LINEAR APPROXIMATION, LIMITS, AND L'HOPITAL'S RULE v.05
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1 LINEAR APPROXIMATION, LIMITS, AND L'HOPITAL'S RULE v.05 Linear Approimation Nearby a point at which a function is differentiable, the function and its tangent line are approimately the same. The tangent line at that point on the curve is called the linearization of the curve on the neighborhood of the point. EXAMPLE Consider the function f 2 nearby. The equation of the tangent line at this point is y f f ' or y 2. The graph below shows the graph of the function and its derivative in a window centered at.if you change the size of the window to shrink it around, you will see that the tangent line approimates the function better. Zoom of function at 0 2 EXERCISE 0 2 a. What is the concavity of the function at? Justify your answer using the second derivative. b. When the linear approimation at is used to estimate the function for values in a neighborhood of, are those values underestimates or overestimates? Why? c. Use the linear approimation to estimate the value of y 2 at.0 and 0.98 d. How good was the estimate? In other words, what is the error in the estimate? Look at the graph and then verify it algebraically. e. You are asked to estimate the value of the function at 2 using the linearization at =. Is it going to be a good estimate? Justify your answer. f. Find the equation of the tangent line to the graph of y 2 at, call it y Tan. Now find y y Tan
2 2 Workbook 7 L'Hopital's v05.nb EXERCISE 2 a. Find the linearization of y 2 at 2. b. Are the values obtained with this linearization and overestimate/underestimate? Justify your answer graphically and using derivatives. c. What is the maimum error estimating the value of function one can made using its linearization at 2on the interval 2.2,.8? Justify your answer. The graphic may help to determine it. d. Find the equation of the tangent line to the graph of y 2 at 2., call it y Tan. Now find y y Tan A function f that is differentiable at a can be approimated locally by a linear function. This function corresponds to the tangent line to the function at a, and has equation f f a a f a When we say that the function can be approimated locally it means that on a neighborhood of a we can estimate the value of the function using its linearization. There in a error involved in the approimation when the function is not linear. EXERCISE 3 Find the linear approimation of y overestimate? underestimate? Why? at 3. Use this to estimate the function at 3.2. Is this an a. Find the linearization of y at 2. b. Determine the interval for which the error made using the linearization is less than 0.00 c. Verify that 2 EXERCISE 3 A y y Tan "" Calculate the limits below if you can use the quotient rule. If you can not use the quotient rule indicate so. a sin b. 0 c d. 0 EXPRESSIONS FOR WHICH WE CAN NOT USE THE QUOTIENT RULE FOR LIMITS The following limits can not be calculated using the quotient rule mentioned above. Try them and you will see it. a. 2 2 sin b. 0 sin c. Ln d. 0
3 Workbook 7 L'Hopital's v05.nb L'Hopital's rule EXERCISE 6 Consider the functions a. 0 sin b. log Verify that the conditions to apply L'Hopital's rule are met. Evaluate each of the limits by hand using L'Hopital's rule. Note: L'Hopital's rule also holds if we replace the limiting point a for, -, a, a. Some times we need to use L'Hopital's rule more than once. In that case we will use a generalized form of L'Hopital's rule EXERCISE 7 a Calculate the following limits. a. b. c. d e ln 4 e f. = g. 2 f g a f g Other it Forms when L'Hopital's rule can be applied There are some limits in which we can use L'Hopital's rule after proper changes are made to obtain the form either 0 or. These are some cases: 0
4 4 Workbook 7 L'Hopital's v05.nb 0 Ln 0 Ln 0 0 Ln Ln Ln Form 0 0 Ln 0 0 Form
5 Workbook 7 L'Hopital's v05.nb Ln Form 0 Ln Form 0 0 L' Hopital 2 2 Ln 0 0 0
6 6 Workbook 7 L'Hopital's v05.nb Ln 0 0 Ln of the form 0 Ln 0 of the form 0 applying L' Hopital' s 0 To recover the original limit we proceed as before : 0 Ln 0 0 EXERCISE 8 Identify the form of the following limits 0 0,, 0, 00 and then proceed to evaluate each of them Ln
7 Workbook 7 L'Hopital's v05.nb APPLICATION Consider the graph of the function y 2 2. Determine: 2. Domain and intercepts with the coordinate ais. 2. Critical points 3. Local maima and local minima. 4. Concavities of the function. 5. End behavior. 6. Produce a window where we can see a "complete" graph of the function. EXERCISE 9 Find the limits below. Use L'Hopital's rule where appropriate. If there is a simpler method to solve the problem, use it. If there are cases when you can not use L'Hopital's indicate why not. a. b. c sin 3 LnLn d. e. 2 e 3 2 f. 0 Ln g h. i. sin 0 Ln Π j. Tan 2 k.
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