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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

Let t = 0 correspond to 1870, and let every decade correspond to an increase in t of 1. Use the table from Exercise 66.(a) Use 1870, 1920, and 1970 to find N, 1870 to find b, and 1920 to find k in the equation(b) Estimate the population of the United States in 2010 and compare your estimate to the
When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. Have you ever wondered how this is done? If the temperature of the medium (air, water, or whatever) has been fairly constant and less
When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. Have you ever wondered how this is done? If the temperature of the medium (air, water, or whatever) has been fairly constant and less
The following table shows the number of students taking the Advanced Placement (AP) Statistics Exam in recent years.Use a calculator with logistic regression capability to do the following.(a) Letting t represent the years since 2000, plot the number of exams y against the number of days t. Discuss
In Section 10.1 we studied three models for growth: exponential growth (Example 5), limited growth (Example 6), and logistic growth. Assume k > 0 in all three models, and N > y0 in the limited growth and logistic models.(a) For what values of t is each model increasing? Decreasing?(b) For
A roast at a temperature of 40°F is put in a 300°F oven. After 1 hour the roast has reached a temperature of 150°F. Newton’s law of cooling states thatwhere T is the temperature of an object, the surrounding medium has temperature TM at time t, and k is a constant. Use Newton’s law to find
Researchers have proposed that the amount a fulltime student is educated (x) changes with respect to the student’s age t according to the differential equationwhere k is a constant measuring the rate that education depreciates due to forgetting or technological obsolescence.(a) Solve the equation
A rumor spreads through the offices of a company with 200 employees, starting in a meeting with 10 people. After 3 days, 35 people have heard the rumor.(a) Write an equation for the number of people who have heard the rumor in t days.(b) How many people have heard the rumor in 5 days?
We saw that the acceleration of gravity is a constant if air resistance is ignored. But air resistance cannot always be ignored, or parachutes would be of little use. In the presence of air resistance, the equation for acceleration also contains a term roughly proportional to the velocity squared.
In Exercise 72, how long does it take for the roast to reach a temperature of 250°F?Exercise 72A roast at a temperature of 40°F is put in a 300°F oven. After 1 hour the roast has reached a temperature of 150°F. Newton’s law of cooling states thatwhere T is the temperature of an object, the
Let ƒ(x, y) = 4x2 - 2y2, and find the following.(a)(b)(c)(d) f(x + h, y) = f(x, y) h
Let ƒ9x, y) = 5x3 + 3y2, and find the following.(a)(b)(c)(d) f(x + h, y) = f(x, y) - h
Find fx(x, y) and fy(x, y).ƒ(x, y) = (y - 2)2ex+2y
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries.z = 8x + 4y + 10; -1 ≤ x ≤ 1, 0 ≤ y ≤ 3
Find all second-order partial derivatives for the following.r = ln |x + y|
The height of a cone is measured as 9.3 cm and the radius as 3.2 cm. Each measurement could be off by as much as 0.1 cm. Estimate the maximum possible error in calculating the volume of the cone.
Let ƒ(x, y) = y2 - 2x2y + 4x3 + 20x2. The only critical points are (-2, 42, 10, 0), and (5, 25). Which of the following correctly describes the behavior of ƒ at these points?(a) (-2, 4): local (relative) minimum(0, 0): local (relative) minimum(5, 25): local (relative) maximum(b) (-2, 4): local
The profit from the sale of x units of radiators for automobiles and y units of radiators for generators is given by P(x, y) = -x2 - y2 + 4x + 8y. Find values of x and y that lead to a maximum profit if the firm must produce a total of 6 units of radiators.
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries.z = 3x + 10y + 20; 0 ≤ x ≤ 3, -2 ≤ y ≤ 1
Find all second-order partial derivatives for the following.k = ln |5x - 7y|
Let ƒ(x, y) = xyex2+y2. Use a graphing calculator or spreadsheet to find each of the following.(a)(b) lim h→0 ƒ(1 + h, 1) − ƒ(1, 1) h
The height of a triangle is measured as 37.5 cm, with the base measured as 15.8 cm. The measurement of the height can be off by as much as 0.8 cm and that of the base by no more than 1.1 cm. Estimate the maximum possible error in calculating the area of the triangle.
Find fx(x, y) and fy(x, y).ƒ(x, y) = ln |2x2 + y2|
A manufacturing firm estimates that its total production of automobile batteries in thousands of units is ƒ(x, y) = 3x1/3y2/3, where x is the number of units of labor and y is the number of units of capital utilized. Labor costs are $80 per unit, and capital costs are $150 per unit. How many units
Consider the function ƒ(x, y) = x2(y + 1)2 + k(x + 1)2y2.(a) For what values of k is the point (x, y) = (0, 0) a critical point?(b) For what values of k is the point (x, y) = (0, 0) a relative minimum of the function?
The following table provides values of the function ƒ(x, y). However, because of potential errors in measurement, the functional values may be slightly inaccurate. Using the statistical package included with a graphing calculator or spreadsheet and critical thinking skills, find the function ƒ(x,
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries.z = x2; 0 ≤ x ≤ 2, 0 ≤ y ≤ 5
Find all second-order partial derivatives for the following.z = x ln |xy|
Suppose there is a maximum error of a% in measuring the radius of a cone and a maximum error of b% in measuring the height. Estimate the maximum percent error in calculating the volume of the cone, and compare this value with the maximum percent error in calculating the volume of a cylinder.
We found the least squares line through a set of n points(x1 , y1), (x2 , y2) ,....,(xn , yn) by choosing the slope of the line m and the y-intercept b to minimize the quantity S(m, b) = ∑(mx + b - y)2, where the summation symbol ∑ means that we sum over all the data points. Minimize S by
Find fx(x, y) and fy(x, y).ƒ(x, y) = ln |2 - x2y3|
For another product, the manufacturing firm in Exercise 35 estimates that production is a function of labor x and capital y as follows: ƒ(x, y) = 12x3/4y1/4. If $25,200 is available for labor and capital, and if the firm’s costs are $100 and $180 per unit, respectively, how many units of labor
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries.z = √y ; 0 ≤ x ≤ 4, 0 ≤ y ≤ 9
Production of a digital camera is given bywhere x is the amount of labor in work-hours and y is the amount of capital. Find the following.(a) What is the production when 32 work-hours and 1 unit of capital are provided?(b) Find the production when 1 work-hour and 32 units of capital are
Find all second-order partial derivatives for the following.z = (y + 1) ln |x3y|
Suppose that in measuring the length, width, and height of a box, there is a maximum 1% error in each measurement. Estimate the maximum error in calculating the volume of the box.
Find fxx(x, y) and fxy(x, y).ƒ(x, y) = 5x3y - 6xy2
Find the volume under the given surface z = f(x, y) and above the rectangle with the given boundaries. z = x√x² + y; 0≤x≤ 1,0 ≤ y ≤ 1
A farmer has 500 m of fencing. Find the dimensions of the rectangular field of maximum area that can be enclosed by this amount of fencing.
Suppose a function z = ƒ(x, y) satisfies the criteria for the test for relative extrema at a point (a, b), and ƒxx(a, b) > 0, while ƒyy(a, b) < 0. What does this tell you about ƒ(a, b)? Based on the sign of ƒxx(a, b) and ƒyy(a, b), why does this seem intuitively plausible?
For the functions defined as follows, find all values of x and y such that both fx(x, y) = 0 and fy(x, y) = 0.ƒ(x, y) = 9xy - x3 - y3 - 6
Use the region R with the indicated boundaries to evaluate each double integral. [f.x3y² dx dy; R bounded R by y = x, y by y = 2x, x = 1
For the functions defined as follows, find all values of x and y such that both fx(x, y) = 0 and fy(x, y) = 0.ƒ(x, y) = 2200 + 27x3 + 72xy + 8y2
The reaction to x units of a drug t hours after it was administered is given by R(x, t) = x2(a - x)t2e-t, for 0 ≤ x ≤ a (where a is a constant). Find the following.(a) ∂R/∂x(b) ∂R/∂t(c) ∂2R/∂x2(d) ∂2R/∂x∂t(e) Interpret your answers to parts (a) and (b).
Refer to Exercise 17. Which of the three cases applies to these functions?(a) ƒ(x) = x2; [0, 3](b) ƒ(x) = 2x; [0, 9](c)Exercise 17Suppose that ƒ(x) > 0 and ƒ′(x) > 0 for all x between a and b, where a 6 b. Which of the following cases is true of a trapezoidal approximation T for the
Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration. Vy Jo Jy x dx dy
For the integral in Exercise 11, apply the midpoint rule with n = 4 and Simpson’s rule with n = 8 to verify the formula S = (2M + T)/3.Exercise 11In Exercises, use n = 4 to approximate the value of the given integrals by the following methods dx x² X
The results from a research study in psychology were as follows.Repeat parts (a) and (b) of Exercise 23 for these data.Exercise 23we estimated the total U.S. wind energy consumption (in trillion BTUs) for the 12-year period from 2001 to 2019 using rectangles and the following data.(a) Approximate
Repeat the instructions of Exercise 21 using the integral in Exercise 12.Exercise 12In Exercises, use n = 4 to approximate the value of the given integrals by the following methods: 4 J2 3 x³ dx
1. According to the formula for C(n), what is the unit price of the 280th unit, to the nearest thousand dollars?2. Suppose that instead of using natural logarithms to compute b, we use logarithms with a base of 10 and define b = (log r)/(log 2). Does this change the value of b?3. All power
Determine whether each improper integral converges or diverges, and find the value of each that converges. J3 1 (x + 1)³° 3 dx
Determine whether each improper integral converges or diverges, and find the value of each that converges. 8 00 4 2 √x dx
Find each integral, using techniques from this or the previous chapter. x(8 - x)³/2 dx
Find a formula for the volume of an ellipsoid. See Exercises 23–25 and the following figures.Exercises 25The function defined by y = √r2 - x2 has as its graph a semicircle of radius r with center at (0, 0) (see the figure on the next page). In Exercises, find the volume that results when each
The rumen is the first division of the stomach of a ruminant, or cud-chewing animal. An article on the rumen microbial system reports that the fraction of the soluble material passing from the rumen without being fermented during the first hour after its ingestion could be calculated by the
The graph of 6x - 2y + 7z = 14 is a plane.Determine whether each of the following statements is true or false, and explain why.
The partial derivative with respect to x of a function z = ƒ(x, y) is the rate of change of z with respect to x if y is held constant.Determine whether each of the following statements is true or false, and explain why.
If ƒx(a, b) = 0 and ƒy(a, b) = 0, then ƒ has a relative maximum or minimum at (a, b).Determine whether each statement is true or false, and explain why.
The Lagrange function is a function of λ, x, and y, and possibly other variables.Determine whether each statement is true or false, and explain why.
The double integral of ƒ(x, y) over the region a ≤ x ≤ b and c ≤ y ≤ d can be written asDetermine whether each statement is true or false, and explain why. bed a C f(x, y) dx dy.
The expression z = ƒ(x, y) is a function of two variables if one or more values of z are obtained from each ordered pair of real numbers (x, y).Determine whether each statement is true or false, and explain why.
1. The general quadratic function of two variables has six terms. How many terms are in the general cubic function of two variables?2. Use the contour plot of orange-banana flavor to estimate the “flavor coordinates” of the best-tasting drink.3. Find the maximum on the flavor response surface
The differential dz can be used to approximate a small change in z, Δz.Determine whether each statement is true or false, and explain why.
The graph of 2x + 4y = 10 is a plane that is parallel to the z-axis.Determine whether each of the following statements is true or false, and explain why.
The partial derivative ƒx(a, b) gives the slope of the curve z = ƒ(x, b) at the point (a, b, ƒ(a, b)).Determine whether each of the following statements is true or false, and explain why.
If ƒx(a, b) = 0 and ƒy(a, b) = 0, then (a, b) is a critical point for ƒ.Determine whether each statement is true or false, and explain why.
The volume of the solid under the graph of a nonnegative function ƒ and over the region R isDetermine whether each statement is true or false, and explain why. f(x, y) dx dy. JR
The variable λ is the Lagrange multiplier in the Lagrange function.Determine whether each statement is true or false, and explain why.
The graph of ax + by + cz = d is a plane parallel to the xy-plane if a and b are both 0.Determine whether each statement is true or false, and explain why.
The differential for a function z = ƒ(x, y) is used to approximate a function by its tangent line.Determine whether each statement is true or false, and explain why.
If the partial derivatives with respect to x and y at some point are both 0, the tangent plane to the function at that point is horizontal.Determine whether each of the following statements is true or false, and explain why.
If ƒx(a, b) = 0, ƒy(a, b) = 0, ƒxx(a, b) = 0, ƒyy(a, b) = 0, and ƒxy(a, b) ≠ 0, then ƒ has a saddle point at (a, b).Determine whether each of the following statements is true or false, and explain why.
The method of Lagrange multipliers gives only the relative extrema, not the absolute extrema.Determine whether each statement is true or false, and explain why.
At a point on the graph of a function of two variables, z = ƒ(x, y), there may be many tangent lines, all of which lie in the same tangent plane.Determine whether each of the following statements is true or false, and explain why.
The double integral of ƒ(x, y) over the region g(y) ≤ x ≤ h(y) and m(x) ≤ y ≤ n(x) can be written asDetermine whether each statement is true or false, and explain why. n(x) ph(y) m(x) Jg(y) f(x, y) dx dy.
A level curve for a paraboloid could be a single point.Determine whether each of the following statements is true or false, and explain why.
The second-order partial derivatives ƒxy(x, y) and ƒyx(x, y) are always equal when they are both continuous.Determine whether each of the following statements is true or false, and explain why.
If the discriminant D = 0 at (a, b), then ƒ has a saddle point at (a, b).Determine whether each statement is true or false, and explain why.
The second derivative test from the previous section can be used to determine if the solution to a Lagrange multiplier problem is a minimum or a maximum.Determine whether each statement is true or false, and explain why.
The xy-trace of the function z = x2 + y2 is a single point.Determine whether each statement is true or false, and explain why.
A double integral can always be written both in the formdy/dx and the formDetermine whether each statement is true or false, and explain why. cb ch(x) a Jg(x) f(x, y)
The expression dV/V is used to estimate the percent error in calculating a volume V.Determine whether each statement is true or false, and explain why.
The circle x2 + y2 = 4 is a level curve for the ellipsoid x2 + y2 + z2/4 = 1.Determine whether each statement is true or false, and explain why.
For a function w = ƒ(x, y, z), calculating the differential dw requires calculating the partial derivative of w with respect to x, y, and z.Determine whether each statement is true or false, and explain why.
If ƒ(x, y) = 3x2 + 2xy + y2, then ƒ(x + h, y) = 3(x + h)2 + 2xy + h + y2.Determine whether each of the following statements is true or false, and explain why.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = xy + y - 2x
Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = 4xy, subject to x + y = 16
Let z = ƒ(x, y) = 6x2 - 4xy + 9y2. Find the following using the formal definition of the partial derivative.(a) ∂z/∂x(b) ∂z/∂y(c) ∂f/∂x(2, 3)(d) ƒy (1, -2)
Let ƒ(x, y) = 2x - 3y + 5. Find the following.(a) ƒ(2, -1) (b) ƒ(-4, 1)(c) ƒ(-2, -3) (d) ƒ(0, 8)
Evaluate dz using the given information.z = 2x2 + 4xy + y2; x = 5, y = -1, dx = 0.03, dy = -0.02
For a function z = ƒ(x, y), suppose that the point (a, b) has been identified such that ƒx(a, b) = ƒy(a, b) = 0. We can conclude that a relative maximum or a relative minimum must exist at (a, b).Determine whether each of the following statements is true or false, and explain why.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 3xy + 6y - 5x
Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = 2xy + 4, subject to x + y = 20
Let z = g(x, y) = 8x + 6x2y + 2y2. Find the following using the formal definition of the partial derivative.(a) ∂g/∂x(b) ∂g/∂y(c) ∂z/∂y(-3, 0)(d) gx(2, 1)
Let g(x, y) = x2 - 2xy + y3. Find the following.(a) g(-2, 4) (b) g(-1, -2)(c) g(-2, 3) (d) g(5, 1)
Evaluate dz using the given information.z = 5x3 + 2xy2 - 4y; x = 1, y = 3, dx = 0.01, dy = 0.02
A saddle point can be a relative maximum or a relative minimum.Determine whether each of the following statements is true or false, and explain why.
In Exercises, find all points where the functions have any relative extrema. Identify any saddle points.ƒ(x, y) = 3x2 - 4xy + 2y2 + 6x - 10
Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = xy2, subject to x + 2y = 15
In Exercises, find fx(x, y) and fy(x, y). Then find fx (2, -1) and fy (-4, 3). Leave the answers in terms of e in Exercises.ƒ(x, y) = -4xy + 6y3 + 5

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