The Secrets Behind Calculus

The first book to bring you tips and helpful, easy to understand calculus

topics in simple English to guarantee a higher AP Calculus

Exam Grade

By: Ashley MartinezAnd

Mariah Apollon*results may vary

Table of ContentsChapter 1:

Limits and Continuity

Chapter 2:Derivatives

Chapter 3:Antiderivatives

Chapter 4: Related Rates

Chapter 6a: Slope Fields

Chapter 6b: Separation of Variables

Chapter 7: Area Under the Curve

About the Authors

CHAPTER 1: LIMITS AND CONTINUITY What is a Limit? A limit occurs when x doesn’t have an value for f(x) (or it’s “Y” coordinate) however when the graph is observed, there is a value that is approached and this is called a “Limit” . A limit is defined as

And is written out as “ the limit as x approaches a , f(x) approaches f(a) But for a limit to exist, it needs to approach the same value from the left and front the right (from both directions it must be equal)

Example:

From -55, we can observe that the graph is continuous. For example at x=3, f(x)=11 from both directions so we would write “ the limit as x approaches 3, f(x) approaches 11.”

f(x)=2x+5

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

𝑦=2 𝑥+5

CHAPTER 1 CONTINUED However, a limit is not always continuous, which means it’s discontinuous

.At the value of x, there are more than one value for f(x)

.From the left and right it approaches a value however it is not defined at that point(simply there is an open circle or gap at a point)(it still is considered continuous however it is not differentiable, which will be covered in another chapter)

f(x)=2/(x-2)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

x

y

At x=2, we can see that the function as x approaches 2, the function has a vertical tangent at f(2), there fore the function is discontinuous at x=2

The function at x=-2 is not continuous as it has an open gap at x=-2 however it is continuous as from the left and right it approaches -5.

CHAPTER TWO: DERIVATIVESWhat is a derivative? A derivative can be defined as the slope of a line or function at a particular point. Also known as the velocity, or the rate of a function, the derivative can be demonstrated using two different equation forms:

Definition of Derivative as a limit:

Definition of derivative as a slope:

EXAMPLE:

CHAPTER TWO: DERIVATIVES (CON.)

Rules for finding the derivative:

The derivative graph shows the rate at which something is increasing and decreasing at a given point. For example: When the graph is at four units the graph is increasing at a rate of three units.

Function name

Function

Derivative

f (x) f '(x)

Constant const 0

Linear x 1

Power x a a x a-1

Exponential e x e x

Exponential a x a x ln a

Natural logarithm ln(x)

Logarithm logb(x)

Sine sin x cos x

Cosine cos x -sin x

CHAPTER 3ANTIDERIVATIVE EQUATIONS

First, you have to know that a function (y=f(x)) is known as a solution.

For example, we know how to antiderive for 2x

This would be known as the general solution as it will show you every value for c

But what if we want to find the particular solution?

Particular solution is obtained from an initial condition where you are finding the function at a particular coordinate pair and only shows you one function for a specific value

For example, the general solution for 2x would be +c (accounting for any value of c (for example) where c equals 1:2:3:4)

Now lets say we were given the particular solution (3,1) we easily substitute 3 for x and 1 for y to find the value of “c”

So our particular solution for (3,1) would be y=x²-8

X

Y

-2.5-2.25 -2 -1.75-1.5-1.25 -1 -0.75-0.5-0.25 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

-9.6-8.64-7.68-6.72-5.76-4.8

-3.84-2.88-1.92-0.96

0.961.922.883.844.8

5.766.727.688.649.6

0

Created with a trial version of Advanced Grapher - http:/ /www.alentum.com/agrapher/

CHAPTER 4: RELATED RATES

Related Rate problems deal with use of a primary and secondary equation.1. One must use the primary

equation given by the object described in the original problem.

2. One must identify the secondary equation and manipulate it accordingly for the problem.

3. Substitute the secondary equation back into the primary equation to solve for the desired value.

Chapter 6a: Slope Fields

So a slope field is a ‘direction’ field for a differential equation of the form F’(x, y). A slope field that consists of small line segments at the coordinates for a function F(x, y), and these line segments are the tangent slopes at the particular coordinate.

For example, lets sketch the slope field for y’=x+y

Slope FieldsYou can also use slope fields for particular solution Let’s find the particular solution of y’=x+y at (1,1)y’=(1)+(1)y’=2

X

Y

-2.5-2.25 -2 -1.75-1.5-1.25 -1 -0.75-0.5-0.25 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

-0.6-0.54-0.48-0.42-0.36-0.3

-0.24-0.18-0.12-0.06

0.060.120.180.240.3

0.360.420.480.540.6

0

Created with a trial version of Advanced Grapher - http:/ /www.alentum.com/agrapher/

X

Y

-2.5-2.25 -2 -1.75-1.5-1.25 -1 -0.75-0.5-0.25 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

-0.6-0.54-0.48-0.42-0.36-0.3

-0.24-0.18-0.12-0.06

0.060.120.180.240.3

0.360.420.480.540.6

0

Created with a trial version of Advanced Grapher - http:/ /www.alentum.com/agrapher/

CHAPTER 6B:SEPARATION OF VARIABLES

Steps to Finding Particular Solution Using Separation of Variables:

1.Combine like terms( move variables with respect to y to one side and variables with respect to x to another) and we will find the anti-derivative of both sides.

2.After anti-deriving, get y by itself

*Since C is a constant they can be combined to make one value for C

3.Substitute values for x and y and solve for C

4. Using the found solution, write the particular solution

Now let’s say we were given the differential equation (the slope) but not the general equation, how do we solve for a particular solution using the differential equation?

Let’s use the example At the initial condition (-1,2)

CHAPTER 7:AREA BETWEEN THE CURVES

The are between the curves can be defined as the definite integral between the top function (the one farthest from the axis) and the bottom function (the one nearest to the axis). And is represented by the function below.

Find the area between y = 7 – x2 and the x-axis between the values x = –1 and x = 2.

EXAMPLE:

About the Author: Ashley Martinez

Click icon to add picture Never being the typical “math nerd”, Ashley wasn’t always the top student in her math classes, yet she was able to grasp the concept of any chapter she learned. Putting her skills she attained from her AP Calculus AB class, Ashley was able to take the concepts most students have difficulty with and write them in a manner that anyone can understand. Currently being a Junior in High School, Ashley is aspiring to head to University( most probably in Florida) and have one day have a master’s or Ph.D. in Forensic Science(minoring in Psychology)

About the Author: Mariah Apollon

Having skipped a grade, Mariah Apollon was always considered a top student in all of her math classes. Since a child Mariah has always wanted to be an astronaut, and although her interest in physics had always been low she figured her ability to do math well in addition to her interest in geochemistry would be enough to carry her along that path anyway. Currently being a Junior in High School, Mariah Apollon plans on attending Rutgers University- New Brunswick to study chemical engineering. She is a spunky, hilarious, educated, very weird teenager who is really excited about college!!!