MATA32H3 Exam Solution: Binary Logarithm, Implicit Function, Marginal Cost

EXERCISES 11.1 Find the derivatives of the functions in Problems 1-10 I. f(x) = 4 ln x 3, y=In 8x 23. Find d if y In(Vx1). 24. Finddy ify = In [r(t-x + i)) In Problems 25-38, find y'. 2. y 3 In 4, y = ln 5x 6. f(x) In 8. y=ln (6x + 1) 'In (4x +9) 26, y=x2 ln (2x + 3) 10, y=In (8x3-2x)-2x 11. Find dp/dq ifp=ln (q2+1). 25, y=x-ln x 28,y=l+Inx 30, y ln (3x + 1) 32, y= (in x) 34. y=VIn(3x+1) ds 12. Find ifs In + 1 29, y=ln (x' + 3)? 31, y = (In x)4 In each of Problems 13-18, find the derivative of the function in part (a). Then find the derivative of the func- tion in part (b) or show that the function in part (b) is the same function as that in part (a). 13. (a) y In x In (x 1) 33. y=[In(x' + 3)? 35-y = log, x 37. y 36, y=log,x 38 y = lo& (1-1-r) log (x4-4x3 + 1) In Problems 39-42, find the relative maxima and relative minima, and sketch the graph with a graphing utility to check your results. 39, y=xln x 41. y_x'-81n x (b) y = In 14. (a) y = ln (x-1) + ln (2x + 1) 15. (a) y=11n(x2-1) 16. (a) y-31n (x"-1) 17. (a) y=ln (4x-1)-31nx (b) y= ln [(x-1)(2x + 1)] 40, y=x2 ln x 42, y=ln x-x (b) y In (x"-1)" APPLICATIONS 4x-1 (b) y = In 18. (a) y 3In 43. Marginal cost Suppose that the total cost (in dollas) for a product is given by In (x + 1) C(a) 1500+ 200 In (2x + 1) (b) y= ln ( where x is the number of units produ (a) Find the marginal cost function. (b) Find the marginal cost when 200 units are po duced, and interpret your result ducing more items costs more. What thept (c) Total cost functions always increase D ds dt dy 20. Find ifs = In [r(t2-1)]. of the marginal cost function? Does it this problem? t + 3 21. Find-if)_ In dt 2)/4 44. Investment If money is invested at the the time to increase the investment by a 22. Find a ify=ln(

ader 1. A cup of coffee at 90°C is put into a 20°C room when t = 0, The coffee's temperature is changing at a rate of r(t) 7(09) °C per minute, with t in minutes. Esti- mate the coffee's temperature when t10 2. If the marginal cost function C (a) is measured in dol- lars per ton, and q gives the quantity in tons, what are the units of measurement for 100 C"(a) de? What does this integral represent? 3. The marginal cost of drilling an oil well depends on the depth at which you are drilling; drilling becomes more expensive, per meter, as you dig deeper into the earth. The fixed costs are 1,000,000 riyals (the riyal is the unit of currency of Saudi Arabia), and, if z is the depth in meters, the marginal costs are C,(x) = 4000 + 10r riyals/meter. Find the total cost of drilling a 500-meter well. 7. 4. The population of Tokyo grew at the rate shown in Fig- ure 5.65,. Estimate the change in population between 1970 and 1990 rate of growin (b (e o.5 P(0)