Please help and answer only questions: 5, 7, 8, 9, 12, 18, 22, 24, 33, 45, 46, 58, 59, 67, 68, 69! and show worn

7.1 EXERCISES 1. What must be true of F(x) and G(x) if both are antiderivatives 4. Explain why the restriction n # -1 is necessary in the rule of f(x)? x" dx = *"+1 n+ + c. 2. How is the antiderivative of a function related to the function? 3. Explain what is wrong with the following use of the power rule: Find the following. ,zdx = - + C. 5 . 6 dk 6. 9dy x 3/361. 62.. 63. Flour Beetles A model for describing the population of adult our beetles involves evaluating the integral J' ell} X dz. where gfx) is die perunit-abundance growth rate for a popula- tion of size .t. The researchers consider the simple case in which g(x) = a 7 bx for positive constants c: and b. Find the integral in this case. Source: Ecology. Concentration of a Solute According to Fick's law' the dif- fusion of a solute across a cell membrane is given by em=%mam. m where A is the area of the cell membrane, V is the volume of the cell, ch) is the concentration inside the cell at time t. C is the concentration outside the cell. and k is a constant. [f q, repre- sents the concentration of the solute inside the cell when r = 0' then it can be shown that (5(3) = (CH _ C33 MIN 'l' C. [2) (3) Use the last result to nd 6(1). (1]) Substitute back into Equation (1) to show that (2) is indeed d'te correct antiderivative of [1}. Cell Growth Under certain conditions' the number of cancer cells NU) at time 1 increases at a rate N' (r) = Ac\386 CHAPTER 7 Integration 7 . ( 2 2 + 3 ) dz 8 . ( 3x - 5 ) dx 46. C'(x) = 0.2x2 + 5x; fixed cost is $10 47. C'(x) = 0.03eO.Olx, fixed cost is $8 9 . ( 61 2 - 81 + 7 ) dt 10 . ( 5x 2 - 6x + 3 ) dx 48. C'(x) = x'/2; 16 units cost $45 49. C'(x) = x23 + 2; 8 units cost $58 11 . (423 + 322 + 2z - 6) dz 12. (16y3 + 9yz - by + 3) dy 50. C'(x) = x + 1/x2; 2 units cost $5.50 51. C'(x) = 5x - 1/x; 10 units cost $94.20 13 . (5 V/ 2 + V2 ) dz 14 . ( 1 1 /4 + # 1 / 4 ) dt 52. C'(x) = 1.2*(In 1.2); 2 units cost $9.44 15 . 5 x ( x 2 - 8 ) dx 16 . x 2 ( x 4 + 4x + 3 ) dx Demand Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is 0. 17 . ( 4 V/ v - 3 v 3 12 ) dv 18 . ( 15 x V/ x + 2V/x) dx 53. R'(x) = 175 - 0.02x - 0.03x2 54. R' (x) = 50 - 5x-213 19 . ( 10 3/2 - 141 312 ) du . (56 +512 + 181 712 ) dt 55. R'(x) = 500 - 0.15 Vx 21. dz 56. R'(x) = 600 - 5 0.0002x 57. Chinese Patents The approximate rate of change in the num- ber of patent applications received in China in recent years is 23. 3 - V") dy 24 . ( Vu + * ) du given by p'(t) = 43.14t - 143.5, 25 . ( - 9: -25 - 21-1 ) dt 26 . ( 10x 3.5 + 4x-1 ) dx where t represents the number of years since 2000, and p rep resents the number of patents (in thousands). In 2008, 828,328 27 . 2 dx 28. 2 4 dx patent applications were received. Source: State Intellectual Property Office of the P.R.C. 29. 3e-0.2x dx 30. -4e 0.2v dv a) Find the function that gives the total number (in thousands) of patent applications received in China in year t. 31. 3+ 4e dax + ell ) dx 32. (2 - 3e Dar ) do (b) According to this function, how many patent applications were received in 2013? Compare this with the actual num- 34. [ 2y 1/2 - 3y2 ber of 2,377,061. 33. (1 + 20 dt 6y - dy 58. Profit The marginal profit of a small fast-food stand is given, in thousands of dollars, by 35 . ( @ 2 + 4u) du 36. ( v 2 - @ 37 ) dv P' (x ) = Vx+, 37 . ( x + 1 ) 2 dx 38 . ( 2y - 1 ) 2 dy where x is the sales volume in thousands of hamburgers. The "profit" is -$1000 when no hamburgers are sold. Find the 39. (Vx+1 dx [1 - 2Vz dz profit function. 40. Vx Vz 59. Profit The marginal profit in dollars on Brie cheese sold at a cheese store is given by 41. 10* dx 42. 32x dx P'(x) = x(50x2 + 30x), 43. Find an equation of the curve whose tangent line has a slope of where x is the amount of cheese sold, in hundreds of pounds. f' (x) = x23 The "profit" is -$40 when no cheese is sold. given that the point (1, 3/5 ) is on the curve. (a) Find the profit function. 44. The slope of the tangent line to a curve is given by (b) Find the profit from selling 200 lb of Brie cheese. f' (x) = 6x2 - 4x + 3. Life Sciences If the point (0, 1) is on the curve, find an equation of the curve. 60. Biochemical Excretion If the rate of excretion of a bio- chemical compound is given by f' (t) = 0.0le-0.01 APPLICATIONS the total amount excreted by time t (in minutes) is f (t). Business and Economics (a) Find an expression for f(t). Cost Find the cost function for each marginal cost function. (b) If 0 units are excreted at time t = 0, how many units are 45. C'(x) = 4x - 5; fixed cost is $8 excreted in 10 minutes