Title Unit 1 Notes - Limits and Continuity - 2024.pdf ✅

THINK ABOUT IT The concept of average velocity is rather intuitive. For example, the distance from Mr. Record’s driveway in Avon, IN is exactly 700 miles from the Florida/Alabama border. If it takes him 10 hours to make the trip, what is his average velocity? What assumption could you make about Mr. Record’s driving if the above assumption is true? Example 1: Suppose a ball is dropped from the top story of the Burj Khalifa in Dubai, UAE, 2717 feet from the ground. Find the velocity after 5 seconds. By conducting experiments in the late 1500’s from atop the Leaning Tower of Pisa, the Italian scientist Galileo Galilei discovered that the distance traveled by a freely falling object is directly proportional to the square of the time it has been falling. In this case, the distance the stone has fallen, s(t), after t seconds, can be given by 2 s t t ( ) 16  For centuries, it was always possible to compute the average velocity of this ball given a time interval. a.) Use your calculator to compute the average velocity of the stone over the time interval t = 1 second to t = 5 seconds. For centuries, computing the instantaneous velocity at a specific time, t, was impossible. However, scientists and mathematicians could compute the instantaneous velocity of the falling object by selecting a time interval very close to the time at which they want to compute the instantaneous velocity. For example, we could compute the instantaneous velocity of this stone at t  5 by computing the average velocity over a short time interval, for example t  5 to t  5.1. b.) Use your calculator to obtain three other estimates that are progressively more accurate for the instantaneous CHA AP CALCULUS AB AVON HIGH SCHOOL MR. RECORD 2 Section Number: 1.1 Introducing Calculus: Can Change Occur at an Instant? Day: 2 2 Topic: 1.1 Introducing Calculus: Can Change Occur at an Instant? Day: 2 Learning Objective CHA-1.A: Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant. UNIT 1 Limits and Continuity The Burj Khalifa in Dubai, completed in 2009, is currently the world’s tallest building. Scan the QR Code above to watch a video covering the Introduction and Example 1
velocity of the falling stone at time t  5 . Fill in the table below with your findings. AP® CALCULUS AB FREE RESPONSE QUESTION Let’s look at an actual AP Calculus exam Free Response Questions from a few years ago. (No calculator allowed) t (years) 2 3 5 7 10 H t( ) (meters) 1.5 2 6 11 15 The height of a tree at time t is given by a twice differentiable function, H, where H t( ) is measured in meters and t is measured in years. Selected values of H t( ) are given in the table above. (a) Use the data in the table to estimate H(6) . Using correct units, interpret the meaning of H(6) in the context of the problem. * Note: The term “twice differentiable” and the functions with the symbols H (6) are concepts that will be taught later. For now, think of H (6) as asking for an instantaneous rate of change approximation using an appropriate average rate of change. What Can Calculus Do For You? Quick TI-nSpire Tutorial When evaluating a function for several values of x (or in this case, t), it can be helpful to store the function in your calculator. One of the many ways to do this is pressing the /t buttons to access Ï. Next, you can enter multiple values for t by using the “such that” notation, |, found by pressing /= and using braces, {}, after entering t = . 2018 AB/BC 4 Scan the QR Code above to watch a video covering the solution to 2018 AB/BC 4 and the Conclusion of the lesson. t s(t) ( ) (5) 5 s t s t  
The following tables illustrate the powerful questions that calculus can answer. Many of these you will learn in Calculus AB (Calculus 1) but others will not be discussed until Calculus BC (Calculus 2) or beyond.
THINK ABOUT IT Consider the function 3 1 ( ) , 1 1 x f x x x     Look at its graph to the right. What is f(x) approaching as x approaches 1? Use your fingers and trace along the graph on both sides of x = 1. Do your fingers seem to come to a place where they are “fairly close” together? The question above can be rewritten symbolically as 3 1 1 lim x 1 x  x   We can find the above limit numerically (Using a Table of Values) Complete the table below for the function given above. Round all decimals to three places. x .9 .99 .999 1 1.001 1.01 1.1 f(x) The informal definition of a limit is read like this: “What is happening to y as x gets close to a certain number?” DESMOS ACTIVITY. 2 Topic: 1.2 Defining Limits and Using Limit Notation Day: 2 2 Topic: 1.4 Estimating Limits from Tables Learning Objective LIM-1.A: Represent limits analytically using correct notation. Learning Objective LIM-1.C: Estimate limits of functions. Quick TI-nSpire Tutorial There are two ways to generate a table of values. Method 1: Go to a Graph Scratchpad. (Press » twice). Press e to bring up function entry line. After entering the function, press ·. Press / + T to bring up the table. Your screen will now be split. You may have to change the setting of the table my pressing b, Option 2: Table; Option 5: Edit Table Settings... change Independent from “Auto” to “Ask.” I’ll show you Method 2 later. -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 x y Scan the QR Code above to watch a video covering this Think About It Activity