, constant, negative then it’s a relative maximum

• If it’s negative, constant, positive then it’s a relative minimum

Concavity

• The graph of f is concave up when f’(x) is increasing

• The graph of f is concave down when f’(x) is decreasing

• If f’’(x) is positive then the graph of f is concave up

• If f’’(x) is negative then the graph of f is concave down

* Points of Inflection

• Occurs when f(x) changes concavity
• Determined by a sign change for f’’(x)

Second Derivative Test

Example:

Critical Numbers

• Critical numbers are points on the graph of a function where there’s a change in direction.

• To find critical numbers, you use the first derivative of the function and set it to zero.

Riemann Sums

You use Riemann sums to find the actual area underneath the graph of f(x).

Trapezoidal Reimann Sum

Integration

The area under the curve of derivatives of F from A to B is equal to the change in y-values of the
function F from A to B, given f is:

• Continuous in interval [a,b]

• F is any function that satisfies F(x)=f’(x)

What is an indefinite integration?
Given y' or f '(x), the anti-derivative can be thought of as the
original function, f(x). Integration
is
used to find the original function.

• The operation of finding all solutions to this equation is called antidifferentiation or
  indefinite integration
  .

• Detonated by an integral sign: ∫

* f '(x) = integrand

• dx = variable of integration

• f(x) = antiderivative

• c = constant of integration

• ∫ = integral

Reminder: ALWAYS add +C when you’re solving for an INDEFINITE integral!

Reminder: Differentiation and integration are inverses!

Basic Integration Rules (w_ examples)

Antiderivative Trig Function

C is still a constant when multiplied by 2 (a constant multiplied by a constant is still a constant)

• HINT: How I memorize antiderivatives by using derivatives of trigonometric functions
  • EX: d_dx sinx = cosx and for the antiderivative, you just switch the twotrigonometric functions and add +C since it’s an indefinite integration.
  • EX: d_dx cscx = -cscxcotx and for the antiderivative, just switch the twotrigonometric functions and add +c since it’s an indefinite integration. Also, if thederivative was negative, then the anti-derivative is also negative!

Integration by U-substitution

Natural Log Function for Integration (Log rule for integration)

Use this rule when ‘x’ becomes DNE

Integrals of the 6 Basic Trig Functions

* HINT: For ∫ tan u du, I memorized it like this: ∫ tan u du = ∫ sin u_cos u du because of the trigonometric identities. After that, I just did u-substitution with cos u being u.

• If you work it out, it looks like this:

Integration rules for “e”

● With e, it’s just the same thing as regular u-substitution but with the additional ‘e’.

Integration Rule for Exponential Functions

Differential Equations (Separate the integral)

Differential Equation with Initial condition

Slope Field

A visual depiction of a differential equation of dy_dx.

• Example of what a slope field looks like

Average Value

• To find the average value, integrate the function by using the fundamental theorem of
  calculus
  

• After that, divide the answer by the length of the interval

Total Displacement

• The difference between the starting position and ending position

• Interval [a,b]

• Can be negative

• Formula:

Total Distance

• Total distance traveled by a particle is the sum of the amounts it displaces betweenthe start, all of the stop(s), and the end.

• Distance can’t be negative

• Formula:

Area of a Region Between Two Curves

The Disk Method

If a region in the plane revolves about a line, the resulting solid is a solid of revolution, and
the line is called the
axis of revolution. The simplest solid is a right circular cylinder or disk,
which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle.

• Rotate Around x-axis
  • The horizontal axis of revolution
• 
• Rotate Around y-axis
  • The vertical axis of revolution
• 

The Washer Method

• Horizontal Line of Rotation:

* Vertical Line of Rotation

The Washer Method Calculating Volume Using Integration

Step One: Draw a picture of your graph→ shade appropriate region

Step Two: Identify whether you are rotating about a vertical or horizontal line

• Vertical
  • Get everything in terms of y

• Horizontal
  • Get everything in terms of x

Step Three: Set up your Integral

Step Four: Simplify

Step Five: Integrate Definite Integral

• Parametric functions: a pair of functions that allow you to find the coordinates of both x

and y, usually written in terms of ‘t’

• Parametric equations are usually written in terms of ‘t’.
• Calculating the derivative of parametric functions:

• Calculating the second derivative of parametric equations:

• Using the example from above, to find the second derivative of that parametric function, take the derivative of $\frac{dy}{dx} \rightarrow \frac{d^2y}{dx^2}=\frac{(3/2)}{2t}=\frac{3}{4t}$
• Finding the arc length of parametric curves:
• 
  • From t=𝛼 to t=𝛽
• Ex. (from Princeton book): Find the length of the curve defined by x=sin(t) and

y=cos(t), from t=0 to t=$\pi$

• 1) find (dx/dt) and (dy/dt) → $\frac{dx}{dt}=cos(t)$ and $\frac{dy}{dt}=-sin(t)$
• 2) plug into the formula →
• 
• Vectors
• Vectors are quantities that have both a magnitude and direction
• Vector functions are similar to parametric functions because they describe the

motion of a particle on a plane

• Vector functions have similar relationships between position, velocity, and

acceleration as we’ve seen earlier.

• If a vector function P = position of a particle, then the derivative of P gives

you velocity, and the derivative of velocity gives you the particle’s acceleration

• Both x(t) and y(t) have to be differentiable for the vector function

to be differentiable as well

• To find speed of a vector function, take the magnitude of the

velocity vector function

• The magnitude of a vector function = $\sqrt{x^2+y^2}$
• Denotation of vectors:
• 
• ‘i’ and ‘j’ are the unit vectors in the x and y directions
• x(t) and y(t) are real valued functions of the variable ‘t’
• Ex. (from Saxon Calc BC book):
• 
• Integrating vector valued functions:
• Similar to how you take the derivative of vector valued functions by

differentiating x(t) and y(t), and leaving the ‘i’ and ‘j’, to find the integral of vector-valued functions, integrate the real-valued functions and leave the ‘i’ and ‘j’

• Integrating a velocity vector gives you the displacement of the particle

over the interval of time, and integrating a speed vector gives you the total distance travelled by the particle in that interval.

• Ex.:
• 
• Polar curves:
• Polar coordinates: used to describe the location of points on a polar curve/graph
• x = rcos𝜃and y = rsin𝜃
  • *x and y are defined parametrically, in terms of ‘𝜃′
• Differentiating in polar form:
  • To find $\frac{dy}{dx}$, use the equation for finding the slope of a parametric equation → $\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta}$
  • 
• Finding the area of a single polar curve:
• 
• 
  • *to find polar area, you will often need to know these trig

identities:

• 
• To find the limits of the integral, find the𝜃points where r consists of that region.
• In this case, it takes from 0 to 2𝜋for “r” to be drawn completely, which is why those are the limits
• Finding the area of a region bounded by 2 polar curves:
• A = $\int_a^b \frac{1}{2}(r_2^2-r_1^2)d\theta$ with $r_2(\theta) > r_1(\theta)$
• Ex. (from Princeton book): find the area inside the circle r = 4 and outside of the curve r = 4 - cos𝜃(the shaded region in the graph below)
• 
• 1) To find the limits of the integral, set the two equations equal to each other, and solve for 𝜃→ 4 = 4 - cos𝜃 = cos𝜃 = 0 at 𝜃= -𝜋/2 and 𝜋/2
• 2) find which function is on the “outside”, and in this case, r = 4 is
• 
• 3) integrate
• Connecting polar, vector, and parametric functions:
• A function can be written in parametric, rectangular/Cartesian, or polar form
• Ex. (on the next page):
• 

• Difference between sequences and series:
  • $a_n$ = sequence
  • S = $\Sigma_{n=1}^\infty a_n$ = series
  • Sequences are lists of the terms, whereas series are lists of the addends of the series.
    • Series: to find the second term of the series, you ADD $𝑎_1$ and $𝑎_2$ together
      • This is known as partial sums → 2nd partial sum = $S_2 = a_1 + a_2$
        • nth partial sum = the sum of the first n terms of the series
    • Convergence/Divergence:
      • A series converges only if the SEQUENCE OF PARTIAL SUMS (𝑆𝑛) converges to a finite number. If it doesn’t converge, then it diverges.
        • 
• Geometric series:
  • Defined as a series with a constant ratio between each term
  • 
  • If |r| < 1, then the geometric series converges to $|frac{a}{1-4}$
    • 
  • If |𝑟| > 1, 𝑜𝑟 = 1,then the series diverges
    • 
  • The nth term test for divergence:
    • Used to determine whether or not a function DIVERGES; it CANNOT tell you if a function converges
    • 
    • 
  • Integral test for convergence:
    • 
  • Harmonic series and p series:
    • 
    • 
    • Alternating series for convergence:
      • This is used to test a series that alternates from negative↔positive
        • 
      • 
    • Ratio Test for convergence:
    • 
    • 
    • Absolute convergence:
      • If an alternating series converges after taking the absolute value of the series function, then that series is said to converge absolutely
        • 
      • 
    • Conditional convergence:
      • If an alternating series converges, and does NOT converge absolutely, then it’s said to converge conditionally
      • An alternating series that conditionally converges must follow these conditions:
        • 
    • Alternating Series Error Bound:
      • 
      • 
  • Taylor polynomials:
    • 
    • 
  • Lagrange Error Bound
    • 
    • 
  • Radius and interval of convergence of power series:
  • 
    • You have to check the endpoints of the interval to see if the function converges at the endpoints! If it does converge at an endpoint, place the interval in brackets, and if not, place the interval in parentheses.
      • If it converges at 1 endpoint and not the other, denote it like this: [a,b) or (a,b], with the bracket indicating convergence at that endpoint
  • Radius of convergence:
    • Similar to how the radius of a circle describes the length from the outer edge of the circle to the centerpoint, the radius of convergence describes the distance between the midpoint and the endpoint of the interval of convergence.
    • In the example above, the interval of convergence was [-1,1]. Thus, the radius of convergence is 1, since the midpoint of that interval is 0, and the distance between 0 (the midpoint) and 1 (the endpoint) is 1.
  • Representing functions as Power series:
    • Differentiation of power series
      • 
    • Substitution of power series
    •