Application of derivatives Questions and Answers

This question is designed to be answered with a calculator When the volume of a sphere is 200 cubic inches in the radius of the sphere is increasing by what rate is the volume of the sphere changing Note The formula for the volume of a sphere is v is v O 3 989 in hour O 11 397 in hour O 82 694 in hour 165 388 in 3 hour inch per hour A

Problem 4 10 Points Words Words Words part 1 5 points each You boss decides to launch a new product cerebrum vastatorem and you are tasked with finding the optimal price and quantity The revenue function is R 5p 1000p where R is the revenue and p is the price Find the revenue maximizing price and the maximum revenue and graph it Remember that you can label the scale for your graph

1000 The population of deer in a park can be modeled by P t where P t is the 1 60e t number of deer and t 0 is time in years since the deer were introduced to the park A Eventually how many deer will be in the park Explain your reasoning B Evaluate P 8 P 3 8 3 problem using correct units Explain the meaning of this value in the context of the

Bryan has 180 feet of fencing to fence off his rectangular garden He will use the wall of his house as one side of the garden garden 1 Write an equation to find the area of the garden 2 What dimensions create the garden with the greatest area 3 What is the greatest area UPLOAD

8 A radioactive isotope is decaying at a rate of 18 every hour Currently there are 100 grams of the substance Part B How much will be left one day from now Round your answer to the nearest hundredth

A rice producer fills boxes with the same weights of different rice varieties to produce rice blends The machinery filling each box is designed to put 42 ounces of rice in each box But the machinery is imperfect The actual weight of each box varies by a certain number of bunces What mathematical model could the manufacturer use to represent the weights of the boxes

A twice differentiable function f is defined for all real numbers x The derivative of the function f and its second derivative have the properties and various values of x as indicated in the table 6 f x positive DNE negative 0 f x positive DNE positive 0 X 0 x 3 3 3 x 6 6 x 9 negative negative Part A Find all values of x at which f has a relative extrema on the interval 0 9 Determine whether f has a relative maximum or a relative minimum at each of these values Justify your answer 10 points Part B Determine where the graph of f is concave up and where it is concave down Support your answer 10 points Part C Find any points of inflection of f Show the analysis that leads to your answer 10 points

The function h represents the height of an object t seconds after it is launched into the air. The function is defined by h(t) = -5t² + 20t+18. Height is measured in meters. 1) When will the object hit the ground? (round to the nearest hundredth) seconds 2) At what 2 times does the object occupy a height of 33 meters? seconds and seconds 3) When does the object reach its maximum height? seconds 4) What is the maximum height that the object reaches? meters

The number of mosquitoes M(x), in millions, in a certain area depends on the June rainfall x, in inches: M(x) = 4x-x². What rainfall produces the maximum number of mosquitoes? A. 16 inches B. 0 inches C. 2 inches D. Maximum number of mosquitoes does not exist E. 4 inches

Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. f(x) = (7-5x)4/3 +7 A. Local minimum at (7/5,7) B. Local minimum at (0,7) C. Local maximum at (7/5,7) D. No local extrema

Find the producer's surplus if the supply function of some item is given by S(x)=x²+2x+8. Assume that supply and demand are in equilibrium at x = 30. A. 12,700 B. 17,200 C. 18,900 D. 19,800

Given that the variable cost is given by 2x² + 4x and fixed cost is 5000. Determine the: A. total cost function B. marginal cost function C. average cost function D. average cost when x = 5

Taylor is on one side of a river that is 50 meters wide and wants to reach a point 200 meters downstream on the opposite side as quickly as possible by swimming in a diagonal line across the river and then running the rest of the way. Where should Taylor exit the water if she can swim at 1.5 meters per second and run at 4 meters per second.

Find the minimum value of the objective function f(x, y) = x² + y² subject to the constraint y = r +1 (a) Use the constraint to rewrite the objective function with only a single variable. Then use single-variable calculus techniques to find the absolute minimum of the new function.

Given the function g(x) = 6x³ + 45x² + 72x, find the first derivative, g'(x) = 0 when a = 4, that is, g'(-4) = 0. Now, we want to know whether there is a local minimum or local maximum at z = 4, so we will use the second derivative test. Find the second derivative, g''(x). g''(x) = Evaluate g''(-4). g''(-4)= Based on the sign of this number, does this mean the graph of g(x) is concave up or concave down at a = - 4? At = - 4 the graph of g(x) is concave Based on the concavity of g(x) at x = At z = - 4 there is a local

Let'f be a twice-differentiable function with derivative given by f '(x) = 4x³ - 24x². Part A: Find the x-coordinate of any possible critical points of f. Show your work. Part B: Find the x-coordinate of any possible inflection points of f. Show your work. Part C: Use the Second Derivative Test to determine any relative extrema and inflection points. Justify your answers. Part D: If f has only one critical point on the interval [5, 8], what is true about the function f on the interval [5, 8]? Justify your answers.