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mathematics
applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

Explain how to find a second partial derivative of a function of two variables.
Find the formula (of the type in Check Your Understanding Problem 1) that gives the least-squares error for the points (8, 4), (9, 2), and (10, 3).
Calculate the following iterated integrals. LL'xew dy) dx xexy -1
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = -3x2 + 7xy - 4y2 + x + y
Let f (x, y, λ) = xy + λ(5 - x - y). Find f (1, 2, 3).
Find δf/δx and δf/δy for each of the following functions.f (x, y) = 2x2ey
Solve the following exercises by the method of Lagrange multipliers.Maximize x2 + xy - 3y2, subject to the constraint 2 - x - 2y = 0.
Find the least-squares error E for the least-squares line fit to the five points in Fig. 6. ∞ y 8 7 6 5 4 3 2 1 y = -1.3x + 8.3 1 2 3 4 5 Figure 6 X
Let g (x, y, z) = x/(y - z). Compute g (2, 3, 4) and g (7, 46, 44).
Find the formula (of the type in Check Your Understanding Problem 1) that gives the least-squares error for the points (2, 6), (5, 10), and (9, 15).
Calculate the following iterated integrals. LL'xy dx) dy
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum.f (x, y) = x2 - 5xy + 6y2 + 3x - 2y + 4
If A dollars are deposited in a bank at a 6% continuous interest rate, the amount in the account after t years is f (A, t) = Ae0.06t. Find and interpret f (10, 11.5).
Find all points (x, y) where f (x, y) has a possible relative maximum or minimum. - f(x, y) = ½x² + y² − 3x + 2y − 5
Explain how to find a first partial derivative of a function of two variables.
Find δf/δx and δf/δy for each of the following functions.f (x, y) = x2 - y2
Solve the following exercises by the method of Lagrange multipliers.Maximize x2 - y2, subject to the constraint 2x + y - 3 = 0.
Let g (x, y) = 2x2 + 2y2. Compute g (1, 1), g (0,-1), and g(a, b).
Calculate the following iterated integrals. L'ody) exty dy dx
Let f (x, y, z) = x2ey/z. Compute f (-1, 0, 1), f (1, 3, 3), and f (5, -2, 2).
Give an example of a level curve of a function of two variables.
Find the least-squares error E for the least-squares line fit to the four points in Fig. 5. 8 7 6 2.0 y 5 4 3 2 1 1 2 3 4 Figure 5 y = 1.1x + 3 X
Find δf/δx and δf/δy for each of the following functions.f (x, y) = 5xy
Solve the following exercises by the method of Lagrange multipliers.Minimize x2 + 3y2 + 10, subject to the constraint 8 - x - y = 0.
Let f(x, y) = x2 - 3xy - y2. Compute f (5, 0), f (5,-2), and f(a, b).
Let f (x, y) = x√y/(1 + x). Compute f (2, 9), f (5, 1), and f (0, 0).
Suppose that the interval 0 ≤ x ≤ 1 is divided into 100 subintervals with a width of Δx = .01. Show that the sum [3e-0.01] Ax + [3e-0.02] Ax + [3e-0.03] Ax +. + [3e-¹] Ax is close to 3(1-e¹).
The function g(x) in Fig. 9 resulted from shifting the graph of f (x) up 2 units. What is the derivative of h(x) = g(x) - f (x)? y Figure 9 121 y = g(x) y = f(x) X
The function g(x) in Fig. 8 resulted from shifting the graph of f(x) up 3 units. If f′(5) = 1/4, what is g′(5)? Y Figure 8 3 5 y = g(x) y = f(x) X
A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is y(t) = -32t feet per second.(a) Find s(t), the height of the rock above the ground at time t.(b) How long will the rock take to reach the ground?(c) What will be its velocity when it hits the ground?
Find the volume of the solid of revolution generated by revolving about the x-axis the region under the curve y = 1 - x2 from x = 0 to x = 1.
A ball is thrown upward from a height of 256 feet above the ground, with an initial velocity of 96 feet per second. From physics it is known that the velocity at time t is y(t) = 96 - 32t feet per second.(a) Find s(t), the function giving the height above the ground of the ball at time t.(b) How
Find the average value of f (x) = 1/x3 from x = 1/3 to x = 1/2.
Figure 7 contains an antiderivative of the function f (x). Draw the graph of another antiderivative of f (x). Figure 7 y [I]
Which of the following is ∫x√x + 1 dx?(a)(b) } (x + 1)5/² - (x + 1)³/² + C
Compute the area under the graph offrom 0 to 5. y 1 2 1 + x²
Three thousand dollars is deposited in the bank at 4% interest compounded continuously. What will be the average value of the money in the account during the next 10 years?
Evaluate a Riemann sum to approximate the area under the graph of f(x) on the given interval, with points selected as specified.f (x) = √1 - x2; -1 ≤ x ≤ 1, n = 20, left endpoints of subintervals
Find the consumers’ surplus for the demand curve p = √25 - .04x at the sales level x = 400.
Evaluate a Riemann sum to approximate the area under the graph of f(x) on the given interval, with points selected as specified.f (x) = x√1 + x2; 1 ≤ x ≤ 3, n = 20, midpoints of subintervals
Use a Riemann sum with n = 2 and midpoints to estimate the area under the graph ofon the interval 0 ≤ x ≤ 2. Then, use a definite integral to find the exact value of the area to five decimal places. f(x) 1 x + 2
Which of the following is ∫ln x dx?(a)(b) x · ln x - x + C(c) 1 - + C
Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves.y = 1/x, y = 3 - x
Use a Riemann sum with n = 5 and midpoints to estimate the area under the graph of f (x) = e2x on the interval 0 ≤ x ≤ 1. Then, use a definite integral to find the exact value of the area to five decimal places.
Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves.y = √x + 1, y = (x - 1)2
The area under the graph of the function e-x2 plays an important role in probability. Compute this area from -1 to 1.
In Exercises, we show that, as the number of subintervals increases indefinitely, the Riemann sum approximation of the area under the graph of f(x) = x2 from 0 to 1 approaches the value 1/3, which is the exact value of the area.Partition the interval [0, 1] into n equal subintervals of length Δx =
Redo Exercise 47 using right endpoints.Exercise 47Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph in Fig. 1 for 0 ≤ x ≤ 2. y 20 10 0 (.5, 14) Figure 1 .5 (1, 10) 1 (1.5, 6) 1.5 (2, 4) 2 X
In Exercises, we show that, as the number of subintervals increases indefinitely, the Riemann sum approximation of the area under the graph of f(x) = x2 from 0 to 1 approaches the value 1/3, which is the exact value of the area.Verify the given formula for n = 1, 2, 3, 4: 1² +2²+ + n² n(n +
Figure 5 shows the graphs of several functions f (x) for which f′(x) = 1/3. Find the expression for the function f (x) whose graph passes through (6, 3). 8 6 4 y ++ Figure 5 2 (6, 3) 6 8 10 +x 12
Figure 4 shows the graphs of several functions f (x) for which f′(x) = 2/x. Find the expression for the function f (x) whose graph passes through (1, 2). (1, 2) 8 6 4 2 -2 -4 y Figure 4 4 6 8 10 F. X 12
Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph in Fig. 1 for 0 ≤ x ≤ 2. y 20 10 0 (.5, 14) Figure 1 .5 (1, 10) 1 (1.5, 6) 1.5 (2, 4) 2 X
A farmer wants to divide the lot in Fig. 18 into two lots of equal area by erecting a fence that extends from the road to the river as shown. Determine the location of the fence. Figure 18 Road 0 10 20 30 40 50 60 70 80 06 ft 40 ft 35 ft 30 ft 25 ft 23 ft 22 ft 25 ft 30 ft 36 ft 42 ft River
Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves.y = 5 - (x - 2)2, y = ex
Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves.y = ex, y = 4x + 1
Find all functions f(x) that satisfy the given conditions. f'(x) = x² + √x, f(1) = 3
A rock thrown straight up into the air has a velocity of y(t) = -9.8t + 20 meters per second after t seconds.(a) Determine the distance the rock travels during the first 2 seconds.(b) Represent the answer to part (a) as an area.
Estimate the area (in square feet) of the residential lot in Fig. 17. 106 ft 0 20 Figure 17 101 ft 60 100 ft 100 113 ft 140 160
Money is deposited steadily so that $3000 is deposited each year into a savings account. After 10 years the balance is $36,887. What interest rate, with interest compounded continuously, did the money earn?
Find all functions f(x) that satisfy the given conditions. f'(x)=√x + 1, ƒ(4) = 0
The velocity of an object moving along a line is given by v(t) = t2 + t - 2 feet per second.(a) Find the displacement of the object as t varies in the interval 0 ≤ t ≤ 3. Interpret this displacement using area under the graph of v(t).(b) Find the total distance traveled by the object during the
Use a Riemann sum with n = 5 and midpoints to estimate the area under the graph of f (x) = √1 - x2 on the interval 0 ≤ x ≤ 1. The graph is a quarter circle, and the area under the graph is .78540, to five decimal places. See Fig. 16. Carry out the calculations to five decimal places and
A drug is injected into a patient at the rate of f (t) cubic centimeters per minute at time t. What does the area under the graph of y = f (t) from t = 0 to t = 4 represent?
Money is deposited steadily so that $1000 is deposited each year into a savings account.(a) Find the expression (involving r) that gives the (future) balance in the account at the end of 6 years.(b) Find the interest rate that will result in a balance of $6997.18 after 6 years.
The velocity of an object moving along a line is given by v(t) = 2t2 - 3t + 1 feet per second.(a) Find the displacement of the object as t varies in the interval 0 ≤ t ≤ 3.(b) Find the total distance traveled by the object during the interval of time 0 ≤ t ≤ 3.
If the marginal revenue function for a company is 400 - 3x2, find the additional revenue received from doubling production if 10 units are currently being produced.
The graph of the function f(x) = √1 - x2 on the interval -1 ≤ x ≤ 1 is a semicircle. The area under the graph is 1/2 π(1)2 = π/2 = 1.57080, to five decimal places. Use a Riemann sum with n = 5 and midpoints to estimate the area under the graph. See Fig. 15. Carry out the calculations to
Find all functions f(x) that satisfy the given conditions. f'(x) = 8x¹/3, f(1) = 4
Cars A and B start at the same place and travel in the same direction, with velocities after t hours given by the functions vA(t) and vB(t) in Fig. 29.(a) What does the area between the two curves from t = 0 to t = 1 represent?(b) At what time will the distance between the cars be greatest? Miles
Find all functions f(x) that satisfy the given conditions. f'(x) = x, f(0) = 3
A conical-shaped tank is being drained. The height of the water level in the tank is decreasing at the rate h′(t) = - t/2 inches per minute. Find the decrease in the depth of the water in the tank during the time interval 2 ≤ t ≤ 4.
A single deposit of $100 is made into a savings account paying 4% interest compounded continuously. How long must the money be held in the account so that the average amount of money during that time period will be $122.96?
An airplane tire plant finds that its marginal cost of producing tires is .04x + $150 at a production level of x tires per day. If fixed costs are $500 per day, find the cost of producing x tires per day.
A saline solution is being flushed with fresh water in such a way that salt is eliminated at the rate r(t) = -(t + 1/2) grams per minute. Find the amount of salt that is eliminated during the first 2 minutes.
A single deposit of $1000 is to be made into a savings account, and the interest (compounded continuously) is allowed to accumulate for 3 years. Therefore, the amount at the end of t years is 1000ert.(a) Find an expression (involving r) that gives the average value of the money in the account
Suppose that the interval 0 ≤ x ≤ 1 is divided into 100 subintervals of width Δx = .01. Show that the following sum is close to 5/4. [2(.01) + (.01)³]Ax + [2(.02) + (02)³]Ax + + [2(1.0) + (1.0)³] A.x.
Two rockets are fired simultaneously straight up into the air. Their velocities (in meters per second) are v1(t) and v2(t), and v1(t) ≥ v2(t) for t ≥ 0. Let A denote the area of the region between the graphs of y = v1(t) and y = v2(t) for 0 ≤ t ≤ 10. What physical interpretation may be
Find all functions f(x) that satisfy the given conditions. f'(x) = 2x ex, f(0) = 1 -
Use a Riemann sum with n = 4 and right endpoints to estimate the area under the graph of f (x) = 2x - 4 on the interval 2 ≤ x ≤ 3. Then, repeat with n = 4 and midpoints. Compare the answers with the exact answer, 1, which can be computed from the formula for the area of a triangle.
Let k be a constant, and let y = f (t) be a function such that y′ = kty. Show that y = Cekt2/2, for some constant C.
A sample of radioactive material with decay constant .1 is decaying at a rate R(t) = -e-0.1t grams per year. How many grams of this material decayed after the first 10 years?
The marginal profit for a certain company is MP1(x) = -x2 + 14x - 24. The company expects the daily production level to rise from x = 6 to x = 8 units. The management is considering a plan that would have the effect of changing the marginal profit to M2(x) = -x2 + 12x - 20. Should the company adopt
Suppose that the interval 0 ≤ x ≤ 3 is divided into 100 subintervals of width Δx = .03. Let x1, x2,....., x100 be points in these subintervals. Suppose that in a particular application we need to estimate the sumShow that this sum is close to 9. (3 − x₁)²Ax + (3 − x₂)² Ax + ··· +
Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph of f (x) = 4 - x on the interval 1 ≤ x ≤ 4. Then repeat with n = 4 and midpoints. Compare the answers with the exact answer, 4.5, which can be computed from the formula for the area of a triangle.
Describe all solutions of the following differential equations, where y represents a function of t.(a) y′ = 4t(b) y′ = 4y(c) y′ = e4t
Using the data from the previous exercise, find P(t).
Use a Riemann sum to approximate the area under the graph of f(x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles.1 ≤ x ≤ 7, n = 3, midpoints of subintervals 9 8 7 6 5 4 3 2 1 0 0 1 y = f(x) 2 3 4 5 6 7 8 9 Figure 14
Find all functions f(x) that satisfy the given conditions. f'(x) = .5e-0.2.x, f(0) = 0
Use a Riemann sum to approximate the area under the graph of f(x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles.4 ≤ x ≤ 9, n = 5, right endpoints 9 8 7 6 5 4 3 2 1 0 0 1 y = f(x) 2 3 4 5 6 7 8 9 Figure 14
After an advertising campaign, a company’s marginal profit was adjusted up from M1(x) = 2x2 - 3x + 11, before advertising, to M2(x) = 2x2 - 2.4x + 8, after advertising. Here x denotes the number of units produced, and M1(x) and M2(x) are measured in thousands of dollars per unit. Determine the
The rate of change of a population with emigration is given bywhere P(t) is the population in millions, t years after the year 2000.(a) Estimate the change in population as t varies from 2000 to 2010.(b) Estimate the change in population as t varies from 2010 to 2040. Compare and explain your
Find the function f (x) for which f′(x) = e-5x, f (0) = 1.
Deforestation is one of the major problems facing sub-Saharan Africa. Although the clearing of land for farming has been the major cause, the steadily increasing demand for fuel wood has also become a significant factor. Figure 28 summarizes projections of the World Bank. The rate of fuel wood
You took a $200,000 home mortgage at an annual interest rate of 3%. Suppose that the loan is amortized over a period of 30 years, and let P(t) denote the amount of money (in thousands of dollars) that you owe on the loan after t years. A reasonable estimate of the rate of change of P is given by
For the Riemann sums on an interval [a, b], in Exercises, determine n, b, and f (x).[3(.3)2 + 3(.9)2 + 3(1.5)2 + 3(2.1)2 + 3(2.7)2] (.6); a = 0
Find all functions f(t) that satisfy the given condition.f′(t) = t2 - 5t - 7
Use a Riemann sum to approximate the area under the graph of f(x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles.3 ≤ x ≤ 7, n = 4, left endpoints 9 8 7 6 5 4 3 2 1 0 0 1 y = f(x) 2 3 4 5 6 7 8 9 Figure 14
Find the function f(x) for which f′(x) = (x - 5)2, f (8) = 2.
Find all functions f(t) that satisfy the given condition. f'(t) 4 6 + t
For the Riemann sums on an interval [a, b], in Exercises, determine n, b, and f (x).[(5 + e5) + (6 + e6) + (7 + e7)] (1); a = 4

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