In Exercises 35-38, find the derivative of each function and evaluate f' (x) at the given value of x. 35. f ( x) = (2x - 1) (x2 + 3); x = 1 36. f(x) = 2x+ 1 Tix = 2 2x - 1'06 10 anoillim ni benuacom ai & owow sqo naged emaleve vinugo2 20A slab, and pl zbrogan 37. f ( x) = - X x4 - 2x2 - 1 ;x = -1 blow kal well anoil o issy brioose ?'vnegoros ard to 38. f (x) = (Vx+ 2x) (x312 - x);x = 4 In Exercises 39-42, find the slope and an equation of the tangent line to the graph of the function fat the specified point. 39. f (x) = (x3 + 1)(x2 - 2); (2, 18) 40. f(x) = * 2 2 , 4 41. f(x) = x+ 1 x+1 x 2 + 1; ( 1 , 1 ) bail , A (1)'S = (Do.E = (1)TH .s 42. f ( x ) = 1 + 2x1/2 ()1+ x3/2 4. 43. Suppose g(x) = x2f(x) and it is known that f(2) = 3 and f' (2) = -1. Evaluate g'(2). 44. Suppose g(x) = (x2 + 1)f(x) and it is known that f(2) = 3 and f'(2) = -1. Evaluate g'(2). 45. Find an equation of the tangent line to the graph of the function f(x) = (x3 + 1) (3x2 - 4x + 2) at the point (1, 2). 46. Find an equation of the tangent line to the graph of the 3x function f(x) = 2 _ , at the point (2, 3). 47. Let f(x) = (x2 + 1)(2 - x). Find the point(s) on the graph of f where the tangent line is horizontal. (x )1 .25 48. Let f(x) = X x2 + 1. Find the point(s) on the graph of f where the tangent line is horizontal. 49. Find the point(s) on the graph of the function f(x) = (x2 + 6)(x - 5) where the slope of the tangent line is equal to -2. x+1 50. Find the point(s) on the graph of the function f(x) = x - 1 where the slope of the tangent line is equal to 51. A straight line perpendicular to and passing through the point of tangency of the tangent line is called the normal to the curve at that point. Find the equation of the tangent line and the normal to the curve nine () ebns SE 1 y 1+ x261. LEARNING CURVES From experience, Emory Secretarial School knows that the average student taking Advanced AMPLI Computer Typing will progress according to the rule to the An 60t + 180 attacks spends on the time N (t ) t + 6 (t2 0) where N(t) measures the number of words per minute the student can type after t weeks in the course. me ed in hou a. Find an expression for N' (t). b. Compute N'(t) for t = 1, 3, 4, and 7, and interpret your results. c. Sketch the graph of the function N. Does it confirm the results obtained in part (b)? pret Its obta d. What will be the average student's typing speed at the soc end of the 12-week course?rural community of Glen Cove. As 19. f(x) = _ 1 01 20. f(u) =du lx2 + x + 2 puladel Cin the u + 1 aouagum $2 - 4 en by *3 - 2 ( S. 1 ) 21. f(s ) = rigs + 1 of onibigs 22. f(x ) = 23. f(x ) = - Vx + 1 Iniog off) is 15 40- (7 ) nononu] d x2 + 1 at which G 24. f(x ) =- Vx+ 2 25. f ( x ) = _x- + 2 28 26. f(x)_*+ 1 rate * + x+ 1 ulation be increas 2x2 + 2x + 3 27. f (x) = (x + 1)(x2 + 1) prl bail. se that the deny ()7 Ja1 . 8A mind equation for duct is px - 2 goshed al.quilhasgastadi guardWided. If x units of 28. f(x) = (3x2 - 1)(x2 xD(x) (number X nuts sole Wof Isbipolar Sniffer X x - 1 29. f(x ) = x2 - 4 x2 + 4 find 1 -ba Use lgeng all no (2 )injog and bar .0a the case of 30. f(x) x + V 3xicantinagast ard to sqola/bdrewordwed b 3x - 1 starts eng bes of aslysibagging sail idgisue A. .ta In Exercises 31-34, suppose f and g are functions that are differ- entiable at x = 1 and that f( 1 ) = 2, f' (1) = -1, 9(1) = -2, and g' (1) = 3. Find the value of h'(1). 31. h(x) =f(x)g(x)pa 32. h(x) = (x2 + 1)g(x) time xf (x) 61 milford. How -Capita 33. h(x) = f (x)9 ( x ) try be 34. h (x) = x + g (x) income at times f (x ) - g(x) is TheIn Exercises 1-30, find the derivative of each function. of [ nob 1 . f (x ) = 2x ( x2 + 1 ) I novit i susaob a'blinds 2. f(x) = 3x2(x - 1) (1)a 3. f (t ) = (t - 1) (2+ + 1 ) 4. f(x) = (2x + 3)(3x -4)ven goldasiquairis 5 . f ( x ) = ( 3x + 1 ) ( x2 - 2 ) lo all of losgeon rhiw ogbeob onerob a'blurlo s to egnado 6. f ( x) = (x + 1)(2x2 - 3x + 1) blinks blo-1Box-d s 101 7. f(x ) = (x3 - 1)(x+ 1) andT 30DM3DA8 30 123793 .20 8. f(x) = (x3 - 12x)(3x2 + 2x) in him \\Stullup fiends vd asvig at boonbound 9 . f (w ) = (w 3 - w 2 + w - 1 ) ( w 2 + 2 ) S, and 5 year 1 10 . f ( x ) = = x5 + ( x 2 + 1 ) (x2 - x - 1 ) + 28 cretarial 11. f(x) = (5x2 + 1)(2V/x - 1)haloing gild bailed 12. f (t) = (1 + Vt)(2+2 - 3) men & bam aim thewillgo ni sitslosd To nouslugog erb ai madW .beoub 13. f ( x) = (x2 - 5x + 2) ( x - only 162 \\ nimn S bas nim I squ X -onul basmob ofd'T 23HDTAN 2THO a bisore affhor mory the 14. f ( x ) = (x3 + 2x + 1 ) ( 2 + Course (05 = 1 2 0) 1 3 15 . f ( x ) = - 16. g(x) = + 2x2 usUp S x - 2 2x + 4 nsauorh s to elinu ni bay riellob 1 2x - 1 1 - 2t 17 . f (x ) 18. f(t) = bait 2x + 1 1 + 3t the alluest mov bris .(21)'s bus . (or) b .dDO 53. COST OF REMOVING TOXIC WASTE A city's main water reser- voir was recently found to be contaminated with @ vin trichloroethylene, a cancer-causing chemical, as a result of an abandoned chemical dump leaching chemicals into the the water. A proposal submitted to the city's council umped is members indicates that the cost, measured in millions of dollars, of removing x% of the toxic pollutant is given by show nwo wox m nous 0.5x lib to alu edi sisie .I olu I ins C(x) = oluI joubor .s 100 - X le of 6.8% per day. the rate is d Find C'(80), C'(90), C'(95), and C'(99). What does your result 10 tell you about the cost of removing all of the pollutant? ES Thomas Young has suggested the follow