2.1 Rates of Change and Limits AP Calculus

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1 .1 Rates of Change and Limits AP Calculus.1 RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important concepts in calculus, derivatives and integrals. Limits can be found using substitution, graphical investigation, numerical approimation, algebra, or some combination of these. Average and Instantaneous Speed In pre calculus courses, ou used the formula d = rt to determine the speed of an object. What ou found was the object s average speed. A moving bod s average speed during an interval of time is found b dividing the total distance covered b the elapsed time. h If an object is dropped from an initial height of 0, we can use the position function s t = 16t + h0 to model the height, s, (in feet) of an object that has fallen for t seconds. Eample: Wile E. Coote, once again tring to catch the Road Runner, waits for the nastil speed bird atop a 900 foot cliff. With his Acme Rocket Pac strapped to his back, Wile E. is poised to leap from the cliff, fire up his rocket pack, and finall partake of a juic road runner roast. Seconds later, the Road Runner zips b and Wile E. leaps from the cliff. Alas, as alwas, the rocket malfunctions and fails to fire, sending poor Wile E. plummeting to the road below disappearing into a cloud of dust. a) Find Wile E. s average speed for the first 3 seconds. b) Find Wile E. s average speed between t = and t = 3 seconds. c) Find Wile E. s speed at the instant t = 3 seconds. The problem with the part c is that we are tring to find the instantaneous velocit. Without the concept of a it, we could not find the answer to part c. Using a it to solve this problem involves studing what happens as we get close to 3 seconds. Eample: Find the average speed between t =.5 and t = 3 seconds. Eample: Find the average speed between t =.9 and t = 3 seconds. Eample: Find the average speed between t =.99 and t = 3 seconds. Eample: Find the average speed between t =.999 and t = 3 seconds. So, even though we cannot find the average velocit at eactl t = 3 seconds, we can discover what Wile E. s speed is approaching at t = 3 seconds. 1

2 .1 Rates of Change and Limits AP Calculus Eample: Sketch the graph of 4 f = ;. a) What happens at =? b) Complete the table of values below to determine what happens as gets close to. approaches from the left approaches from the right f () Informal Definition of a Limit Suppose a function f is defined on an interval around = c, but possibl not at the point = c itself. Suppose that as becomes sufficientl close to c, f () becomes as close to a single number L as we please. We then sa that the it of f () as approaches c is L, and we write f c = L. c) Appl this definition to the function from above to find the f. Eample: Use the graph to find g, where g is defined as 1, g = 0, =

3 .1 Rates of Change and Limits AP Calculus When we sa f () becomes as close to L as we please in the informal definition, we mean that we can specif a maimum distance between f () and L. This distance is given b f L = Distance between f () and L. We use the Greek letter ε (epsilon) to stand for the maimum distance, so we require f L < ε. Similarl, we interpret becomes sufficientl close to c to mean c < δ, where the Greek letter δ (delta) tells us how close must be to c. Then f = L c means that we can make the distance f L between the function values and L as small as we like (less than an number ε > 0 ) b making the distance c between and c sufficientl small (less than some δ > 0 ). Formal Definition of a Limit Let c and L be real numbers. The function f has a it L as approaches c if, given an positive number ε, there is a positive number δ such that for all, We write 0 < c < δ f L < ε. f = L c (Just for fun ) Smbolicall this can be written as follows: = ( ε > )( δ > ) ( < < δ) ( <ε ) f L c f L c Eample: Consider the following function. Graphicall show the definition of a it. L c When Limits Do Not Eist If there does not eist a number L satisfing the condition in the definition, then we sa the f c Limits tpicall fail for three reasons: 1. f () approaches a different number from the right side of c than it approaches from the left side.. f () increases or decreases without bound as approaches c. 3. f () oscillates between two fied values as approaches c. Eample: Investigate (use a graph and/or table) the eistence of the following its. 3 does not eist.

4 .1 Rates of Change and Limits AP Calculus (a) f () (b) 1 1 ( 1) f () (c) 1 sin 0 Tr graphing this on our calculator. First convince ourself that as ou move to the right in the chart below is actuall getting closer and closer to 0. f () π 3π 5π 7π 9π 11π 13π As 0 4

5 .1 Rates of Change and Limits AP Calculus Properties of Limits For man well behaved functions, evaluating the it can be done b direct substitution. That is, f ( c ) = f ( c) Such well behaved functions are continuous at c. We will stud continuit of a function in.3. The following theorems describe its that can be evaluated b direct substitution. Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following its. f = L and g = K b = b c c c c c = f ± g = L± K c = = f g LK c f L = ;provided K 0 c g K c b f bl r s r s f = L c provided r and s are integers and s 0 You should realize that all the properties in the bo above basicall sa that ou can evaluate a it as approaches c b plugging c into the equation. Eample: Find each it (a) li m (b) (c) (d) π sec 7 6 Eample: Use the given information to evaluate the its: f g (a) 5g c = and = 3 c c (b) li m f + g c (c) li m f g c (d) f c g 5

6 .1 Rates of Change and Limits AP Calculus A Strateg for Finding Limits If a it cannot be found using direct substitution, then we will use other techniques to evaluate the it. : Keep in mind that some functions do not have its. If direct substitution ields the meaningless result 0, then ou cannot determine the it in this form. 0 The epression that ields this result is called an Indeterminate Form. When ou encounter this form, ou must rewrite the fraction so that the new denominator does not have 0 as its it. One wa to do this is to cancel like factors, and a second wa is to rationalize the numerator. Eample: Find the it: Eample: Find the it (if it eists): Eample: Find the it (if it eists): ( + )

7 .1 Rates of Change and Limits AP Calculus One Sided Limits One of the reasons a it did not eist is because the function approached a different value from the left than it did from the right. Suppose we have the graph below. L K c Earlier, we would have said that the it as approaches c does not eist because as approaches c from the left, the function approaches K, and as approaches c from the right, the function approaches L. However, sometimes we are interested in what the function approaches as approaches onl from the right or left of c. We can sa this using the following notation: + c c f = L the it of f () as approaches c from the right is L f = K the it of f () as approaches c from the left is K Thus, we can sa that the it of a function as approaches an number c eists if and onl if the it as approaches c from the right is equal to the it as approaches c from the left. Using it notation we have = f ( ) f eists f c + c c Eample: Let f ( ) = int this is the greatest integer function. a) Graph f () b) Find f + c) Find f. Eample: Find f, f, and + c c f c for c = 0, 1,, 3, 4. If the it does not eist, eplain wh. The Sandwich Theorem (a.k.a. The Squeeze Theorem) 7

8 .1 Rates of Change and Limits AP Calculus Eample: Investigate the sin 0 b sketching a graph and making a table. You must understand that while using a graph and/or a table, we ma be able to determine what a it is, we have not proved it until we algebraicall confirm the it is what we think it is. The proof of the above it requires the use of the sandwich theorem. The Sandwich Theorem If g f h for all c in some interval about c, and then g = h = L, c c f = L. c In other words, if we sandwich the function f between two other functions g and h that both have the same it as approaches c, then f is forced to have the same it too. h L f g c 8

9 .1 Rates of Change and Limits AP Calculus sin Eample: Prove that = 1. To do this, we are going to use the figure below. Admittedl, the toughest part of 0 using the sandwich theorem is finding two functions to use as bread. First, we need to find sin π. In order to do this we need to restrict so that 0 < <. Wh are we able to do this? + 0 a) Find the area of OAP. T P b) Find the area of sector OAP. 1 c) Find the area of OAT. O Q A (1, 0) 1 d) Set up an inequalit with the three areas from parts a, b, and c. e) Divide all three parts b 1 sin. Wh do the inequalit signs sta the same? f) Make the middle term sin. Hint: If our middle term doesn t look anthing like this, start over! g) Use the Sandwich Theorem to show that sin = h) Show that f ( ) sin = is an even function. i) Since f ( ) sin = is an even function, what can ou conclude about sin 0? j) Eplain wh we can conclude that sin =

10 .1 Rates of Change and Limits AP Calculus Eample: Find sin 0 Eample: Find sin 5 0 Eample: Find sin 0 5 Eample: Find sin sin Eample: Find Eample: Find tan 0 10