Application of derivatives Questions and Answers

Solve the problem. Records indicate that x years after 2008, the average property tax on a three bedroom home in a certain community was T(x)=20x^2 + 60x + 600 dollars. At what rate was the property tax increasing with respect to time in 2008?  $60 per year $600 per year  $100 per year $ 40 per year

A rectangular box has a height of 5 feet. The sum of the length and width of the box is 8 feet. (a) Express the volume V (in ft³) of the box as a function of the length (b) Find the maximum volume (in ft3) that such a box could have.

The height y (in feet) of a ball thrown by a child is y =-1/12 x² + 2x + 3 where x is the horizontal distance in feet from the point at which the ball is thrown. (a) How high is the ball when it leaves the child's hand? (b) What is the maximum height of the ball? (c) How far from the child does the ball strike the ground? Round your answers to the nearest 0.01.

The population of a certain town is growing at a constant rate. Suppose the population is 1900 initially, and after 2 years, the population is 2020. Which of these expresses the rate at which the population is increasing? Select the correct answer below: -60 people per year 60 people per month 30 people per month 1900 people per year 120 people per year 60 people per year

Given g(x) = -2x³ +96x and g'(x) = -6x² +96, select the local maximum and minimum of g(x). A. Local Max: (-4,0); Local Min: (4,0) C. Local Max: (-4,256); Local Min: (0,0) B. Local Max: (4,256); Local Min: (-4,-256) D. Local Max: (4,236); Local Min: (-4,-236) E. Local Max: (0,0); Local Min (4,-256)

Use logarithmic differentiation to find dy/dx. y = (2 + x)^3/x , x > 0 dy/dx=

A farmer is building a fence to endose a rectangular area against an existing wall, shown in the figure below. Three of the sides will require fencing and the fourth wall already exists. If the farmer has 144 feet of fencing, what are the dimensions of the region with the largest area? Select the correct answer below: 36 feet by 72 feet 44 feet by 64 feet 24 feet by 84 feet 32 feet by 76 feet 20 feet by 88 feet

When the annual rate of inflation averages 4% over the next 10 years, the approximate cost C of goods or services during any year in that decade is given below, where t is the time in years and P is the present cost. C(t) = P(1.04)^t (a) The price of an oil change for your car is presently $22.24. Estimate the price 10 years from now. (Round your answer to two decimal places.) C(10) = $ (b) Find the rates of change of C with respect to t when t = 1 and t = 9. (Round your coefficients to three decimal places.) At t = 1 At t = 9 (c) Verify that the rate of change of C is proportional to C. What is the constant of proportionality?

The monthly profit for a company that makes decorative picture frames depends on the price per frame. The company determines that the profit is approximated by f(p) =-80p²+2560p-17,600, where p is the price per frame and f(p) is the monthly profit based on that price. (a) Find the price that generates the maximum profit. (b) Find the maximum profit. (c) Find the price(s) that would enable the company to break even. If there is more than one price, use the "and" button.

The power P required to propel a ship varies directly as the cube of the speed s of the ship. If 6500 hp will propel a ship at 13.0 mi/h, what power is required to propel it at 16.0 mi/h? Set up the general variation equation. P= (Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.) The power required to propel the ship at 16.0 mi/h is hp. (Round to the nearest hundred as needed.)

What is the average rate of change of f(x) from x₁ = -1.6 to x₂ = 4.7? Please write your answer rounded to the nearest hundredth. f(x) = -6/7x + 4

Assume that a snowball is in the shape of a sphere. If a snowball melts so that its surface area decreases at a rate of 8 cm2/min, find the rate at which the diameter decreases when the diameter is 11 cm. (Round your answer to four decimal places.)

For the quadratic function f(x) = 2x² + 2x-6, answer parts (a) through (f). (a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down. The vertex is

Ying is a professional deep water free diver. His altitude (in meters relative to sea level), a seconds after diving, is modeled by: D(x) = 1/36(x-60)² - 100 How many seconds after diving will Ying reach his lowest altitude?

What is the expected profit if Jesaki Inc. produces and sells 31,006 widgets? Round to the nearest dollar. Note: The profit may be negative, if Jesaki Inc. experiences a loss.

Consider the following function. f(x) = (x-8)e^x (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) (c) Apply the First Derivative Test to identify all relative extrema. (If an answer does not exist, enter DNE.)

The number of strikeouts per game in Major League Baseball can be approximated by the function f(x) corresponds to one year of play. Step 2 of 2: What is the value of f'(5) and what does it represent? Answer f'(5) = What does f '(5) represent? = 0.065x+ 5.09, where x is the number of years after 1977 and The rate of change in expected strikeouts per game was f'(5) in 1982. The total change between 1977 and 1982 for expected strikeouts per game is f'(5). The average change between 1977 and 1982 for the expected number of strikeouts per game is f'(5). The expected strikeouts per game was f'(5) in 1982.

A local internet provider is constructing a function that will model its profits (P) in thousands of dollars (USD) as a function of the number of internet subscribers (s) in hundreds of people. To accomplish this, they have already constructed the function R(s) for revenue and E(s) for expenses, both measured in thousands of dollars. R(s) = 40 ln(s + 1) E(s) = 10e0.1s Find how many hundreds of subscribers are necessary to maximise the company's profit.

The position of an object moving along a line is given by the function s(t)= - 10t² +80t. Find the average velocity of the object over the following intervals. (a) [1,9] (b) [1, 8] (c) [1.7] (d) [1, 1+h] where h> 0 is any real number.. (a) The average velocity of the object over the interval [1, 9] is (b) The average velocity of the object over the interval [1, 8] is.

A closed box has height h cm and a square base of side x cm. It has volume 100 cm³. What is the value of x which minimises the surface area of the box? Give your answer in cm correct to 3 significant figures.

The Mean Value Theorem applies to the given function on the given interval. Find all possible values of c. Then sketch the graph of the given function on the given interval. f(x)=5x²+2x-3; [-3,3]

At a certain factory, it is determined that an output of Q units is to be expected when L worker-hours of labor are employed, where Q(L) = 3600√L a. Find the average rate of change of output as the labor employment changes from L = 3025 worker-hours to 3125 worker-hours. b. Use calculus to find the instantaneous rate of change of output with respect to labor level when L = 3025. A. about 32.46 units per worker-hour; about 32.2 units per worker-hour B. about 32.6 units per worker-hour; about 32.2 units per worker-hour C. about 32.46 units per worker-hour; about 32.73 units per worker-hour D. about 32.6 units per worker-hour; about 32.73 units per worker-hour

The gross annual earnings of a certain company were A(t) = 0.3t² + 12t + 21 thousand dollars t years after its formation in 2008. At what percentage rate were the gross annual earnings growing with respect to time in 2012? 18.74% per year 20.44% per year 19.51% per year 17.89% per year

Analyze and sketch a graph of the function over the given interval. Label any intercepts, relative extrema, points of inflection, and asymptotes. (Order your answers from smallest to largest x, then from smallest to largest y. If an answer does not exist, enter DNE. Round your intercept value to three decimal places.) Find the equations of the asymptotes. (Enter your answers as a comma-separated list of equations. Include asymptotes that fall within the closed interval 0 ≤ x ≤. If an answer does not exist, enter DNE.)

Malik, the business manager of a company that produces in-ground outdoor spas, determines that the cost of producing x spas is C thousand dollars, where C(x) = 0.06x² + 2.1x + 50 a. If Malik decides to increase the level of production from x= 10 to x= 11 spas, what is the corresponding average rate of change of cost? b. Next, Malik computes the instantaneous rate of change of cost with respect to the level of production x. What is this rate when * = 10?

Records indicate that x years after 2008, the average property tax on a three bedroom home in a certain community was T(x)=30x2 + 45x + 400 dollars. At what rate was the property tax increasing with respect to time in 2008? $60 per year $400 per year $105 per year $45 per year

If a formula for a linear function is given by f(x) = mx + b, find the best match below: b< 0 b=0 m < 0 m y-intercept is below the x-axis The line passes through the origin f decreases This is the average rate of change between any two points in this line

The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline. Find the point of diminishing returns. The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline, that is, when the graph turns from concave up, to concave down. This is its (No Response) point of inflection. The point of diminishing returns occurs when

Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Begin by finding the intercepts. (If an answer does not exist, enter DNE.) To find the y-intercept, substitute x = 0 and solve for To find the x-intercept, substitute y = 0 and solve for x.

Consider the following function. f(x) = 4 - |x-4| (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.). (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.)

Consider the following function. f(x) = 4|x - 4| (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) (-∞0,4) [4,00) increasing decreasing (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) 4,4 relative maximum (x, y) = relative minimum (x, y) = DNE

The function y=c1 cos 5x + c2 sin 5x is a two-parameter family of solutions of the second-order differential equation y" + 25y = 0. Find the values of c1 and c2 so that the solution satisfies the following boundary conditions. y(π) = 2, y(π/6) = 2.

Suppose the monthly cost for manufacturing bar stools is C(x)=517 +31x, where x is the number of bar stools produced each month. Find and interpret the marginal cost for the product. A. The marginal cost is $31 per bar stool which means that manufacturing one additional bar stool decreases the cost by $31. B. The marginal cost is $31 per bar stool which means that manufacturing one additional bar stool increases the cost by $31. C. The marginal cost is $517 per bar stool which means that manufacturing one additional bar stool decreases the cost by $517. D. The marginal cost is $517 per bar stool which means that manufacturing one additional bar stool increases the cost by $517.

A certain bacterial culture is growing so that it has a mass of 1/3 t² +8 grams after t hours. a) How much did it grow during the interval 2 ≤ t ≤ 2.01? b) What was the average growth rate during the interval 2 ≤t 2.01? c) What was its instantaneous growth rate at t = 2?

Consider the following function. f(x) = 4 - |x - 4| (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.)

Andy needs to create an open topped box to carry a catapult to school for physics class. He uses a piece of cardboard that is 40 inches by 50 inches to make the box, and plans to cut out square corners of measure x inches. What is the maximum possible volume of the box? 8,209 cubic inches 4,818 cubic inches 6,048 cubic inches 6,564 cubic inches

Identify the critical points and find the maximum value and minimum value on the given interval.. f(x)=x²-162x² + 162; 1=[-10,10] The critical points are The maximum value is The minimum value is

A 12-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 2 ft/sec, how fast is the top of the ladder moving down when the foot of the ladder is 4 feet from the wall? The top of the ladder is moving down at a rate of sec when the foot of the ladder is 4 feet from the wall.

A patient receives an injection of 1.9 milligrams of a drug, and the amount remaining in the bloodstream t hours later is A(t) = 1.9e-0.05t. Find the instantaneous rate of change of this amount at the following intervals. (a) just after the injection (at time t = 0). mg per hr (b) after 8 hours (Round your answer to three decimal places.) mg per hr

Which statement is true for the graph of f(x) = 2x³-6x²-48x + 24? (4, -140) is a relative minimum; (-2, 77) is a relative maximum (4, -136) is a relative minimum; (-2, 80) is a relative maximum (-2, 80) is a relative minimum; (4, -136) is a relative maximum (-2, 77) is a relative minimum; (4, -140) is a relative maximum

Find the polynomial function, f, such that there is a local minimum at x = 1, a local maximum at x = 2, a maximum value f(0) = 18, and a minimum value (root) is f(3) = 0. All on the interval [0,3]

Consider the function below. f(x) = -5x² + 30x + 5 Exercise (a) Find the critical numbers of f. Step 1 Differentiate f(x) = -5x² + 30x + 5 with respect to x. f'(x) = -10x +30 Step 2 To find the critical numbers, set f'(x) = 0 and solve for x. (Enter your answers as a comma-separated list.) -10x + 30 = 0

Find the average rate of change of g(x) = 5x² + 5/x4 on the interval [-4, 3].

The function f(x) = 2x³ 30x² +96x + 2 has one local minimum and one local maximum. Use a graph of the function to estimate these local extrema. This function has a local minimum at x = with output value: and a local maximum at x = with output value:

Find dy/dx by implicit differentiation. cot(y) = 6x - 5y

This question is designed to be answered with a calculator. The rate at which the value of a bank account changes over time is given by dY/dt = KV, where k is a constant. If the initial value of the account is Vo dollars and the value is expected to double in 35 years, which function gives the value of the account after t years? V(t) = Voe-0.020t V(t)= Voe0.020t V(t)= Vo 2-0.020t V(t)= Vo 20.020t

All edges of a cube are expanding at a rate of 6 centimeters per second. (a) How fast is the surface area changing when each edge is 4 centimeters? (b) How fast is the surface area changing when each edge is 11 centimeters?

Use the concavity theorem to determine the interval f(x)= (4x-5)² Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The function is concave up on (Type your answer in interval notation.) B. The function is not concave up on any interval. Previous Question on is concave up and where it is concave down. Also find all inflection points.

A point is moving along the graph of the given function at the rate dx/dt. Find dy/dt for the given values of x. dx y = 4x² + 1; dt (a) X = -1 dy dt (b) x = 0 dy dt = (c) X = 1 dy dt = = 3 centimeters per second cm/sec cm/sec cm/sec

Let D be the distance between the origin and a moving point on the graph of y = sin 7x. D =√ + y²