4.4 L Hospital s Rule

Transcription

1 CHAPTER 4. APPLICATIONS OF DERIVATIVES L Hospital s Rule ln() Eample. Find!. ln() Solution. Check:! ln() X ln()!! 0 0 cos() Eample 2. Find.!0 sin() Solution. WRONG SOLUTION:!0 sin(0) 0. There are four mistakes here. First: we need to check for or 0 0.Secondly, in this case we don t have one of these its, so it s wrong to use L Hospital s Rule. Third, the original it can be figured out without L Hospital s Rule. Fourth, the answer we just got was wrong! RIGHT SOLUTION:!0 cos() 2 Eample 3. Find! e Solution. Check:! e e 0.0 e X. cos(0) 0 0 ±.! 2 e e 0.0 X. 2! 0.0e 0.0

2 CHAPTER 4. APPLICATIONS OF DERIVATIVES 03 So we should use L Hospital s Rule again. (0.0) 2 e (0.0) 2 e 2 0! 2 cos( 0 ) Eample 4. Find!0 20. Hint: Do once, simplify s, and then do again. Compare your answer to the graph of cos Solution. This is one of my favorite eamples, for reasons I ll give below. cos( 0 ) Check:!0 20 cos(0) X cos( 0 )!0 20!0 sin( 0 ) SIMPLIFY!!0 sin( 0 ) 2 0 sin(0) 0!0 0 0 X cos( 0 ) 0 9! SIMPLIFY! cos( 0 ) cos(0)! One of the reasons I like this eample so much is that involved both simplification and doing L Hospital twice. But the other reason I like it is that your calculator cannot do this problem. In the picture below we show a simulation of a calculator graph of this function. graphics/emulate_ti_calc_it_cos_frac-eps-converted-to.pdf

3 CHAPTER 4. APPLICATIONS OF DERIVATIVES 04 According to the graph, the it looks like it will be 0 at 0. Doyou know what is going on? Is the calculator right and Calculus wrong? No way! The calculator is running into round error. The numbers on top of the original fraction are the same out to the 4th decimal place (or so), but these di erences are what s keeping the it from really being equal to 0. But the 4th decimal place is the it of the calculator s accuracy, so the calculations are not quite able to give the right values. Eample 5. Find ln()!0 + Solution. Check: ln()!0 + 0 ln(0 + ) 0 ( )X Use L Hospital: ln()!0 + ln()!0 + / / 2!0 + SIMPLIFY!! !0 + Eample 6. Find ln() tan( /2).! + Solution. Check: ln()tan( /2)! + ln()tan( /2 + ) 0 ( )X Use L Hospital:! + ln()tan( /2) ln()! + / tan( /2) ln()! +! + 2 cot( /2) csc 2 ( /2) /2 csc 2 ( /2) /2 /2

4 CHAPTER 4. APPLICATIONS OF DERIVATIVES 05 Optional Variations on L Hospital s Rule Eample 7. Find! /2 (sec() tan()) Solution. Check:!( /2) (sec() tan()) sec( /2 ) tan( /2 ) X (note: you need to know what the vertical asymptotes of sec() andtan() look like, see the graphs in front cover of the book.) Use L Hospital: (sec() tan()) WRONG:!( /2)!( /2) sec()tan() sec 2 (). This is wrong because we first need to put this function into a fraction, then take the derivative of the top and bottom. sin()!( /2) cos() cos()!( /2)!( /2)!( /2) 0 0. sin() cos() 0 cos() sin() cos() sin() Eample 8. Find cot()).!0 +(csc() Solution. Check:!0 +(csc() cot()) X (Again, check the graphs to see that each function gives positive infinity.) Rewrite as fraction, us :!0 +(csc() cot()) cos()!0 + sin() sin()!0 + cos() sin()!0 + 0+sin() cos() 0+sin(0) cos(0) 0 Eample 9. Find, then simplify.) Solution.! + Check:! + (Hint: get a common denominator, use ln() ln()

5 CHAPTER 4. APPLICATIONS OF DERIVATIVES 06 ln() X (note: because the it is! +,withtheplussignmeaning >, we have that is + and similarly is +.) ln() Rewrite as fraction and use L Hospital:! + ln() ln() ( )! + ( ) ln() ( ) ln()! + ln() ( ) ( ) ln() ln() +! + ( ) ln() ln()+! + ln()+( ) ln()! + ln()+ ln() + ln() Do again!!! ln()! + ln()+! Eample 0. If you have a debt of P (in dollars), compounded n times per year, at an annual interest rate of r, then the amount you will owe after t years is A P + r n nt. (a) Suppose you have $00,000 of debt, at 7% interest, for 5 years. How much will you owe if it s compounded monthly? (b) Daily? (c) Compounded every instant? Solution. (a) $40, (b) $4, (c) We want to find n n! n. This will be 0000 times 5n. We ll leave the 0000 alone for now, and just worry about n! +.07 n this it.

6 CHAPTER 4. APPLICATIONS OF DERIVATIVES 07 Check: +.07 n! n 5 5n +.07 (+0) X. We need to use L Hospital to figure this out. y +.07 ln(y) ln ln(y) 5 ln 5 ln!! ln( +.07/) /(5) L.H.!! / 5 5! (0 +.07/.07/2 ) y e ln(y)! e 5(.07) e 5(.07)! Total debt: 0000e 5(.07) Note: the final formula was 0 5 e 5(.07).Thus,wehaveseen/provedthat as n!. P + r nt! Pe rt n Eample. Find + sin(4))cot()!0 +( Solution. Check:!0 +( + sin(4))cot() (+sin(0)) cot(0) X Take ln( ), rewrite as a product, rewrite as a fraction, use. y (+sin(4)) cot() ln(y) ln ( + sin(4)) cot() ln(y) cot()ln(+sin(4)) cot()ln(+sin(4))!0 + ln( + sin(4))!0 + tan() cos(4)4 +sin(4)!0 + sec 2 ()