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QUEST TO CALCULUS

QUEST TO CALCULUS. By Ivan Del Aguila & Alondra Espino. ABOUT THE AUTHORS. Alondra Espino

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QUEST TO CALCULUS

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  1. QUEST TO CALCULUS By Ivan Del Aguila & Alondra Espino

  2. ABOUT THE AUTHORS Alondra Espino She is 17 and lives in NYC with her 40 cats and 10 dogs. She is a top athlete who loves to run and swim and relieve stress from all the calculus that she does. She plans to pursue her scholarly endeavors at The University of Wisconsin- Madison on a full scholarship and plans on becoming a successful business woman. Ivan Del Aguila (The Eagle) 17 years young, Ivan lives in NYC. A top athlete in his own right, he loves soccer and plays constantly. He is a gifted musician and plans to pursue mastery at Hunter college this fall. He hopes to be an entrepreneur in many roles of society. Both took the AP Calculus exam this year and managed to do very well despite the reputation of the test being a hard one to pass. They love cats, dogs and music. They also love math, especially Calculus. They are the best of friends. Be on the look out for more of their literary works such as: • I’m A Mexican, Why Aren’t You • A Dog Named Cat • Where is My Guitar • I Can Beat You In A Race

  3. TABLE OF CONTENTS About the Authors Page 2 Chapter 1: Limits And Continuity Pages 3- 7 Chapter 2: Derivatives Pages 8- 12 Chapter 3: Antiderivatives Pages 13- 17 Real World Applications Pages 18- 19

  4. CHAPTER 1: Limits & Continuity Welcome to calculus 101 part 1. Here, you will go over what are commonly known as limits and how to locate them when chance presents itself. Let us begin. What in the World is a Limit? In mathematics, the limit is the value that a function or sequence approaches as the input or sequence approaches a particular value. Limits are essential in calculus as we all know and they will help us define continuity, derivatives, and integrals.

  5. FINDING LIMITS ALGEBRAICALLY So, what do we do when we get a question that asks us to find a limit without the help of a graph or calculator? Well there are three steps to follow.   • First, we must plug in the value of x into the equation. Sometimes, the answer is straightforward and we will get an answer as soon as we plug in for the value of x. sometimes, however, we will get no answer, such as a zero in the denominator. In other words, the answer does not exist. • Second, we will factor as much as possible and cancel out. This will help mostly when we are looking for a limit of a fraction. Most of the time we will be able to factor both the numerator and denominator and cancel out. • Finally, we will finish by substituting in the value of x again. Hopefully, this will bring us to the answer that we have been looking for. Either that or the final answer will be -∞, ∞, or DNE.

  6. FINDING LIMTS AS X INFINITY There will come a time when those crazy test makers ask you to find the limit of a function as x approaches to infinity. While the concept of infinity may be unfathomable to you, it is quite easy to do what those test makers ask. There are some basic shortcuts to follow.   • If the highest power of x appears in the denominator, then the limit is Zero • If the highest power of x appears in the numerator, then the limit DNE. • If the highest power of x appears in both the top and the bottom, then the limit is equal to the coefficients of those x values.

  7. FINDING LIMITS GRAPHICALLY Limits only get easier when we are able to use a graph to find it. So say they ask us to find a limit and they give us a graph. There are two rules to find the limit. • First, find the limit from the left and find the limit from the right. This is very important because sometimes those darn test makers will give you a graph and ask you to find a limit at a point that is not defined. So you will have to approach the point from the right and then from the left getting closer and closer to a value, which will be your limit. • Second, your left and right limits must equal. You must make sure that the limits from the right and from the left are the same. This is important because the next step will be finding the continuity of a function. Sometimes, those test makers will give you a graph that is not continuous, this will also mean that at that un-continuous point, there is no limit because the right and left limit are not the same.

  8. FINDING CONTINUITY What do you do when they ask you if a certain function is continuous? This is relatively easy to do when you have a graph. You must make sure that the graph does not have: • Jumps • Breaks • Holes If the graph does not have that, then the graph is said to be continuous. • But what if the question asks you to solve it algebraically? This will mostly be true if the question asks if said graph is continuous at a specific point. So say they ask you if the graph of f(x) is continuous at point c. Follow these three rules to find out. • First, does f(c) exist? Sometimes, there will be a hole in the graph at point c and this easy test of just plugging in for x and seeing if the answer exists or not can let you know if it is continuous. • Second, does the limit of f(x) as x approaches c exist? This is where the testing of the limit from the left and limit from the right helps a bunch. Test makers will give you a graph that has a limit that exists from both the left and from the right but those values aren’t equal. This means that the graph probably has a jump or break. • Finally, does number 1 and number 2 equal to each other? This I pretty much straight forward. If they do, then they are continuous. If they don’t then they aren’t. Simple right?

  9. CHAPTER 2: DERIVATIVES Deriva-what? What are derivatives? A derivative is a measure of how a function changes in accordance to how an input changes. Without much more to say let’s continue reviewing When is There a Derivative? Sometimes, those test makers will ask if the derivative exists at a certain point. Or the questions will need you to determine if there is a derivative at that point in order to answer the question. So when do the derivative not exist? • In a corner or cusp. Here is where two lines of different slopes intersect. The derivative or slope cannot be equal to two different slopes. • In the vertical tangent of a graph. The slope of a graph cannot equal infinity. I mean, it can equal anything it wants but for our intents and purposes, it won’t exist. • In a point of discontinuity. This is the same concept as that of number one. There cannot be two slopes at the same x-value.

  10. DEFINITION OF DERIVATIVE What is the “Definition of a Derivative Through Limit Process”? Very often on the test, we will get a question that asks us to give the formal definition of a derivative using limit process. Well here is the answer. You may be asking yourself if we know a way to learn and memorize this. Well guess what? We don’t so start reciting this and learn it for that AP. Good Luck!

  11. DERIVATIVE RULES Rules for Differentiation. When we are asked to find a derivative, we are also being asked to differentiate. They are both one and the same. Don’t be scared of the ever-elusive math terms that those test creator’s use. Now we have a couple of rules that are used when we differentiate. Remember them and you will forever get the right answer. • The derivative of a constant is zero. This is very similar to the “anything to the zero power is 1”. Very straightforwardly, if there is not ambiguous value such as an x, then the derivative is zero. • The power rule. Very simple once you get the hang of it. The derivative of a function is when we have an x raised to an n value. We drop the n value to the front and subtract one from the n value of the exponent. So the derivative of x^3 is 3x^2. Simple right? • The sum and subtraction rule. We know how those lovely test makers love to make stuff difficult. It is also overwhelming when they ask us to find the derivative of a function with multiple terms. Don’t be scared. This rule tells us to treat each one as a separate function. So if we have f(x)+g(x), find the derivative of each one independently. • The quotient rule. Here is where it gets a little long but actually not hard. The quotient rule is only necessary when we have a fraction function in its simplest form. This is what we do. Low-Derivative of high minus high-Derivative of low all over low-low. All the dashes represent when we must multiply. Quite more simply, LoDiHi-HiDiLo/LoLo. Simple. • The last rule that will help you a whole lot is the chain rule. This is most useful when we have a composite function. Say they ask us to find the derivative of f(g(x)). We must find the derivative of the outside function and work our way in, finding the derivatives of those pesky inside functions. • Finally. There are a lot of trig function derivatives that you must memorize. Good luck.

  12. RULES, SIMPLIFIED Just in case you didn’tunderstand that last slide…

  13. IMPLICIT DIFFERENTIATION Lastly and Probably Not the Easiest: Implicit Differentiation. Implicit differentiation is where you differentiate a function with 2 variables. Here are a few simple steps to do this the correct way. • Differentiate the x values first. They are done the exact same way as they always have been. • Then you differentiate the y values. After each time you differentiate a y value, you must add dy/dx because that is showing differentiation. • When you run into a term that is both y and x values, you must use a quotient or product rule. Then you must combine both steps one and two. • Finally we must rearrange the function and move all the dy/dx to one side. This is probably very easy. Finally, we have reached the end of a productive derivative review. Calculus can only get easier if we learn these basics. Have fun studying and good luck. We must move on to Integrals. The integral part of getting a 5 on that darn test in May.

  14. CHAPTER 3: Antiderivatives You are very familiar with taking the derivative of a function. Now we are going to go the other way around, if given a derivative of a function, can you come up with a possible original function. In other words, taking the anti-derivative! As the name suggests, anti-differentiation is an “undoing” of a differentiation. What exactly ARE anti-derivatives? An anti-derivative of a function f is a function whose derivative is f. In other words, F is an antiderivative of f if F' = f . To find an antiderivative for a function f , we can often reverse the process of differentiation. For example, if f = x4, then an antiderivative of f is F = x5, which can be found by reversing the power rule. Not only is x5 an antiderivative of f , but so are x5 + 4 , x5 + 6 , etc. In fact, adding or subtracting any constant would be acceptable.

  15. INDEFINITE & DEFINITE INTEGRALS Definite integral • Definite Integrals are written as ∫f(x)dx lower limit b, upper limit a which is equal to the signed area under the curve on the interval [a, b]. These can be evaluated by the fundamental theorem of Calculus. Indefinite Integral The notation used to refer to antiderivatives is the indefinite integral, ∫f(x)dx. This means the antiderivative of f with respect to x. If F is an antiderivative of f, we can write ∫f(x)dx= F+c, where c is the constant of integration.

  16. FTC, LEFT, RIGHT, MIDPOINT & TRAPEZOIDAL SUMS Fundamental Theorem of Calculus- a theorem that provides a way to calculate definite integrals. It states: If F is an antiderivative of f, then ∫ lower limit a, upper limit b f(x)dx= F(b) - F(a) Riemann Sums- A method of approximating the area under a curve by dividing the region of interest into rectangles whose individual areas are calculated and summed to approximate the whole.

  17. CONTINUED… Left Hand Approximation- A type of Riemann sum in which the rectangles are of equal width, and in which the value of the function at the left endpoint of each subdivision is used as the height of the corresponding rectangle. Right hand Approximation- A type of Riemann sum in which the rectangles are of equal width, and in which the value of the function at the right endpoint of each subdivision is used as the height of the corresponding rectangle. Midpoint Approximations- A type of Riemann sum in which the rectangles are of equal width, and in which the value of the function at the midpoint of each subdivision is used as the height of the corresponding rectangle. Trapezoidal Rule- This is another method of calculating areas under curves by making subdivisions. Instead of using rectangles, the trapezoid rule uses trapezoids to generate the area of a desired region.

  18. TO MAKE YOUR LIFE A LITTLE EASIER 

  19. REAL WORLD APPLICATIONS • Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed. This research can help increase the rate of growth of necessary bacteria, or decrease the rate of growth for harmful and potentially threatening bacteria. • A physicist uses calculus to find the center of mass of a sports utility vehicle to design appropriate safety features that must adhere to federal specifications on different road surfaces and at different speeds.

  20. REAL WORLD PROBLEM This problem would apply to the physics departments because many physicists work with these types of vehicles on daily basis. When Neil Armstrong went to the moon there must have been a strong emphasis on how fast the spaceship was traveling, where calculus was a must. The use of calculus is important in physics for safety measures, experimental measures and more. They conduct experiments and many times have to apply Calculus when trying to figure out the velocity or acceleration of a certain object that they are using.

  21. REFERENCES http://www.math.com/tables/integrals/tableof.htm http://math.about.com/od/formulas/ss/calculusform.htm http://www-math.mit.edu/~djk/calculus_beginners/chapter01/section02.html http://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx

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