Review Book: Chapters 1-4 at the tip of your fingers

Review Book: Chapters 1-4 at the tip of your fingers Authors Diana Mendez Xyna Ronquillo

Introduction Hello. My name is Diana Mendez. I’m 16 and a junior in High School for Environmental Studies. Some people calls me Didi Bear (well only Xyna .)))) )In the future I want to become an environmental engineer. I currently live in New York with my parents and 2 older brothers. When I’m not helping Ms. Zhao, I go to work or f\go to the boxing gym…so don’t mess with me O;<

Hi! My name is Xyna Ronquillo. A few people(mostly teachers) calls me Xyna the Warrior Princess because my name came from a show (Xena the Warrior Princess). I’m 16 almost 17 (in 2 days) and currently a junior in HSES. In the future I want to become a veterinarian because I’m a huge animal lover so why not help them! I live in New York with my mother, sister and 10 babies (my pets)!! When I’m not in school…chances are I’m either sleeping or eating! Yum Yum (:

Chapter 1: Limits And Continuity

Chapter One: Limits and Continuity Definition To better understand limits, let’s work an example. Suppose you had the function The function tells us that when x is equal to 1, the function does not exists. Limits can be used to define the function. If we place numbers that approach 1 from the left and from the right, the value of the function results in values close to the limit. • A limit is defined as the value of a function as it approaches a specific x-value. • The limit may or may not exist. • A limit exists when… exist exist

Chapter One • The function tells us that when x is equal to 1, the function does not exists. • Limits can be used to define the function. If we place numbers that approach 1 from the left and from the right, the value of the function results in values close to 1. • For this specific problem, limit is written as: Using the chart above, we can infer that as the function approaches to x=1, the limit will be about 3.

Graphical Understanding Limits Point C Point A illustrates a limit because the limit from the left and the limit from the right meet at the same point, 1. So the right and left limits equal each other. Point B Points B and C each illustrate one-sided limits because the right limit doesn’t equal the limit from the left. There is no limit as x approaches 2. Point A

Numerically Understanding Limits • Suppose you are given the limit • In order to solve this we’ll have to do the following steps: First factor. Simplify. Substitute

How Limits Impact the Continuity of a Function • A function is continuous when • f(c) exists • f (c) equals • A function is not continuous when there are gaps, holes, or jumps. Ex:

Chapter 2: Derivatives

Chapter 2: Derivatives Definition Derivative Formula • A function is differentiable if it is defined on an open interval, c, and if the limit exist, then the line passing through (c,f(c)) with slope m is the tangent line of function f. • The derivative of f(x) is given by • Different notations of derivative • f’(x)

Finding Derivative by Limit Process • Find the derivative of by using the limit process Substitute into f(x) Simplify. Substitute

Finding derivative by using rules • Find the derivative of • Find the slope of f’ at x=2 Original function First do the power rule/chain rule and constant rule Thensimplify

Critical Numbers and Point of Inflection • Derivative can be used to find critical numbers and point of inflection. • A critical number is the point where the derivative changes from increasing to decreasing or vise versa. Critical numbers can be found when the first derivative is set equal to 0 and we solve for x. But to make sure it is a critical number the 1st derivative has to be done to make sure that the x-value is the point where the derivative changes from increasing to decreasing and vise versa. • A point of inflection is the x-value where the graph changes concavity(concave up to concave down or vise versa). To find the point of inflection, the second derivative must be equaled to 0 and then solve for x. In this case, the 2nd derivative test must be done to assure that there is a change in concavity.

Derivative in graph • As seen in this graph, when , the derivative graph is below the x-axis because it indicates that at these intervals f(x) is decreasing. • This means that when −1 ≤ 𝑥 ≤ 2, f’(x) because it is below the x-axis. • In this case the critical point is 0.5 because:

Chapter 3: Anti-Derivatives

Anti-derivatives • Suppose you are told to find the derivative of • You will find that the derivative of each expression is . Now consider the question: is the derivative of what function? You will find that all the functions above will correctly answer the question, no matter what constant is added. This means that the function that will always satisfy the question is x^3+C. The C in the equation represents any constant that maybe added to the function. • A simpler way to express when to find the anti-derivative of a function is using the integration sign, ∫dx. • So the previous example can be written as • The whole setup above is known as indefinite integral.

Common Integration Rules

Indefinite Integral Definite Integral A definite integral is when the integral is restricted to specific points or limits. Basically represents the area underneath the curve between two endpoints. It’s express as The arepresents the lower limit or smaller number and the brepresents the upper limit or larger number. In terms of definite integrals, They never require an addition of a constant (+C) f (x) is continuous on [a,b] • An indefinite integral is not restricted by lower or upper limit. • It is expressed as • It is used to simply find the ant- derivative and always contains +C.

Fundamental Theorem of Calculus 1stFTC 2nd FTC f (x) is continuous on the closed interval [a,b]so that g(x)= when continuous and Example: f(x) = • When f (x) is continuous on the closed intervals [a,b]and f(X) is the integral of f(x), then • Example:

Approximation of Definite Integrals • All sub-intervals must be of equal value • Left Riemann Sum: Use the left endpoint of each sub-interval • Right Riemann Sum: Use the right endpoint of each sub-interval • Trapezoid Sum: • Midpoint Sum:

Sums Left Hand Sum Right Hand Sum Midpoint Sum Trapezoidal Sum

Accumulation Problem • What is the position of the particle at time, t=4? • =2 • C=2 • What is the velocity of the particle at time, t=2? • v (t) = • v (2) = =16+16(8)=16+128=144 Q. A particle is moving at a velocity of v (t) = . The initial condition is given as s (0)= 2. • What is the acceleration of the particle at time, t=6? • v (t) = • a (t) = • a (6) =

Chapter 4: Application Problem

Calculus Related to Physics • Didi bear and Papa bear were driving their private jet air planes at a constantly changing velocity v(t) because they were avoiding clouds. (The position of x is measured in miles, and time t is measured in hours.)The velocity of the jet airplanes is recorded for selected values of t, in hours, over the intervals 0≤t≤5. • Find the acceleration at t=2.5. Include units. • Write but do not evaluate an integral that gives the average value of the velocity for 0≤𝑡≤5. • For 0≤t≤5, what is the least amount of times that Didi bear and Papa bear changed direction. Justify your answer.

Steps by Steps: • For 0≤t≤5, what is the least amount of times that Didi bear and Papa bear changed direction. Justify your answer. • Between t=1 and t=2 there was a change of direction because the velocity went from positive to negative. Between t=2 and t=3 a change of direction occurred because the velocity went from negative to positive. Between t=4 and t=5 there was also a change of direction because the velocity went from positive to negative again. This means that Didi Bear and Papa Bear changed directions 3 times. • Find the acceleration at t=2.5. Include units. • Write but do not evaluate an integral that gives the average value of the velocity for 0≤𝑡≤5.

Hope You Enjoyed it !

Bibliography • http://www.dummies.com/how-to/content/how-to-approximate-area-with-midpoint-rectangles.html • http://www.zweigmedia.com/RealWorld/summarypic/cs5_9.gif • http://www.mesacc.edu/~davvu41111/Riemannimages/RiemannStudent16.gif • https://www.math.duke.edu//education/ccp/materials/diffcalc/accumul/area.gif • https://www.somath.com/calculus/diff/der12/der12.html

Review Book: Chapters 1-4 at the tip of your fingers