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 Martinez And Mariah Apollon *results may vary

Table of Contents Chapter 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

CHAPTER 1 CONTINUED 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)

CHAPTER TWO: DERIVATIVES What 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 FunctionDerivative f (x)f '(x) Constant const0 Linear x1 Power x a a x a-1 Exponential e x Exponential a x a x ln a Natural logarithm ln(x) Logarithm log b (x) Sine sin xcos x Cosine cos x-sin x

CHAPTER 3 ANTIDERIVATIVE EQUATIONS

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 Fields You 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

CHAPTER 6B: SEPARATION OF VARIABLES

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 – x 2 and the x-axis between the values x = –1 and x = 2. EXAMPLE:

About the Author: Ashley Martinez 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!!!