3. To approximate a function by a quadratic function near a number , it is best to write in the form Show that the quadratic function that satisfies conditions (i), (ii), and (iii) is 4. Find the quadratic approximation to near . Graph , the quadratic approximation, and the linear approximation from Example 2 in Section 3.10 on a common screen. What do you conclude? 5. Instead of being satisfied with a linear or quadratic approximation to near , let’s try to find better approximations with higher-degree polynomials. We look for an th-degree polynomial such that and its first derivatives have the same values at as and its first derivatives. By differentiating repeatedly and setting , show that these conditions are satisfied if , and in general where . The resulting polynomial is called the th-degree Taylor polynomial of centered at . 6. Find the 8th-degree Taylor polynomial centered at for the function . Graph together with the Taylor polynomials in the viewing rectangle [H110025, 5] by [H110021.4, 1.4] and comment on how well they approximate .f T 2 , T 4 , T 6 , T 8 f f H20849xH20850 H33527 cos xa H33527 0 afn T n H20849xH20850 H33527 f H20849aH20850 H11001 fH11032H20849aH20850H20849x H11002 aH20850 H11001 f H11033H20849aH20850 2! H20849x H11002 aH20850 2 H11001 H11080H11080H11080 H11001 f H20849nH20850 H20849aH20850 n! H20849x H11002 aH20850 n k! H33527 1 H11554 2 H11554 3 H11554 4 H11554 H11080H11080H11080 H11554 k c k H33527 f H20849kH20850 H20849aH20850 k! c 0 H33527 f H20849aH20850, c 1 H33527 fH11032H20849aH20850, c 2 H33527 1 2 f H11033H20849aH20850 x H33527 a nfx H33527 anT n T n H20849xH20850 H33527 c 0 H11001 c 1 H20849x H11002 aH20850 H11001 c 2 H20849x H11002 aH20850 2 H11001 c 3 H20849x H11002 aH20850 3 H11001 H11080H11080H11080 H11001 c n H20849x H11002 aH20850 n n x H33527 af H20849xH20850 fa H33527 1f H20849xH20850 H33527sx H11001 3 PH20849xH20850 H33527 f H20849aH20850 H11001 fH11032H20849aH20850H20849x H11002 aH20850 H11001 1 2 f H11033H20849aH20850H20849x H11002 aH20850 2 PH20849xH20850 H33527 A H11001 BH20849x H11002 aH20850 H11001 CH20849x H11002 aH20850 2 PaPf 254 |||| CHAPTER 3 DIFFERENTIATION RULES HYPERBOLIC FUNCTIONS Certain even and odd combinations of the exponential functions and arise so fre- quently in mathematics and its applications that they deserve to be given special names. In many ways they are analogous to the trigonometric functions, and they have the same relationship to the hyperbola that the trigonometric functions have to the circle. For this reason they are collectively called hyperbolic functions and individually called hyperbolic sine, hyperbolic cosine, and so on. DEFINITION OF THE HYPERBOLIC FUNCTIONS coth x H33527 cosh x sinh x tanh x H33527 sinh x cosh x sech x H33527 1 cosh x cosh x H33527 e x H11001 e H11002x 2 csch x H33527 1 sinh x sinh x H33527 e x H11002 e H11002x 2 e H11002x e x 3.11 The graphs of hyperbolic sine and cosine can be sketched using graphical addition as in Figures 1 and 2. Note that has domain and range , while has domain and range . The graph of is shown in Figure 3. It has the horizontal asymptotes . (See Exercise 23.) Some of the mathematical uses of hyperbolic functions will be seen in Chapter 7. Applications to science and engineering occur whenever an entity such as light, velocity, electricity, or radioactivity is gradually absorbed or extinguished, for the decay can be rep- resented by hyperbolic functions. The most famous application is the use of hyperbolic cosine to describe the shape of a hanging wire. It can be proved that if a heavy flexible cable (such as a telephone or power line) is suspended between two points at the same height, then it takes the shape of a curve with equation called a cate- nary (see Figure 4). (The Latin word catena means “chain.”) Another application of hyperbolic functions occurs in the description of ocean waves: The velocity of a water wave with length moving across a body of water with depth is modeled by the function where is the acceleration due to gravity. (See Figure 5 and Exercise 49.) The hyperbolic functions satisfy a number of identities that are similar to well-known trigonometric identities. We list some of them here and leave most of the proofs to the exercises. HYPERBOLIC IDENTITIES coshH20849x H11001 yH20850 H33527 cosh x cosh y H11001 sinh x sinh y sinhH20849x H11001 yH20850 H33527 sinh x cosh y H11001 cosh x sinh y 1 H11002 tanh 2 x H33527 sech 2 xcosh 2 x H11002 sinh 2 x H33527 1 coshH20849H11002xH20850 H33527 cosh xsinhH20849H11002xH20850 H33527 H11002sinh x t v H33527 H20881 tL 2H9266 tanh H20873 2H9266d L H20874 dL y H33527 c H11001 a coshH20849xH20862aH20850 y H33527 H110061tanh H208511, H11009H20850H11938coshH11938H11938sinh FIGURE 3 y=tanh x y 0 x y=_1 y=1 FIGURE 1 y=sinh x= ´- e–® 1 2 1 2 1 2 y= ´ y=_ e–® 1 2 y=sinh x 0 y x FIGURE 2 y=cosh x= ´+ e–® 1 2 1 2 y= e–® 1 2 1 2 y= ´ y=cosh x 1 0 y x SECTION 3.11 HYPERBOLIC FUNCTIONS |||| 255 FIGURE 4 A catenary y=c+a cosh(x/a) y 0 x L d FIGURE 5 Idealized ocean wave EXAMPLE 1 Prove (a) and (b) . SOLUTION (a) (b) We start with the identity proved in part (a): If we divide both sides by , we get or M The identity proved in Example 1(a) gives a clue to the reason for the name “hyper- bolic” functions: If is any real number, then the point lies on the unit circle because . In fact, can be interpreted as the radian measure of in Figure 6. For this reason the trigonometric functions are sometimes called circular functions. Likewise, if is any real number, then the point lies on the right branch of the hyperbola because and . This time, does not represent the measure of an angle. However, it turns out that represents twice the area of the shaded hyperbolic sector in Figure 7, just as in the trigonometric case rep- resents twice the area of the shaded circular sector in Figure 6. The derivatives of the hyperbolic functions are easily computed. For example, We list the differentiation formulas for the hyperbolic functions as Table 1. The remaining proofs are left as exercises. Note the analogy with the differentiation formulas for trigono- metric functions, but beware that the signs are different in some cases. DERIVATIVES OF HYPERBOLIC FUNCTIONS d dx H20849tanh xH20850 H33527 sech 2 x d dx H20849coth xH20850 H33527 H11002csch 2 x d dx H20849cosh xH20850 H33527 sinh x d dx H20849sech xH20850 H33527 H11002sech x tanh x d dx H20849sinh xH20850 H33527 cosh x d dx H20849csch xH20850 H33527 H11002csch x coth x 1 d dx H20849sinh xH20850 H33527 d dx H20873 e x H11002 e H11002x 2 H20874 H33527 e x H11001 e H11002x 2 H33527 cosh x t t tcosh t H33356 1cosh 2 t H11002 sinh 2 t H33527 1x 2 H11002 y 2 H33527 1 PH20849cosh t, sinh tH20850t H11028POQtcos 2 t H11001 sin 2 t H33527 1 x 2 H11001 y 2 H33527 1PH20849cos t, sin tH20850t 1 H11002 tanh 2 x H33527 sech 2 x 1 H11002 sinh 2 x cosh 2 x H33527 1 cosh 2 x cosh 2 x cosh 2 x H11002 sinh 2 x H33527 1 H33527 4 4 H33527 1H33527 e 2x H11001 2 H11001 e H110022x 4 H11002 e 2x H11002 2 H11001 e H110022x 4 cosh 2 x H11002 sinh 2 x H33527 H20873 e x H11001 e H11002x 2 H20874 2 H11002 H20873 e x H11002 e H11002x 2 H20874 2 1 H11002 tanh 2 x H33527 sech 2 xcosh 2 x H11002 sinh 2 x H33527 1V 256 |||| CHAPTER 3 DIFFERENTIATION RULES FIGURE 7 0 y x ≈-¥=1 P(cosh t, sinh t) FIGURE 6 O y x P(cos t, sin t) ≈+¥=1 Q The Gateway Arch in St. Louis was designed using a hyperbolic cosine function (Exercise 48). © 2006 Getty Images EXAMPLE 2 Any of these differentiation rules can be combined with the Chain Rule. For instance, M INVERSE HYPERBOLIC FUNCTIONS You can see from Figures 1 and 3 that and are one-to-one functions and so they have inverse functions denoted by and . Figure 2 shows that is not one- to-one, but when restricted to the domain it becomes one-to-one. The inverse hyper- bolic cosine function is defined as the inverse of this restricted function. The remaining inverse hyperbolic functions are defined similarly (see Exercise 28). We can sketch the graphs of , , and in Figures 8, 9, and 10 by using Figures 1, 2, and 3. Since the hyperbolic functions are defined in terms of exponential functions, it’s not surprising to learn that the inverse hyperbolic functions can be expressed in terms of log- arithms. In particular, we have: EXAMPLE 3 Show that . SOLUTION Let . Then x H33527 sinh y H33527 e y H11002 e H11002y 2 y H33527 sinh H110021 x sinh H110021 x H33527 ln(x H11001sx 2 H11001 1) tanh H110021 x H33527 1 2 ln H20873 1 H11001 x 1 H11002 x H20874 H110021 H11021 x H11021 15 cosh H110021 x H33527 ln(x H11001sx 2 H11002 1) x H33356 14 sinh H110021 x H33527 ln(x H11001sx 2 H11001 1) x H20678 H119383 FIGURE 8 y=sinh–! x domain=R range=R 0 y x FIGURE 9 y=cosh–! x domain=[1, `} range=[0, `} 0 y x 1 FIGURE 10 y=tanh–! x domain=(_1, 1) range=R 0 y x 1_1 tanh H110021 cosh H110021 sinh H110021 y H33527 tanh H110021 x &? tanh y H33527 x y H33527 cosh H110021 x &? cosh y H33527 x and y H33356 0 y H33527 sinh H110021 x &? sinh y H33527 x 2 H208510, H11009H20850 coshtanh H110021 sinh H110021 tanhsinh d dx (cosh sx ) H33527 sinh sx H11554 d dx sx H33527 sinh sx 2sx SECTION 3.11 HYPERBOLIC FUNCTIONS |||| 257 N Formula 3 is proved in Example 3. The proofs of Formulas 4 and 5 are requested in Exercises 26 and 27. so or, multiplying by , This is really a quadratic equation in : Solving by the quadratic formula, we get Note that , but (because ). Thus the minus sign is inadmissible and we have Therefore (See Exercise 25 for another method.) M DERIVATIVES OF INVERSE HYPERBOLIC FUNCTIONS The inverse hyperbolic functions are all differentiable because the hyperbolic functions are differentiable. The formulas in Table 6 can be proved either by the method for inverse functions or by differentiating Formulas 3, 4, and 5. EXAMPLE 4 Prove that . SOLUTION 1 Let . Then . If we differentiate this equation implicitly with respect to , we get Since and , we have , so dy dx H33527 1 cosh y H33527 1 s1 H11001 sinh 2 y H33527 1 s1 H11001 x 2 cosh y H33527s1 H11001 sinh 2 y cosh y H33356 0cosh 2 y H11002 sinh 2 y H33527 1 cosh y dy dx H33527 1 x sinh y H33527 xy H33527 sinh H110021 x d dx H20849sinh H110021 xH20850 H33527 1 s1 H11001 x 2 V d dx H20849tanh H110021 xH20850 H33527 1 1 H11002 x 2 d dx H20849coth H110021 xH20850 H33527 1 1 H11002 x 2 d dx H20849cosh H110021 xH20850 H33527 1 sx 2 H11002 1 d dx H20849sech H110021 xH20850 H33527 H11002 1 xs1 H11002 x 2 d dx H20849sinh H110021 xH20850 H33527 1 s1 H11001 x 2 d dx H20849csch H110021 xH20850 H33527 H11002 1 H11341 x H11341 sx 2 H11001 1 6 y H33527 lnH20849e y H20850 H33527 ln(x H11001sx 2 H11001 1) e y H33527 x H11001sx 2 H11001 1 x H11021sx 2 H11001 1x H11002sx 2 H11001 1 H11021 0e y H11022 0 e y H33527 2x H11006s4x 2 H11001 4 2 H33527 x H11006sx 2 H11001 1 H20849e y H20850 2 H11002 2xH20849e y H20850 H11002 1 H33527 0 e y e 2y H11002 2xe y H11002 1 H33527 0 e y e y H11002 2x H11002 e H11002y H33527 0 258 |||| CHAPTER 3 DIFFERENTIATION RULES N Notice that the formulas for the derivatives of and appear to be identical. But the domains of these functions have no numbers in common: is defined for , whereas is defined for H11341 x H11341 H11022 1.coth H110021 x H11341 x H11341 H11021 1tanh H110021 x coth H110021 xtanh H110021 x SOLUTION 2 From Equation 3 (proved in Example 3), we have M EXAMPLE 5 Find . SOLUTION Using Table 6 and the Chain Rule, we have M H33527 1 1 H11002 sin 2 x cos x H33527 cos x cos 2 x H33527 sec x d dx H20851tanh H110021 H20849sin xH20850H20852 H33527 1 1 H11002 H20849sin xH20850 2 d dx H20849sin xH20850 d dx H20851tanh H110021 H20849sin xH20850H20852V H33527 1 sx 2 H11001 1 H33527 sx 2 H11001 1 H11001 x (x H11001sx 2 H11001 1)sx 2 H11001 1 H33527 1 x H11001sx 2 H11001 1 H20873 1 H11001 x sx 2 H11001 1 H20874 H33527 1 x H11001sx 2 H11001 1 d dx (x H11001sx 2 H11001 1) d dx H20849sinh H110021 xH20850 H33527 d dx ln(x H11001sx 2 H11001 1) SECTION 3.11 HYPERBOLIC FUNCTIONS |||| 259 13. 14. 16. 18. 19. ( any real number) 20. If , find the values of the other hyperbolic functions at . 21. If and , find the values of the other hyperbolic functions at . 22. (a) Use the graphs of , , and in Figures 1–3 to draw the graphs of , , and .cothsechcsch tanhcoshsinh x x H11022 0cosh x H33527 5 3 x tanh x H33527 12 13 n H20849cosh x H11001 sinh xH20850 n H33527 cosh nxH11001 sinh nx 1 H11001 tanh x 1 H11002 tanh x H33527 e 2x tanhH20849ln xH20850 H33527 x 2 H11002 1 x 2 H11001 1 17. cosh 2x H33527 cosh 2 x H11001 sinh 2 x sinh 2x H33527 2 sinh x cosh x15. tanhH20849x H11001 yH20850 H33527 tanh x H11001 tanh y 1 H11001 tanh x tanh y coth 2 x H11002 1 H33527 csch 2 x1–6 Find the numerical value of each expression. 1. (a) (b) 2. (a) (b) 3. (a) (b) 4. (a) (b) 5. (a) (b) 6. (a) (b) 7–19 Prove the identity. 7. (This shows that is an odd function.) 8. (This shows that is an even function.) 10. 11. 12. coshH20849x H11001 yH20850 H33527 cosh x cosh y H11001 sinh x sinh y sinhH20849x H11001 yH20850 H33527 sinh x cosh y H11001 cosh x sinh y cosh x H11002 sinh x H33527 e H11002x cosh x H11001 sinh x H33527 e x 9. cosh coshH20849H11002xH20850 H33527 cosh x sinh sinhH20849H11002xH20850 H33527H11002sinh x sinh H110021 1sinh 1 cosh H110021 1sech 0 coshH20849ln 3H20850cosh 3 sinh 2sinhH20849ln 2H20850 tanh 1tanh 0 cosh 0sinh 0 EXERCISES3.11 for the central curve of the arch, where and are measured in meters and . ; (a) Graph the central curve. (b) What is the height of the arch at its center? (c) At what points is the height 100 m? (d) What is the slope of the arch at the points in part (c)? 49. If a water wave with length moves with velocity in a body of water with depth , then where is the acceleration due to gravity. (See Figure 5.) Explain why the approximation is appropriate in deep water. ; 50. A flexible cable always hangs in the shape of a catenary , where and are constants and (see Figure 4 and Exercise 52). Graph several members of the family of functions . How does the graph change as varies? A telephone line hangs between two poles 14 m apart in the shape of the catenary , where and are measured in meters. (a) Find the slope of this curve where it meets the right pole. (b) Find the angle between the line and the pole. 52. Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve that satisfies the differential equation where is the linear density of the cable, is the acceleration due to gravity, and is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function is a solution of this differential equation. y H33527 f H20849xH20850 H33527 T H9267t cosh H20873 H9267tx T H20874 T tH9267 d 2 y dx 2 H33527 H9267t T H20881 1 H11001 H20873 dy dx H20874 2 y H33527 f H20849xH20850 y 0 x_7 7 5 ¨ H9258 y xy H33527 20 coshH20849xH2086220H20850 H11002 15 51. a y H33527 a coshH20849xH20862aH20850 a H11022 0acy H33527 c H11001 a coshH20849xH20862aH20850 v H11015 H20881 tL 2H9266 t v H33527 H20881 tL 2H9266 tanh H20873 2H9266d L H20874 d vL H11341 x H11341 H33355 91.20 yx ; (b) Check the graphs that you sketched in part (a) by using a graphing device to produce them. 23. Use the definitions of the hyperbolic functions to find each of the following limits. (a) (b) (c) (d) (e) (f) (g) (h) (i) 24. Prove the formulas given in Table 1 for the derivatives of the functions (a) , (b) , (c) , (d) , and (e) . 25. Give an alternative solution to Example 3 by letting and then using Exercise 9 and Example 1(a) with replaced by . 26. Prove Equation 4. 27. Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 18 with replaced by . 28. For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula sim- ilar to Equation 3. (a) (b) (c) 29. Prove the formulas given in Table 6 for the derivatives of the following functions. (a) (b) (c) (d) (e) 30–47 Find the derivative. Simplify where possible. 30. 31. 32. 33. 34. 36. 37. 38. 39. 40. 41. 42. 43. 44. 46. 47. 48. The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y H33527 211.49 H11002 20.96 cosh 0.03291765x y H33527 coth H110021 sx 2 H11001 1 y H33527 sech H110021 s1 H11002 x 2 , x H11022 0 y H33527 x sinh H110021 H20849xH208623H20850 H11002s9 H11001 x 2 45. y H33527 x tanh H110021 x H11001 ln s1 H11002 x 2 y H33527 tanh H110021 sx y H33527 x 2 sinh H110021 H208492xH20850 GH20849xH20850 H33527 1 H11002 cosh x 1 H11001 cosh x y H33527 H20881 1 H11001 tanh x 1 H11002 tanh x 4 y H33527 arctanH20849tanh xH20850y H33527 sinhH20849cosh xH20850 f H20849tH20850 H33527 sech 2 H20849e t H20850f H20849tH20850 H33527 csch tH208491 H11002 ln csch tH20850 y H33527 e cosh 3x 35.y H33527 x cothH208491 H11001 x 2 H20850 hH20849xH20850 H33527 lnH20849cosh xH20850tH20849xH20850 H33527 coshH20849ln xH20850 f H20849xH20850 H33527 x sinh x H11002 cosh xf H20849xH20850 H33527 tanhH208491 H11001 e 2x H20850 coth H110021 sech H110021 csch H110021 tanh H110021 cosh H110021 coth H110021 sech H110021 csch H110021 yx yx y H33527 sinh H110021 x cothsechcschtanhcosh lim xlH11002H11009 csch x lim x l 0 H11002 coth xlim x l 0 H11001 coth x lim xlH11009 coth xlim xlH11009 sech x lim xlH11002H11009 sinh xlim xlH11009 sinh x lim xlH11002H11009 tanh xlim xlH11009 tanh x 260 |||| CHAPTER 3 DIFFERENTIATION RULES CHAPTER 3 REVIEW |||| 261 55. At what point of the curve does the tangent have slope 1? 56. If , show that . 57. Show that if and , then there exist numbers and such that equals either or . In other words, almost every function of the form is a shifted and stretched hyperbolic sine or cosine function. f H20849xH20850 H33527 ae x H11001 be H11002x H9251 coshH20849x H11001 H9252H20850 H9251 sinhH20849x H11001 H9252H20850ae x H11001 be H11002x H9252 H9251b HS33527 0a HS33527 0 sec H9258 H33527 cosh xx H33527 lnH20849sec H9258 H11001 tan H9258H20850 y H33527 cosh x(a) Show that any function of the form satisfies the differential equation . (b) Find such that , , and . 54. Evaluate .lim xlH11009 sinh x e x yH11032H208490H20850 H33527 6 yH208490H20850 H33527 H110024yH11033 H33527 9yy H33527 yH20849xH20850 yH11033 H33527 m 2 y y H33527 A sinh mx H11001 B cosh mx 53. REVIEW CONCEPT CHECK 3 3. (a) How is the number defined? (b) Express as a limit. (c) Why is the natural exponential function used more often in calculus than the other exponential functions ? (d) Why is the natural logarithmic function used more often in calculus than the other logarithmic functions ? 4. (a) Explain how implicit differentiation works. (b) Explain how logarithmic differentiation works. 5. (a) Write an expression for the linearization of at . (b) If , write an expression for the differential . (c) If , draw a picture showing the geometric mean- ings of and .dyH9004y dx H33527 H9004x dyy H33527 f H20849xH20850 af y H33527 log a x y H33527 ln x y H33527 a x y H33527 e x e e1. State each differentiation rule both in symbols and in words. (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f) The Quotient Rule (g) The Chain Rule 2. State the derivative of each function. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) y H33527 tanh H110021 xy H33527 cosh H110021 x y H33527 sinh H110021 xy H33527 tanh xy H33527 cosh x y H33527 sinh xy H33527 tan H110021 xy H33527 cos H110021 x y H33527 sin H110021 xy H33527 cot xy H33527 sec x y H33527 csc xy H33527 tan xy H33527 cos x y H33527 sin xy H33527 log a xy H33527 ln x y H33527 a x y H33527 e x y H33527 x n Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If and are differentiable, then 2. If and are differentiable, then 3. If and are differentiable, then 4. If is differentiable, then . 5. If is differentiable, then . d dx f (sx ) H33527 fH11032H20849xH20850 2sx f d dx sf H20849xH20850 H33527 fH11032H20849xH20850 2sf H20849xH20850 f d dx H20851 f H20849tH20849xH20850H20850H20852 H33527 fH11032H20849tH20849xH20850H20850tH11032H20849xH20850 tf d dx H20851 f H20849xH20850tH20849xH20850H20852 H33527 fH11032H20849xH20850tH11032H20849xH20850 tf d dx H20851 f H20849xH20850 H11001tH20849xH20850H20852 H33527 fH11032H20849xH20850 H11001tH11032H20849xH20850 tf 6. If , then . 7. 8. 9. 10. 11. If , then . 12. An equation of the tangent line to the parabola at is .y H11002 4 H33527 2xH20849x H11001 2H20850H20849H110022, 4H20850 y H33527 x 2 lim x l 2 tH20849xH20850 H11002tH208492H20850 x H11002 2 H33527 80tH20849xH20850 H33527 x 5 d dx H11341 x 2 H11001 x H11341 H33527 H11341 2x H11001 1 H11341 d dx H20849tan 2 xH20850 H33527 d dx H20849sec 2 xH20850 d dx H20849ln 10H20850 H33527 1 10 d dx H2084910 x H20850 H33527 x10 xH110021 yH11032 H33527 2ey H33527 e 2 TRUE-FALSE QUIZ 262 |||| CHAPTER 3 DIFFERENTIATION RULES 1–50 Calculate . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. If , find .f H11033H208492H20850f H20849tH20850 H33527s4t H11001 1 y H33527 sin 2 (cosssin H9266x )y H33527 cos(e stan 3x ) y H33527 x tanh H110021 sx y H33527 cosh H110021 H20849sinh xH20850 y H33527 ln H20895 x 2 H11002 4 2x H11001 5 H20895 y H33527 lnH20849cosh 3xH20850 y H33527 sin mx x y H33527 x sinhH20849x 2 H20850 y H33527 H20849x H11001 H9261H20850 4 x 4 H11001 H9261 4 y H33527 sx H11001 1 H208492 H11002 xH20850 5 H20849x H11001 3H20850 7 xe y H33527 y H11002 1y H33527 tan 2 H20849sin H9258H20850 y H33527 arctan(arcsin sx )y H33527 sin(tan s1 H11001 x 3 ) y H33527st lnH20849t 4 H20850 y H33527 cotH208493x 2 H11001 5H20850 y H33527 10 tan H9266H9258 y H33527 ln H11341 sec 5x H11001 tan 5x H11341 y H33527 e cos x H11001 cosH20849e x H20850y H33527 x tan H110021 H208494xH20850 y H33527 H20849x 2 H11001 1H20850 4 H208492x H11001 1H20850 3 H208493x H11002 1H20850 5 y H33527 ln sin x H11002 1 2 sin 2 x y H33527 H20849cos xH20850 x y H33527 log 5 H208491 H11001 2xH20850 y H33527ssin sx sinH20849xyH20850 H33527 x 2 H11002 y y H33527 1H20862s 3 x H11001sx y H33527 H208491 H11002 x H110021 H20850 H110021 y H33527 secH208491 H11001 x 2 H20850y H33527 3 x ln x y H33527 lnH20849x 2 e x H20850y H33527 e cx H20849c sin x H11002 cos xH20850 x 2 cos y H11001 sin 2y H33527 xyy H33527 sec 2H9258 1 H11001 tan 2H9258 y H33527 lnH20849csc 5xH20850xy 4 H11001 x 2 y H33527 x H11001 3y y H33527 1 sinH20849x H11002 sin xH20850 y H33527 e 1H20862x x 2 y H33527 H20849arcsin 2xH20850 2 y H33527sx cos sx y H33527 e mx cos nxy H33527 t 1 H11002 t 2 y H33527 e H11002t H20849t 2 H11002 2t H11001 2H20850y H33527 e sin 2H9258 y H33527 e x 1 H11001 x 2 y H33527 2xsx 2 H11001 1 y H33527 3x H11002 2 s2x H11001 1 y H33527sx H11001 1 s 3 x 4 y H33527 cosH20849tan xH20850y H33527 H20849x 4 H11002 3x 2 H11001 5H20850 3 yH11032 52. If , find . 53. Find if . 54. Find if . 55. Use mathematical induction (page 77) to show that if , then . 56. Evaluate . 57–59 Find an equation of the tangent to the curve at the given point. 57. , 58. , 59. , 60–61 Find equations of the tangent line and normal line to the curve at the given point. 60. , 61. , ; 62. If , find . Graph and on the same screen and comment. 63. (a) If , find . (b) Find equations of the tangent lines to the curve at the points and . ; (c) Illustrate part (b) by graphing the curve and tangent lines on the same screen. ; (d) Check to see that your answer to part (a) is reasonable by comparing the graphs of and . 64. (a) If , , find and . ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of , , and . 65. At what points on the curve , , is the tangent line horizontal? 66. Find the points on the ellipse where the tangent line has slope 1. 67. If , show that 68. (a) By differentiating the double-angle formula obtain the double-angle formula for the sine function. (b) By differentiating the addition formula obtain the addition formula for the cosine function. sinH20849x H11001 aH20850 H33527 sin x cos a H11001 cos x sin a cos 2x H33527 cos 2 x H11002 sin 2 x fH11032H20849xH20850 f H20849xH20850 H33527 1 x H11002 a H11001 1 x H11002 b H11001 1 x H11002 c f H20849xH20850 H33527 H20849x H11002 aH20850H20849x H11002 bH20850H20849x H11002 cH20850 x 2 H11001 2y 2 H33527 1 0 H33355 x H33355 2H9266y H33527 sin x H11001 cos x f H11033fH11032f f H11033fH11032H11002H9266H208622 H11021 x H11021 H9266H208622f H20849xH20850 H33527 4x H11002 tan x fH11032f H208494, 4H20850H208491, 2H20850y H33527 xs5 H11002 x fH11032H20849xH20850f H20849xH20850 H33527 xs5 H11002 x fH11032ffH11032H20849xH20850f H20849xH20850 H33527 xe sin x H208490, 2H20850y H33527 H208492 H11001 xH20850e H11002x H208492, 1H20850x 2 H11001 4xy H11001 y 2 H33527 13 H208490, 1H20850y H33527s1 H11001 4 sin x H208490, H110021H20850y H33527 x 2 H11002 1 x 2 H11001 1 H20849H9266H208626, 1H20850y H33527 4 sin 2 x lim tl0 t 3 tan 3 H208492tH20850 f H20849nH20850 H20849xH20850 H33527 H20849x H11001 nH20850e x f H20849xH20850 H33527 xe x f H20849xH20850 H33527 1H20862H208492 H11002 xH20850f H20849nH20850 H20849xH20850 x 6 H11001 y 6 H33527 1yH11033 tH11033H20849H9266H208626H20850tH20849H9258H20850 H33527 H9258 sin H9258 E X E R C I S E S (b) Find , the rate at which the drug is cleared from circulation. (c) When is this rate equal to 0? 87. An equation of motion of the form represents damped oscillation of an object. Find the velocity and acceleration of the object. 88. A particle moves along a horizontal line so that its coordinate at time is , , where and are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction. 89. A particle moves on a vertical line so that its coordinate at time is , . (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval . ; (d) Graph the position, velocity, and acceleration functions for . (e) When is the particle speeding up? When is it slowing down? 90. The volume of a right circular cone is , where is the radius of the base and is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant. 91. The mass of part of a wire is kilograms, where is measured in meters from one end of the wire. Find the linear density of the wire when m. 92. The cost, in dollars, of producing units of a certain com- modity is (a) Find the marginal cost function. (b) Find and explain its meaning. (c) Compare with the cost of producing the 101st item. 93. A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after hours. (b) Find the number of bacteria after 4 hours. (c) Find the rate of growth after 4 hours. (d) When will the population reach 10,000? 94. Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg? t CH11032H20849100H20850 CH11032H20849100H20850 CH20849xH20850 H33527 920 H11001 2x H11002 0.02x 2 H11001 0.00007x 3 x x H33527 4 x x(1 H11001sx ) hr V H33527 H9266r 2 hH208623 0 H33355 t H33355 3 0 H33355 t H33355 3 t H33356 0y H33527 t 3 H11002 12t H11001 3t cbt H33356 0x H33527sb 2 H11001 c 2 t 2 t s H33527 Ae H11002ct cosH20849H9275t H11001 H9254H20850 CH11032H20849tH2085069. Suppose that and , where , , , , and . Find (a) and (b) . 70. If and are the functions whose graphs are shown, let , , and . Find (a) , (b) , and (c) . 71–78 Find in terms of . 71. 72. 73. 74. 75. 76. 77. 78. 79–81 Find in terms of and . 79. 80. 81. ; 82. (a) Graph the function in the viewing rectangle by . (b) On which interval is the average rate of change larger: or ? (c) At which value of is the instantaneous rate of change larger: or ? (d) Check your visual estimates in part (c) by computing and comparing the numerical values of and . 83. At what point on the curve is the tangent horizontal? 84. (a) Find an equation of the tangent to the curve that is parallel to the line . (b) Find an equation of the tangent to the curve that passes through the origin. 85. Find a parabola that passes through the point and whose tangent lines at and have slopes 6 and , respectively. 86. The function , where a, b, and K are positive constants and , is used to model the concentra- tion at time t of a drug injected into the bloodstream. (a) Show that .lim t l H11009 CH20849tH20850 H33527 0 b H11022 a CH20849tH20850 H33527 KH20849e H11002at H11002 e H11002bt H20850 H110022 x H33527 5x H33527 H110021H208491, 4H20850 y H33527 ax 2 H11001 bx H11001 c y H33527 e x x H11002 4y H33527 1 y H33527 e x y H33527 H20851lnH20849x H11001 4H20850H20852 2 fH11032H208495H20850 fH11032H208492H20850fH11032H20849xH20850 x H33527 5x H33527 2 x H208512, 3H20852H208511, 2H20852 H20851H110022, 8H20852H208510, 8H20852 f H20849xH20850 H33527 x H11002 2 sin x hH20849xH20850 H33527 f H20849tH20849sin 4xH20850H20850 hH20849xH20850 H33527 H20881 f H20849xH20850 tH20849xH20850 hH20849xH20850 H33527 f H20849xH20850tH20849xH20850 f H20849xH20850 H11001tH20849xH20850 tH11032fH11032hH11032 f H20849xH20850 H33527tH20849ln xH20850f H20849xH20850 H33527 ln H11341 tH20849xH20850 H11341 f H20849xH20850 H33527 e tH20849xH20850 f H20849xH20850 H33527tH20849e x H20850 f H20849xH20850 H33527tH20849tH20849xH20850H20850f H20849xH20850 H33527 H20851tH20849xH20850H20852 2 f H20849xH20850 H33527tH20849x 2 H20850f H20849xH20850 H33527 x 2 tH20849xH20850 tH11032fH11032 0 g f y x 1 1 CH11032H208492H20850QH11032H208492H20850PH11032H208492H20850 CH20849xH20850 H33527 f H20849tH20849xH20850H20850QH20849xH20850 H33527 f H20849xH20850H20862tH20849xH20850PH20849xH20850 H33527 f H20849xH20850tH20849xH20850 tf FH11032H208492H20850hH11032H208492H20850 fH11032H208495H20850 H33527 11fH11032H208492H20850 H33527 H110022tH11032H208492H20850 H33527 4tH208492H20850 H33527 5f H208492H20850 H33527 3 FH20849xH20850 H33527 f H20849tH20849xH20850H20850hH20849xH20850 H33527 f H20849xH20850tH20849xH20850 CHAPTER 3 REVIEW |||| 263 ; 102. (a) Find the linear approximation to near 3. (b) Illustrate part (a) by graphing and the linear approximation. (c) For what values of is the linear approximation accurate to within 0.1? 103. (a) Find the linearization of at . State the corresponding linear approximation and use it to give an approximate value for . ; (b) Determine the values of for which the linear approxima- tion given in part (a) is accurate to within 0.1. 104. Evaluate if , , and . 105. A window has the shape of a square surmounted by a semi- circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in com- puting the area of the window. 106–108 Express the limit as a derivative and evaluate. 106. 107. 108. 109. Evaluate . 110. Suppose is a differentiable function such that and . Show that . 111. Find if it is known that 112. Show that the length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant. x 2H208623 H11001 y 2H208623 H33527 a 2H208623 d dx H20851 f H208492xH20850H20852 H33527 x 2 fH11032H20849xH20850 tH11032H20849xH20850 H33527 1H20862H208491 H11001 x 2 H20850fH11032H20849xH20850 H33527 1 H11001 H20851 f H20849xH20850H20852 2 f H20849tH20849xH20850H20850 H33527 xf lim x l 0 s1 H11001 tan x H11002s1 H11001 sin x x 3 lim H9258lH9266H208623 cos H9258 H11002 0.5 H9258 H11002 H9266H208623 lim h l 0 s 4 16 H11001 h H11002 2 h lim xl1 x 17 H11002 1 x H11002 1 dx H33527 0.2x H33527 2y H33527 x 3 H11002 2x 2 H11001 1dy x s 3 1.03 a H33527 0f H20849xH20850 H33527s 3 1 H11001 3x x f f H20849xH20850 H33527s25 H11002 x 2 95. Let be the concentration of a drug in the bloodstream. As the body eliminates the drug, decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus , where is a positive number called the elimination constant of the drug. (a) If is the concentration at time , find the concen- tration at time . (b) If the body eliminates half the drug in 30 hours, how long does it take to eliminate 90% of the drug? 96. A cup of hot chocolate has temperature in a room kept at . After half an hour the hot chocolate cools to . (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to ? 97. The volume of a cube is increasing at a rate of 10 . How fast is the surface area increasing when the length of an edge is 30 cm? 98. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of , how fast is the water level rising when the water is 5 cm deep? 99. A balloon is rising at a constant speed of . A boy is cycling along a straight road at a speed of . When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later? 100. A waterskier skis over the ramp shown in the figure at a speed of . How fast is she rising as she leaves the ramp? 101. The angle of elevation of the sun is decreasing at a rate of . How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is ?H9266H208626 0.25 radH20862h 4 ft 15 ft 30 ftH20862s 15 ftH20862s 5 ftH20862s 2 cm 3 H20862s cm 3 H20862min 40H11034C 60H11034C20H11034C 80H11034C t t H33527 0C 0 kCH11032H20849tH20850 H33527 H11002kCH20849tH20850 CH20849tH20850 CH20849tH20850 264 |||| CHAPTER 3 DIFFERENTIATION RULES Before you look at the example, cover up the solution and try it yourself first. EXAMPLE 1 How many lines are tangent to both of the parabolas and ? Find the coordinates of the points at which these tangents touch the parabolas. SOLUTION To gain insight into this problem, it is essential to draw a diagram. So we sketch the parabolas (which is the standard parabola shifted 1 unit upward) and (which is obtained by reflecting the first parabola about the x-axis). If we try to draw a line tangent to both parabolas, we soon discover that there are only two possibilities, as illustrated in Figure 1. Let P be a point at which one of these tangents touches the upper parabola and let a be its x-coordinate. (The choice of notation for the unknown is important. Of course we could have used b or c or or instead of a. However, it’s not advisable to use x in place of a because that x could be confused with the variable x in the equation of the parabola.) Then, since P lies on the parabola , its y-coordinate must be Because of the symmetry shown in Figure 1, the coordinates of the point Q where the tangent touches the lower parabola must be . To use the given information that the line is a tangent, we equate the slope of the line PQ to the slope of the tangent line at P. We have If , then the slope of the tangent line at P is . Thus the condi- tion that we need to use is that Solving this equation, we get , so and . Therefore the points are (1, 2) and (H110021, H110022). By symmetry, the two remaining points are (H110021, 2) and (1, H110022). M EXAMPLE 2 For what values of does the equation have exactly one solution? SOLUTION One of the most important principles of problem solving is to draw a diagram, even if the problem as stated doesn’t explicitly mention a geometric situation. Our pres- ent problem can be reformulated geometrically as follows: For what values of does the curve intersect the curve in exactly one point? Let’s start by graphing and for various values of . We know that, for , is a parabola that opens upward if and downward if . Figure 2 shows the parabolas for several positive values of . Most of them don’t intersect at all and one intersects twice. We have the feeling that there must be a value of (somewhere between and ) for which the curves intersect exactly once, as in Figure 3. To find that particular value of , we let be the -coordinate of the single point of intersection. In other words, , so is the unique solution of the given equa- tion. We see from Figure 3 that the curves just touch, so they have a common tangent aln a H33527 ca 2 xac 0.30.1c y H33527 ln x cy H33527 cx 2 c H11021 0c H11022 0y H33527 cx 2 c HS33527 0 cy H33527 cx 2 y H33527 ln x y H33527 cx 2 y H33527 ln x c ln x H33527 cx 2 c a H33527 H110061a 2 H33527 11 H11001 a 2 H33527 2a 2 1 H11001 a 2 a H33527 2a fH11032H20849aH20850 H33527 2af H20849xH20850 H33527 1 H11001 x 2 m PQ H33527 1 H11001 a 2 H11002 H20849H110021 H11002 a 2 H20850 a H11002 H20849H11002aH20850 H33527 1 H11001 a 2 a H20849H11002a, H11002H208491 H11001 a 2 H20850H20850 1 H11001 a 2 .y H33527 1 H11001 x 2 x 1 x 0 y H33527 H110021 H11002 x 2 y H33527 x 2 y H33527 1 H11001 x 2 y H33527 1 H11001 x 2 y H33527 H110021 H11002 x 2 265 x y P Q 1 _1 FIGURE 1 0 3≈ ≈ 0.3≈ 0.1≈ y=ln x ≈ 1 2 x y FIGURE 2 y=ln x y=c≈ c=? y x0 a FIGURE 3 P R O B L E M S P L U S P R O B L E M S P L U S 266 line when . That means the curves and have the same slope when . Therefore Solving the equations and , we get Thus and For negative values of we have the situation illustrated in Figure 4: All parabolas with negative values of intersect exactly once. And let’s not forget about : The curve is just the -axis, which intersects exactly once. To summarize, the required values of are and . M 1. Find points and on the parabola so that the triangle formed by the -axis and the tangent lines at and is an equilateral triangle. ; 2. Find the point where the curves and are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent. 3. Show that the tangent lines to the parabola at any two points with -coordinates and must intersect at a point whose -coordinate is halfway between and . 4. Show that d dx H20873 sin 2 x 1 H11001 cot x H11001 cos 2 x 1 H11001 tan x H20874 H33527 H11002cos 2x q pxqpx y H33527 ax 2 H11001 bx H11001 c y H33527 3H20849x 2 H11002 xH20850y H33527 x 3 H11002 3x H11001 4 x y P Q A 0B C QP xABCy H33527 1 H11002 x 2 QP PROBLEMS c H33355 0c H33527 1H20862H208492eH20850c y H33527 ln xxy H33527 0x 2 H33527 0c H33527 0 y H33527 ln xcy H33527 cx 2 c c H33527 ln a a 2 H33527 ln e 1H208622 e H33527 1 2e a H33527 e 1H208622 ln a H33527 ca 2 H33527 c H11554 1 2c H33527 1 2 1H20862a H33527 2caln a H33527 ca 2 1 a H33527 2ca x H33527 a y H33527 cx 2 y H33527 ln xx H33527 a y y=ln x x 0 FIGURE 4 P R O B L E M S P L U S 267 5. Show that . 6. A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (see the figure). The car starts at a point 100 m west and 100 m north of the origin and travels in an easterly direction. There is a statue located 100 m east and 50 m north of the origin. At what point on the highway will the car’s headlights illuminate the statue? 7. Prove that . 8. Find the th derivative of the function . 9. The figure shows a circle with radius 1 inscribed in the parabola . Find the center of the circle. 10. If is differentiable at , where , evaluate the following limit in terms of : 11. The figure shows a rotating wheel with radius 40 cm and a connecting rod with length 1.2 m. The pin slides back and forth along the -axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute. (a) Find the angular velocity of the connecting rod, , in radians per second, when . (b) Express the distance in terms of . (c) Find an expression for the velocity of the pin in terms of . 12. Tangent lines and are drawn at two points and on the parabola and they intersect at a point . Another tangent line is drawn at a point between and ; it intersects at and at . Show that 13. Show that where and are positive numbers, , and . 14. Evaluate .lim x l H9266 e sin x H11002 1 x H11002 H9266 H9258 H33527 tan H110021 H20849bH20862aH20850r 2 H33527 a 2 H11001 b 2 ba d n dx n H20849e ax sin bxH20850 H33527 r n e ax sinH20849bx H11001 nH9258H20850 H11341 PQ 1 H11341 H11341 PP 1 H11341 H11001 H11341 PQ 2 H11341 H11341 PP 2 H11341 H33527 1 Q 2 T 2 Q 1 T 1 P 2 P 1 TP y H33527 x 2 P 2 P 1 T 2 T 1 H9258P H9258x H33527 H11341 OP H11341 H9258 H33527 H9266H208623 dH9251H20862dt xP AP lim x l a f H20849xH20850 H11002 f H20849aH20850 sx H11002sa fH11032H20849aH20850a H11022 0af x0 y 11 y=≈ y H33527 x 2 f H20849xH20850 H33527 x n H20862H208491 H11002 xH20850n d n dx n H20849sin 4 x H11001 cos 4 xH20850 H33527 4 nH110021 cosH208494x H11001 nH9266H208622H20850 sin H110021 H20849tanh xH20850 H33527 tan H110021 H20849sinh xH20850 x y FIGURE FOR PROBLEM 6 A P(x, 0) ¨ å FIGURE FOR PROBLEM 11 x y O P R O B L E M S P L U S 268 15. Let and be the tangent and normal lines to the ellipse at any point on the ellipse in the first quadrant. Let and be the - and -intercepts of and and be the intercepts of . As moves along the ellipse in the first quadrant (but not on the axes), what values can , , , and take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is. 16. Evaluate . 17. (a) Use the identity for (see Equation 14b in Appendix D) to show that if two lines and intersect at an angle , then where and are the slopes of and , respectively. (b) The angle between the curves and at a point of intersection is defined to be the angle between the tangent lines to and at (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection. (i) and (ii) and 18. Let be a point on the parabola with focus . Let be the angle between the parabola and the line segment , and let be the angle between the horizontal line and the parabola as in the figure. Prove that . (Thus, by a principle of geo- metrical optics, light from a source placed at will be reflected along a line parallel to the -axis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.) 0 x y F(p, 0) P(⁄, ›) ¥=4px y=› å ∫ x F H9251 H33527 H9252y H33527 y 1 H9252FP H9251FH20849p, 0H20850y 2 H33527 4pxPH20849x 1 , y 1 H20850 x 2 H11002 4x H11001 y 2 H11001 3 H33527 0x 2 H11002 y 2 H33527 3 y H33527 H20849x H11002 2H20850 2 y H33527 x 2 PC 2 C 1 PC 2 C 1 L 2 L 1 m 2 m 1 tan H9251 H33527 m 2 H11002 m 1 1 H11001 m 1 m 2 H9251L 2 L 1 tan H20849x H11002 yH20850 lim xl0 sinH208493 H11001 xH20850 2 H11002 sin 9 x x N x T y T y N 3 2 T N P x y 0 y N x N y T x T PN y N x N Tyxy T x T Px 2 H208629 H11001 y 2 H208624 H33527 1NT P R O B L E M S P L U S 269 19. Suppose that we replace the parabolic mirror of Problem 18 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, is a semicircle with center . A ray of light coming in toward the mirror parallel to the axis along the line will be reflected to the point on the axis so that (the angle of incidence is equal to the angle of reflection). What happens to the point as is taken closer and closer to the axis? 20. If and are differentiable functions with and , show that 21. Evaluate . 22. (a) The cubic function has three distinct zeros: 0, 2, and 6. Graph and its tangent lines at the average of each pair of zeros. What do you notice? (b) Suppose the cubic function has three distinct zeros: , , and . Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros intersects the graph of at the third zero. 23. For what value of does the equation have exactly one solution? 24. For which positive numbers is it true that for all ? 25. If show that . 26. Given an ellipse , where , find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) nega- tive reciprocals. 27. Find the two points on the curve that have a common tangent line. 28. Suppose that three points on the parabola have the property that their normal lines intersect at a common point. Show that the sum of their -coordinates is 0. 29. A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius are drawn using all lattice points as centers. Find the smallest value of such that any line with slope intersects some of these circles. 30. A cone of radius centimeters and height centimeters is lowered point first at a rate of 1 cmH20862s into a tall cylinder of radius centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged? 31. A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is , where is the radius and is the slant height.) If we pour the liquid into the container at a rate of , then the height of the liquid decreases at a rate of 0.3 cmH20862min when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container? 2 cm 3 H20862min l rH9266rl R hr 2 5 rr x y H33527 x 2 y H33527 x 4 H11002 2x 2 H11002 x a HS33527 bx 2 H20862a 2 H11001 y 2 H20862b 2 H33527 1 yH11032 H33527 1 a H11001 cos x y H33527 x sa 2 H11002 1 H11002 2 sa 2 H11002 1 arctan sin x a H11001sa 2 H11002 1 H11001 cos x xa x H33356 1 H11001 xa e 2x H33527 ksx k fa and b cba f H20849xH20850 H33527 H20849x H11002 aH20850H20849x H11002 bH20850H20849x H11002 cH20850 ff H20849xH20850 H33527 xH20849x H11002 2H20850H20849x H11002 6H20850CAS lim xl0 sinH20849a H11001 2xH20850 H11002 2 sinH20849a H11001 xH20850 H11001 sin a x 2 lim x l 0 f H20849xH20850 tH20849xH20850 H33527 fH11032H208490H20850 tH11032H208490H20850 tH11032H208490H20850 HS33527 0f H208490H20850 H33527tH208490H20850 H33527 0tf PR H11028PQO H33527H11028OQRRPQ OC OR P Q ¨ ¨ C A FIGURE FOR PROBLEM 19 James Stewart Stewart - Calculus - Early Transcedentals 6e calculus