AP CALC AB Review

What is a Limit?

The y Value of a function as x approaches some given value

lim x → 0 sinx/x

1

lim x → 0 1-cosx/x

0

Squeeze Theorum

A method used to find limits of functions by squeezing them between two other functions that have the same limit at a given point.

What are the three types of Discontinuties

1. Hole: At a point where a function has a limit but it does not have a limit at that point

2. Jump: When it jumps and there is a gap between two points

3. Infinite: Never able to reach each other like an asymptote

What is the IVT?

How to find continuity with Piece Wise Functions?

Check limits from both sides and ensure they equal the function's value at the point of interest. If they are not equal then it is not continuous.

What is the Tangent Line or Instantaneous Rate of Change?

A slope at one point of the function

Equation: y-y1 = m ( x- x1)

What is the secant line or AROC?

The secant line represents the Average Rate of Change (AROC) of a function over an interval, calculated by finding the slope between two points on the curve.

Equation:

Requirements for Differentiability

1. Continuity (has to agree on both sides, (a,b)

2. No corners or cups

3. It can not have a vertical tangent line meaning its slope is infinity at the point (think of an x³ graph except vertical)

F’ Graph:

Vertical tangent line - Vertical Asymptote

Corner - Jump discontinuity

Turning point - F’ = 0

DIfferentiation Rules

How do you find the Normal Line?

Find the Tangent line and then do the opposite reciprocal of the slope for the tangent line.

Derivative of a^x

a^x lna

Chain Rule

[f(g(x)] = f’(g(x)) times g’(x)

Inverse Derivatives

What is Inverse Derivative for sin^-1(u)

What is Inverse Derivative for cos^-1(u)

What is Inverse Derivative for tan^-1(u)

What is Inverse Derivative for sec^-1(u)

What is Inverse Derivative for csc^-1(u)

What is Inverse Derivative for cot^-1(u)

Derivative of logax

1/xlna

Position Time Graph

When you go up you’re moving forward or to the left

When you go down you are going backwards or to the right

Velocity time graph

Positive Velocity, moving right

Negative velocity, moving left

Acceleration time graph

Take the slope of the velocity graph

Linear Approximation

Find Tangent Line and then plug in the x-value you want to find

Mean Value Theorum

If the function is continuous and differentiable then there is a number c on the open interval from (a,b) and a<c<b

To find it find the derivative of the function using AROC and then find out when the function = AROC

Extreme Value Theorum

If a function is continuous on [a,b] then it has both an absolute minimum and max in the interval

Critical points: When the function has a slope of 0

Set slope equal to 0 and then solve for X and those are the critical points

First Derivative Test

Find Critical Points and then Put them on number line and plug numbers into the derivative before and after to see where the intervals are increasing and decreasing

Second Derivative Test

Get second derivative and test values on both sides by plugging into the second derivative

How to find volume of a area under the curve (disk method)

Washer method

Cross section semi circle

Cross section equilateral triangle