Math Archive: Questions from 2024-05-31

Find the rate of change of the distance D between the origin and a moving point on the graph of y = x2 + 8 if dx/dt = 8 centimeters per second. A) dDdt = 8 x3+136 xx4+16 x2+64 B) dDdt = 16 x3+24 xx4+16 x2+64 C) dDdt = 16 x3+136 xx4+17 x2+64 D) dDdt = 16 x3−136 xx3+17 x2+64 E) dDdt = 8 x3−24 xx3+17 x2+64

Find the rate of change of the distance D between the origin and a moving point on the graph of y = x2 + 4 if dx dt = 2 centimeters per second. dD dt = 4 x3+10 xx4+4 x2+16 dD dt = 4 x3−18 xx3+9 x2+16 dD dt = 4 x3+18 xx4+9 x2+16 dD dt = 2 x3+18 xx4+8 x2+16 dD dt = 2 x3−10 xx3+9 x2+16

The total stopping distance T of a vehicle is T = 3.5x + 4.5x2 where T is in feet and x is the speed in miles per hour. Use differentials to approximate the percent change in total stopping distance as speed changes from x = 34 to x = 39 miles per hour. Round your answer to one decimal place. 332.0% 114.7% 33.3% 14.7% 29.1%

A container holds 6 liters of a 25% brine solution. A model for the concentration C of the mixture after adding x liters of an 0.88% brine solution to the container and then draining x liters of the well-mixed solution is given as C = 6+7x 24+8x. Find x→∞ C. Round your answer to two decimal places.

A plane begins its takeoff at 2:00 P.M. on a 3400-mile flight. After 7.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

An open box of maximum volume is to be made from a square piece of material, s = 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown. ) Use the table to guess the maximum volume. V = (b) Write the volume V as a function of x. V = 0 < x < 12 (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. V = (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

Use a graphing utility to graph the function. f(x) = 7|x| x+2 Find the equations of any horizontal asymptotes. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

The graph of f(x) is shown (see figure). f(x) = 4 x2 x2+5 (a) Find the following limit. L = limx→∞ f(x) = (b) Determine x1 and x2 in terms of ε. x1 = x2 = (c) Determine M, where M > 0, such that |f(x)−L| < ε for x > M. M = (d) Determine N, where N < 0, such that |f(x)−L| < ε for x < N. N =

The graph of f(x) is shown. f(x) = 4x x2+9 (a) Find the following limits. L = limx→+∞ f(x) = K = limx→−∞ f(x) = (b) Determine x1 and x2 in terms of ε. x1 = x2 = (c) Determine M, where M > 0, such that |f(x)−L| < ε for x > M. M = (d) Determine N, where N < 0, such that |f(x)−K| < ε for x < N. N =

Find two positive numbers that satisfy the given requirements. The sum is S and the product is a maximum. (smaller value) (larger value)

Find two positive numbers satisfying the given requirements. The product is 432 and the sum of the first plus three times the second is a minimum. (first number) (second number)

Use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate the limit. Finally, find the limit analytically and compare your results with the estimates. (Round your answers to one decimal place. ) f(x) = x2 − x x2−x limx→∞ =

An employee has two options for positions in a large corporation. One position pays $15.80 per hour plus an additional unit rate of $0.70 per unit produced. The other pays $12.20 per hour plus a unit rate of $1.30. (a) Find linear equations for the hourly wages W in terms of x, the number of units produced per hour, for each option. option 1 W1 = option 2 W2 = (b) Use a graphing utility to graph the linear equations and find the point of intersection. (x, y) = ( ) (c) Interpret the meaning of the point of intersection of the graphs in part (b). When are produced, the for both options is $ per hour. How would you use this information to select the correct option if the goal were to obtain the highest hourly wage? Choose if you think you will produce less than units per hour and choose if you think you will produce more than units.

Find the length and width of a rectangle that has the given area and a minimum perimeter. Area: 3A square centimeters cm (smaller value) cm (larger value)

Find two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 51 and the product is a maximum. (first number) (second number)

From the graph, determine the x- and y-intercepts and the vertical and horizontal asymptotes. (If an answer does not exist, enter DNE.) x-intercept (x, y) = ( ) y-intercept (x, y) = ( ) vertical asymptote x = (smaller value) vertical asymptote x = (larger value) horizontal asymptote y =

Use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate your answer graphically. (Round your answers to five decimal places.) f(x) = 15x 16x2 − 2 limx→∞ f(x) =

From the graph, determine the x- and y-intercepts and the vertical and horizontal asymptotes. (If an answer does not exist, enter DNE. if necessary.) -intercept (x, y) = ( ) y-intercept (x, y) = ( ) vertical asymptote x = (smaller value) vertical asymptote x = (larger value) horizontal asymptote y =

A light source is located over the center of a circular table of diameter 4 feet. Find the height h of the light source such that the illumination I at the perimeter of the table is maximum if I = k(sin⁡α)/s2, where s is the slant height, α is the angle at which the light strikes the table, and k is constant.

Match the function with one of the graphs below using horizontal asymptotes as an aid. f(x) = 7 x x2 + 2

Match the function with one of the graphs below using horizontal asymptotes as an aid. f(x) = 9x2 − 3x + 2 x2 + 1

Find the limit. (Hint: Let x = 1 /t and find the limit as t → 0+.) limx→∞ 7xtan⁡(2/x)

A business has a cost of C = 0.3x + 250 for producing x units. The average cost per unit is given by the following. C¯ = C/x Find the limit of the average cost as x approaches infinity.

Use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point (−5, 8) Slope m is undefined (x, y) = ( ) (x, y) = ( ) (x, y) = ( )

A moving conveyor is built to rise 1 m for each 7 m of horizontal change. (a) Find the slope of the conveyor.(b) Suppose the conveyor runs between two floors in a factory. Find the length of the conveyor if the vertical distance between floors is 10 ft. (Round your answer to three decimal places.) ft

Find the slope and the y-intercept (if possible) of the line. (If an answer is undefined, enter UNDEFINED.) y = 4x + 2 slope y-intercept (x, y) = ( )

You are given the dollar value of a product in 2008 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 0 represent 2000. ) 2008 Value $1750 Rate $450 increase per year

Complete two iterations of Newton's Method for the function using the given initial guess. (Round your answers to four decimal places.) f(x) = tan⁡x, x1 = −0.03

Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. f(x) = 2 − x3 Newton's method: Graphing utility: x = x =

Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. f(x) = x5 + x − 8 Newton's method: Graphing utility: x = x =

Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001 . Then find the zero(s) using a graphing utility and compare the results. f(x) = x3 − 8.9x2 + 23.59x − 19.551 Newton's method: x = Graphing Utility: x = x = (smallest value) x = x = x = (largest value)

Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001 . Then find the zero(s) to three decimal places using a graphing utility and compare the results. f(x) = x3 − cos⁡x Newton's method: Graphing utility: x = x =

Apply Newton's Method to approximate the x-value of the indicated point of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001 . [Hint: Let h(x) = f(x) − g(x).] f(x) = x g(x) = tan⁡(x) x ≈

Apply Newton's Method to approximate the x-value(s) of the given point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001 . [Hint: Let h(x) = f(x) − g(x).] (Round your answer to three decimal places. ) f(x) = 2 arccos⁡x g(x) = arctan⁡x

Approximate the fixed point of the function to two decimal places. [A fixed point x0 of a function f is a value of x such that f(x0) = x0.] f(x) = -2 lnx x0 =

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers to four decimal places and compare the results with the exact value of the definite integral. ∫0 4 x2 dx, n = 4

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. ∫0 4 x3 dx, n = 4 Trapezoidal Simpson's exact

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers to four decimal places and compare the results with the exact value of the definite integral. ∫4 9 x dx, n = 8

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. ∫0 1 2 (x+2)2 dx, n = 4

Approximate the definite integral using the Trapezoidal Rule and Simpson's Rule with n = 4. Compare these results with the approximation of the integral using a graphing utility. (Round your answers to four decimal places.) ∫0 π/2 sin(x2) dx Trapezoidal Simpson's graphing utility

Use the Errors in the Trapezoidal Rule and Simpson's Rule Theorem to estimate the errors in approximating the integral, with n = 4, using the Trapezoidal Rule and Simpson's Rule. (Round your answers to four decimal places.) ∫0 1 sin⁡(5πx) dx (a) the Trapezoidal Rule (b) Simpson's Rule

Use the Errors in the Trapezoidal Rule and Simpson's Rule Theorem to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using the following rules. ∫0 2 x+2 dx

Use Simpson's Rule with n = 14 to approximate the area of the region bounded by the graphs of y = xcos⁡x, y = 0, x = 0, and x = π/2

Find two positive numbers satisfying the given requirements. The product is 1200 and the sum of the first plus three times the second is a minimum. (first number) (second number)

On a given day, the flow rate F (in cars per hour) on a congested roadway is given by the equation below, where v is the speed of traffic in miles per hour. F = v 27 + 0.06v2 What speed will maximize the flow rate on the road?

A farmer plans to enclose a rectangular pasture adjacent to a river. (see figure). The pasture must contain 80,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river? x = m y = m

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 13.5 square centimeters. cm (smallest value) cm cm (largest value)

A right triangle is formed in the first quadrant by the x - and y-axes and a line through the point (1, 2) (see figure below). Part (a) Write the length L of the hypotenuse as a function of x. Part (b) Use a graphing utility to approximate x graphically such that the length of the hypotenuse is a minimum. Part (c) Find the vertices of the triangle such that its area is a minimum.

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 16 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on advertising is shown below. P = −1/10 s3 + 6s2 + 400 Part (a) Find the amount of money the company should spend on advertising in order to yield a maximum profit. Part (b) 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.

(1 point) a) The rectangles in the graph below illustrate a Riemann sum for f(x) = 2sin⁡x on the interval [0, π]. The value of this Riemann sum is b) The rectangles in the graph below illustrate a Riemann sum for f(x) = 2sin⁡x on the interval [0, π]. The value of this Riemann sum is

(a) Estimate the area under the graph of f(x) = 10/x from x = 1 to x = 5 using four approximating rectangles and right endpoints. R4 = (b) Repeat part (a) using left endpoints. L4 = (c) By looking at a sketch of the graph and the rectangles, determine for each estimate whether is overestimates, underestimates, or is the exact area. 1. L4 2. R4

Find the Riemann sum for f(x) = x2 + 3x over the interval [0, 8] (see figure), where x0 = 0, x1 = 1, x2 = 3, x3 = 7, and x4 = 8, and where c1 = 1, c2 = 2, c3 = 5, and c4 = 8.

Write a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = 4 − |x| ∫−2 dx

Find the Riemann sum for f(x) = sin⁡(x) over the interval (0, 2π], where x0 = 0, x1 = π/4, x2 = π/3, x3 = π, and x4 = 2π, and where c1 = π/6, c2 = π/3, c3 = 2π/3, and c4 = 3π/2. (Round your answer to three decimal places.)

Write a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = 25 − x2 ∫−2 dx

Set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = 3e-x ∫0 dx

The radius r of a sphere is increasing at a rate of 7 inches per minute. (a) Find the rate of change of the volume when r = 10 inches. in.3/min (b) Find the rate of change of the volume when r = 36 inches. in.3/min

Sketch the region whose area is given by the definite integral. ∫−4 4 16−x2 dx Then use a geometric formula to evaluate the integral (a > 0, r > 0).

A point is moving along the graph of a given function such that dx/dt is 6 centimeters per second. Find dy/dt for the given values of x. y = 2x2 + 4 (a) x = −3 dy/dt = cm/sec (b) x = 0 dy/dt = cm/sec (c) x = 1 dy/dt = cm/sec

A point is moving along the graph of a given function such that dx/dt is 7 centimeters per second. Find dy/dt for the given values of x. (Round your answers to three decimal places.) y = 9cos⁡(x) (a) x = π/3 dy/dt = (b) x = π/6 dy/dt = (c) x = π/2 dy/dt =

A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat. (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 20 feet of rope out. ft/sec What happens to the speed of the boat as it gets closer to the dock? The speed of the boat as it approaches the dock. (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 20 feet of rope out. ft/sec What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock? The speed at which the winch pulls in rope .

Set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = cos(x) ∫0 dx

Find the Riemann sum for f(x) = sin⁡x over the interval [0, 2π], where x0 = 0, x1 = π/4, x2 = π/3, x3 = π, and x4 = 2π, and where c1 = π/6, c2 = π/3, c3 = 2π/3, and c4 = 3π/2. (Round your answer to three decimal places.)

Sketch the region whose area is given by the definite integral. ∫−2 2 (2 − |x|) dx Use a geometric formula to evaluate the integral.

Find the Riemann sum for f(x) = sin⁡x over the interval [0, 2pi], where x0 = 0, x1 = π/4, x2 = π/3, x3 = π, x4 = 2π, and c1 = π/6, c2 = π/3, c3 = 2π/3, c4 = 3π/2 Use a calculator to find the value accurate to 3 decimal places. Approximately 0.708 Approximately - 10.708 Approximately -0.708 None of these.

The graph of f consists of line segments and a semicircle, as shown in the figure. Evaluate each definite integral by using geometric formulas. (a) ∫0 2 f(x)dx (b) ∫2 6 f(x)dx (c) ∫−4 2 f(x)dx (d) ∫−4 6 f(x)dx (e) ∫−4 6 |f(x)|dx

Sketch the region whose area is given by the definite integral. ∫−1 1 (1 − |x|) dx Use a geometric formula to evaluate the integral.

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. y = 2(x2 − 3x) (a) Find dy/dt when x = 5, given that dx/dt = 1. dy/dt = (b) Find dx/dt when x = 8, given that dy/dt = 1. dx/dt =

A point is moving along the graph of a given function such that dx/dt is 5 centimeters per second. Find dy/dt for the given values of x. y = 4x2 + 2 (a) x = −3 dy/dt = cm/sec (b) x = 0 dy/dt = cm/sec (c) x = 3 dy/dt = cm/sec

Find the rate of change of the distance between the origin and a moving point on the graph y = sin⁡(7x) if dx/dt = 3 centimeters per second.

All edges of a cube are expanding at a rate of 7 centimeters per second. (a) How fast is the surface area changing when each edge is 2 centimeters? (b) How fast is the surface area changing when each edge is 10 centimeters?

At a sand a gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 9 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 27 feet high?

An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 90 miles from the point and has a speed of 450 miles per hour. The other is 120 miles from the point and has a speed of 600 miles per hour. Part (a) At what rate is the distance between the planes changing? Part (b) How much time does the controller have to get one of the airplanes on a different flight path?

The combined electrical resistance R of R1 and R2, connected in parallel, is given by the equation below, where R, R1, and R2 are measured in ohms. R1 and R2 are increasing at rates of 0.8 and 1.5 ohms per second, respectively. 1 R = 1 R1+1 R2 At what rate is R changing when R1 = 54 ohms and R2 = 84 ohms? (Round your answer to three decimal places.) ohm/sec

Sketch the region whose area is given by the definite integral. ∫0 8 3 dx O Use a geometric formula to evaluate the integral.

Sketch the region whose area is given by the definite integral. ∫0 4 (3x + 5) dx Use a geometric formula to evaluate the integral.

Sketch the region whose area is given by the definite integral. ∫0 4 (4 − x) dx Use a geometric formula to evaluate the integral.