MA 302 Assignment number 6

Need help with Calculus Problems

Find both the first and second derivatives. Locate and relative maximum and minimum points and any points of inflection. Determine the intervals on which the function is concave upwards or concave downwards.

14. f(x) =

Find f”(x). Then evaluate f”(0), f”(1), and f”(4), if they exist.

24. f(x) =

Determine the intervals on which each function is concave upward and concave downward. Then locate all points of inflection. Use the information gathered to sketch the function. Confirm the details with a graphing calculator.

44. f(x) =

46. f(x) =

74. Meterology. Meterology records for a certain city suggest that for the month of June, the daily temperature between midnight and 6:00P.M. can be approximated by T(t) = -0.04 + 1.14 – 7.2t = 66 degrees, where t is the number of hours after midnight and 0 t 18.

a. Find the maximum and minimum temperature.

b. Find the maximum rate of increase in the temperature.

Sketch the graph of a continous function that satisfies all the given conditions.

8. Given conditions:

f(10) = 5

f’(5) = 0

f”(5) = 10

f”(x)0 if x 10

f”(x) 0 if x 10

For each of the functions, determine f’(x) and f”(x). Then complete a summary table with the headers of interval or value, derivative, nature of graph. Use this table to sketch the graph of the function.

24. f(x) = 2 – 3 – 12x + 5

44. The productivity rating of an individual worker at the Cruz Corporation assembly line is based on the number of tasks accomplished, mistakes made, and responsiveness to difficukties encountered, The average of all scores allows the company to use a simple model based on time on the floor given by PR = – 0.4 + 2 + 10x + 5, where x is in hours at work. A PR score of 20 is acceptable and a score of 40 is highly unusual.

a. When are workers scores the highest?

b. Design an 8-hour work day where workers do the most demanding jobs for about 6 hours and have 2 hours for less stressful work. Explain your reasoning.

For each of the rational functions find any vertical asymptotes, any horizontal asymptotes, and any oblique asymptotes.

12. f(x) =

Sketch the graph of the rational function. Show any related asymptotes on each graph.

24. f(x) =

30. The sugar level concentration in the bloodstream of a certain diabetes patient is modeled by

S(t) = 1 + , where S is in suitable units and t is the time in hours following a meal of allowed carbohydrate content.

a. Which asymptotes play a role here?

b. For 0 t 6, what is the highest S-value and when does it occur? )if this level exceeds 4, the patient will become ill).

c. Are there any inflection points? What is the meaning in the context of the problem of an inflection point?

8. Minimizing inventory costs. A car dealer expects to sell 1320 cars during the next year. It costs $660 to store one car for one year. To reorder, there is a fixed cost of $225, plus $304 for each car. Find the lot size and number of orders that should be placed so inventory costs will be minimized.

14. Maximizing revenue. Ms. Willis owns a 16-unit apartment complex. The unit rent is currently $400 per month, and all units are rented. Each time rent is increased by $20, one tenant will move out. Find the rental price that will maximize Ms. Wills’ revenue.

26. Suppose the demand for a product is $12 and the total costs are C(x) = 0.3 + 2x + 5.

a. What is the revenue function?

b. What is the profit function?

c. What is the maximum value of the profit?

6. A rectangular box is designed to have a square base and an open top. The volume is to be 256

a. What dimensions will give a minimum surface area?

b. What is the minimum surface area?

12. A front window on a new home is designed as a rectangle with a semicircle top. If the window is designed to let in a maximum amount of light, and the architect fixes the perimeter of the entire window at 600 inches, determine the radius r and rectangular height h so as to maximize the area.

Graph the function.

14. y =

22. y =

34. A colony of bees grows according to the formula P = where is the number present initially and t is the time in days. How many bees will be present after 6 days if there were 1200 present initially?

Use the properties of logarithms to rewrite each expression as the logarithm of a single expression. Be sure to use positive exponents and avoid radicals.

32. 3 ln x + ln(x + 5)

Use the properties of logarithms to rewrite each expression as a sum, difference, or constant multiple of logarithms. Replace all radicals with exponents.

42. ln( x)

Use the properties of logarithms to solve each equation.

54. = 72

Find the derivative of each of the functions.

14. f(x) = ln

26. y = ln ()

Determine a formula for f”(x).

34. f(x) = ln x + + 3x + 2

Use logarithmic differentiation to find the derivatives of the functions.

40. y=

Find the absolute extrema for each of the functions on the indicated interval.

46. f(x) = ;

Sketch the graph of each function.

54. f(x) =

Find the first and second derivative of each of the functions

6. f(x) = -7

Find a formula for f’(x) and determine the slope f’(a) at the point where x = a is given.

14. f(x) = 5; x = 0

Find any critical values, find and hypercritical values, find all intervals of concavity, and sketch the graph of the function.

36. f(x) 3

10. Half-life. The decay rate of a radioactive isotope is 6.5 percent per year. Find it’s half life.

16. The population of a certain economically depressed union is declining exponentially at a rate of 1.5 percent. If the population in 1990 was 30,000, estimate the population in 2010.

22. The number of dairy farmers in a particular state who are feeding a new supplement to their milking cows is given by the function W(t) = 340(1 -, where t is the number of months the supplement has been available. How long will it be before 200 farmers are feeding the supplement to their cows?

For each of the demand functions find, the elasticity function and the value of x that maximizes the revenue.

38. p = D(x) = 21-2