HW 7

i=14 Differentiate. 1. flx) = x%sinx 3. f(x) =e*cosx N . f(x) =xcosx + 2tanx FY . y=128ecx — ¢scx 5. y = secf tanf 6. g(6) = e'(tand — @) cot ¢ 7. y=ccost+ t%sin¢t 8. f(t)=7 X S y=— 10. y = sin8 cos8 + 2 —tanx P ) sin 6 * cos X 11, f(6) = ——— 12, y= ———— 1® 1 + cos 6 ¥ 1 —sinx tsint sin ¢ 13. y = 4 y=—— AT YT T ¥ tans 15. f(6) = 0 cos 6 sin § 16. f(t) = te'cot t d 17. Prove thatd—(csc X) = —csc x cot x. x d 18. Prove that " (sec x) = sec x tan x. d 19. Prove that = (cot x) = —csc'x. 20. Prove, using the definition of derivative, that if S(x) = cos x, then f'(x) = —sin x. 27-2% Find an equation of the tangent line to the curve at the given point, 21. y =sinx + cosx, (0, 1) Zzoy=e'cosx, (0,1) (7T, _1) 23. y = cos x — sin x, 24, y=x+tanx, (mm) 25. (a) Find an equation of the tangent line to the curve ¥ = 2x sin x at the point (7/2, ). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. 26. (a) Find an equation of the tangent line to the curve ¥ = 3x + 6 cos x at the point (7/3, 77 + 3). (b) Ulustrate part (a) by graphing the curve and the tangent line on the same screen. 27. (a) If f(x) = secx — x, find f'(x). ;| (b) Check to see that your answer to part (a) is reasonable by graphing both f and (" for | x| < /2. 28. (a) If f(x) = e"cosx, find f'(x) and f"(x). (b) Check to see that your answers to part (a) are reasonable by graphing f, /', and f". 20 IEH(8) = 6 sind, find H'(6) and H" (). 30. If f(1) = sect, find f"(/4). 31. (a) Use the Quotient Rule to differentiate the function _ tanx — ] sec x (b) Simplify the expression for f(x) by writing it in terms of sin x and cos x, and then find f'(x). (¢) Show that your answers to parts (a) and (b) are equivalent, 32. Suppose f(m/3) = 4 and f'(7/3) = —2, and let g(x) = f(x) sin x and /1(x) = (cos x)/f(x). Find (@) ¢'(7/3) (b) #'(m/3) 33-34 Por what values of x does the graph of f have a horizon- tal tangent? 35 f(x) =x + 2sinx 34, f(x) =e'cosx 35. A mass on a spring vibrates horizontally on a smooth level surface (see the figure). lis equation of motion is x(z) = 8sin 1, where t is in seconds and x in centimeters. (a) Find the velocity and acceleration at time . (b) Find the position, velocity, and acceleration of the mass at time £ = 24r/3. In what direction is it moving at that time? ] equilibrium | posilion | s 0 X /" 36. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos ¢t + 3sint, ¢ = 0, where s is measured in centimeters and 7 in seconds. (Take the positive direction to be downward.) (a) Find the velocity and acceleration at time ¢. (b) Graph the velocity and acceleration functions. (c) When does the mass pass through the equilibrium position for the first time? (d) How far from its equilibrium position does the mass travel? (e) When is the speed the greatest? /. Aladder 10 ft long rests against a vertical wall. Let 8 be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to § when 8 = 7 /39 38. An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object.

754 3 1f the rope makes an angle 8 with the plane, then the magnitude of the force is w F=—" wsin@ + cos6 where g is a constant called the coefficient of friction. (a) Find the rate of change of F with respect to . (h) When is this rate of change equal 1o 07 (¢) Ifw=>501band p = 0.6, draw the graph of F as a function of @ and use it to locate the value of 0 for which dF/df = 0. Is the value consistent with your answer to part (b)? 54-50 Find the limit. g }l—l% 3x 0. }T}) sin mx . i s 2 i, . T aa. i SR 45, Jim e—jrint%l? 46. 11_\'1'(1) csc x sin(sin x) . jin et as. iy " 49 1 = tanx 6. i sin(x — 1) lim —— 5 v—m/4 Sin X — COS X =1 X2t x— 2 2%-22 Find the given derivative by finding the first few deriva- tives and observing the pattern that occurs. 99 15 52. :I.x—gs_(x sin x) 53. Find constants A and B such that the function y = Asinx + B cos x satisfies the differential equation Y +y =2y =sinux 54, (a) Evaluate lim x sin -]— x> X (b) Evaluate lim x sin —1— 1—0 s () Mlustrate parts (a) and (b) by graphing y = x sin(1/x). 3.4 The Chain Rule SECTION 3.4 The Chain Rule 197 55. Differentiate each trigonometric identity to obtain a new (or familiar) identity. sin x (a) tan x = (b) sec x = 08 X COS X . 1 + cotx (c) sinx +cosx =——"— cse x 56. A semicircle with diameter PQ sits on an isosceles triangle POR to form a region shaped like a two-dimensional ice-cream cone, as shown in the figure. If A(8) is the area of the semicircle and B(0) is the area of the triangle, find / 10¢em e \'\6',"{ vV R 57. The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle 6. Find N lim n—ot d X 58. Let f(x) m. (a) Graph f. What type of discontinuity does it appear to have at 0? (b) Calculate the left and right limits of f at 0. Do these values confirm your answer to part (a)? Suppose you are asked to differentiate the function F(x) = x>+ 1 The differentiation formulas you learned in the previous sections of this chapter do not enable you to calculate F'(x).

204 CHAPTER 3 Differentiation Rules 3.4 EXERCISES -6 Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function Ay SO e Lf(b) + &:]lg'(@) + &] A As Ax — 0, Equation 8 shows that Au — 0. So both &, — 0 and &, — 0 as Ax —> 0. Therefore Y. iy 2 dx Ax—0 Ax ' Ax = f(b)g'(a) = f'(9(a)) g'(a) This proves the Chain Rule. 21. F(f} - elsinZl ¥ = f().] Then find the derivative dy/dx. 1.y=3J1 + 4x 3, y=tanmx o y=e'/; 2. y= (24 + 5)* 33. G(x) = 4%/ 4, y = sin(cot x) | — gt 35. y= cos< = > 1 e~ 6.y=\/2——e"' 7. y = cot¥(sin 8) 746 Find the derivative of the function. 7. F(x) = (5x% + 2x%)* 9. flx) =V5x+ 1 11. £(8) = cos(6?) 3y 13. y = x%~ 15. f(f) = e"sin bt 7. flo)=@2x —~3)G* +5+ 1P 18. g(x) = (x? + 1)'(x? + 2)¢ 19. A(t) = (t + )22 — 1)° 20. F(r) = 3t — 1)*Q2r + D7? X 2l = x+1 23, y = 'm0 20~ (1557 8 F() = (1 + x + x)" 39. f(r) = tan(sec(cos #)) 1 41, f() = sin'(e™") 10. f(x) = ey 12, 4(6) = cos*0 14. f(¢) = tsinrt 43, g(x) = Qra'™ + n)* 45, y = cos+/sin(tan mx) li.T() [f,(b) + 82][9'('1) + 81] r} T 34. U(y) = (f" i 1)' 36, y = x2% '/ 38. y=+/1 + xe 40. y = "> + sin(e?) 2. y=wx+Jx+ /x a4, y—2" 32, F(t) = 46. y = [x + (x + sin'x)*]* 16. g(x) = "¢ 47-57 Find y" and y", 1 47. y = in 36 48, y= —— §= sl 2 (1 + tanx)? 49. y = /1 — sect 50, y = ¢ 1 < 22, y= <x + —~) 57-54 Find an equation of the tangent line to the curve at the given ¥ ' point. 24, f(r) = 2" 51. y=2% (0,1) 52, y=./1 +'x% (2,3) 1+ sin¢ 53. y = sin(sinx), (m, 0) e = V 1+ cost — - 54. y=xe, (0,0) 27. r(f) = 103 28, Flz) = glEN i~ I 29. H(r) = T 30. J(6) = tan*(n0) 5. (a) Find an equation of the tangent line to the curve y =2/(1 + ¢™) at the point (0, 1). (b) lustrate part (a) by graphing the curve and the tangent line on the same screen,