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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = x2 − 4x − y; C = −4, C = 5
A manufacturer is planning to sell a new product at the price of $150 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion, approximatelyunits of the product will be sold. The cost of manufacturing the product is $50 per unit. If the
The average retail price per gallon (in cents) of regular unleaded gasoline at 3-year intervals from 1992 to 2010 is given in this table:a. Plot these data on a graph, with the number of years after 1992 on the t axis and the average price of gasoline on the y axis.b. Find the equation of the
In Exercises 1 through 22, find the critical points of the given functions and classify each as a relative maximum, a relative minimum, or a saddle point. f(x, y) = x In X + 3x - xy²
In Exercises 17 through 22, describe the domain of the given function. f(x, y) = ety √x - 2y
In Exercises 17 through 24, find all critical points of the given function and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = 3x2y + 2xy2 − 10xy − 8y2
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = x2 + y; C = 0, C = 4, C = 9
In Exercises 19 through 24, use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = ex, y = 2, and x = 0.
In Exercises 17 through 24, find all critical points of the given function and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = 8xy − x4 − y4
In Exercises 23 through 30, sketch the indicated level curve f(x, y) = C for each choice of constant C.f(x, y) = x + 2y; C = 1, C = 2, C = −3
In Exercises 19 through 24, use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = ln x, y = 0, and x = e.
In Exercises 19 through 24, use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the triangle with vertices (1, 0), (1, 1), and (2, 0).
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0).f(x, y) = x2 + 3y at (1, −1)
In Exercises 1 through 22, find the critical points of the given functions and classify each as a relative maximum, a relative minimum, or a saddle point. f(x, y) = xye -(16x² +9y²)/288
Suppose the manufacturer in Exercise 22 decides to spend $8,100 instead of $8,000 on the development and promotion of the new product. Use the Lagrange multiplier λ to estimate how this change will affect the maximum possible profit.Data from Exercises 22A manufacturer is planning to sell a new
In Exercises 25 through 30, find all interior and boundary critical points and determine the largest and smallest values of the function f(x, y) over the given closed, bounded region R.f(x, y) = yex2−y on the circular region R bounded by x2 + y2 = 2.
In Exercises 25 through 36, evaluate the given double integral for the specified region R.where R is the triangle with vertices (0, 0), (1, 0), and (0, 2). fox. R (x + 2y) dA,
A manufacturer gathers the data listed in the accompanying table relating the level of production x (hundred units) of a particular commodity to the demand price p (dollars per unit) at which all the units will be sold:a. Plot these data on a graph.b. Find the equation of the least-squares line for
In Exercises 1 through 22, find the critical points of the given functions and classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = 4xy − 2x4 − y2 + 4x − 2y
Jennifer has several different kinds of investments, whose total value V(t) (in thousands of dollars) at the beginning of the tth year after she began investing is given in this table, for 1 ≤ t ≤ 10:a. Modify the least-squares procedure, as illustrated in Example 7.4.4, to find a function of
Rajit is an editor who has been allotted $60,000 to spend on the development and promotion of a new book. He estimates that if x thousand dollars are spent on development and y thousand dollars on promotion, approximately S(x, y) = 20x3/2y copies of the book will be sold.a. How much money should
In Exercises 1 through 22, find the critical points of the given functions and classify each as a relative maximum, a relative minimum, or a saddle point. f(x, y) 1 x² + y² + 3x - - 2y + 1
In Exercises 1 through 20, compute all first-order partial derivatives of the given function z = In X + y X
In Exercises 1 through 20, compute all first-order partial derivatives of the given function f(x, y) In(x + 2y) 2 y²
In Exercises 17 through 22, describe the domain of the given function. f(x, y) = X In(x + y)
A manager has been allotted $8,000 to spend on the development and promotion of a new product. It is estimated that if x thousand dollars are spent on development and y thousand dollars on promotion, approximately f(x, y) = 50x1/2y3/2 units of the product will be sold.a. How much money should the
A company’s annual sales (in units of 1 billion dollars) for its first 5 years of operation are shown in this table:a. Plot these data on a graph.b. Find the equation of the least-squares line.c. Use the least-squares line to predict the company’s sixth-year sales.
In Exercises 17 through 22, describe the domain of the given function. 2 f(x, y) = Vx² - y
In Exercises 19 through 24, use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = √x and y = x2
Evaluate the double integrals in Exercises 1 through 18. e ln x xy dy dx
In Exercises 17 through 24, find all critical points of the given function and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = x3 + y3 + 3x2 − 18y2 + 81y + 5
In Exercises 17 through 22, describe the domain of the given function. f(x, y) = √9-x² - y² /9
Evaluate the double integrals in Exercises 1 through 18. 10-y² Jo Jy²/4 xy dx dy
In Exercises 17 through 24, find all critical points of the given function and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = x3 + y3 + 3x2 − 3y2
Repeat Exercise 72 for the curvesData from Exercises 72Use the graphing utility of your calculator to draw the graphs of the curves y = x2e−x and y = 1/x on the same screen. Use ZOOM and TRACE or some other feature of your calculator to find where the curves intersect, and then compute the area
In Exercises 1 through 22, find the critical points of the given functions and classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = 2x4 + x2 + 2xy + 3x + y2 + 2y + 5
In Exercises 13 through 16, modify the least-squares procedure as illustrated in Example 7.4.4 to find a curve of the form y = Aemx that best fits the given data.(5, 33.5), (10, 22.5), (15, 15), (20, 10), (25, 6.8), (30, 4.5)Data from Example 7.4.4.A manufacturer gathers these data relating the
In Exercises 19 through 24, use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = x2 and y = 3x.
In Exercises 17 through 22, describe the domain of the given function. f(x, y) || 5x + 2y 4x + 3y
A manufacturer of television sets makes two models, the Deluxe and the Standard. The manager estimates that when x hundred Deluxe sets and y hundred Standard sets are produced each year, the annual profit will be P(x, y) thousand dollars, whereThe company can produce exactly 30,000 sets each year.
In Exercises 13 through 16, modify the least-squares procedure as illustrated in Example 7.4.4 to find a curve of the form y = Aemx that best fits the given data.(2, 13.4), (4, 9), (6, 6), (8, 4), (10, 2.7)Data from Example 7.4.4.A manufacturer gathers these data relating the level of production x
In Exercises 1 through 20, compute all first-order partial derivatives of the given function Z ty² 2.3 xy² + 1
In Exercises 17 through 24, find all critical points of the given function and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = (x + y + 3)2 − (x + 2y − 5)2
In Exercises 13 through 16, modify the least-squares procedure as illustrated in Example 7.4.4 to find a curve of the form y = Aemx that best fits the given data.(5, 9.3), (10, 10.8), (15, 12.5), (20, 14.6), (25, 17)Data from Example 7.4.4.A manufacturer gathers these data relating the level of
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(u, v) = u ln uv
In Exercises 17 through 24, find all critical points of the given function and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = (x + y)(2x + y − 6)
For each of these functions, find the slope of the indicated level curve at the specified value of x:a. f(x, y) = x2 − y3; f = 2; x = 1b. f(x, y) = xey; f = 2; x = 2
Evaluate the double integrals in Exercises 1 through 18. (2x Loft ex- x ex dy dx
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.z = u ln v
In Exercises 1 through 22, find the critical points of the given functions and classify each as a relative maximum, a relative minimum, or a saddle point.f(x, y) = (x − 4)ln(xy)
For each of these functions, sketch the indicated level curves:a. f(x, y) = x2 − y; f = 2, f = −2b. f(x, y) = 6x + 2y; f = 0, f = 1, f = 2
In Exercises 1 through 16, compute the indicated functional value.f(x, y, z) = xyez + xzey + yzex; f(1, 1, 1), f(ln 2, ln 3, ln 4)
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Find the minimum value of f(x, y, z) = x2 + y2 + z2 subject to 4x2 + 2y2 + z2 = 4.
Find the median of a uniformly distributed random variable over the interval A ≤ x ≤ B.
The median of a random variable X is the real number m such that P(X ≤ m) = 1/2. This term is used in Exercises 52 through 55.Find the median time you have to wait for the light to turn green in Example 6.4.3.Data from Example 6.4.3.A certain traffic light remains red for 40 seconds at a time.
A 2-hour movie runs continuously at a local theater. Seth leaves for the theater without first checking the show times. Use an appropriate uniform density function to find the probability that he will arrive at the theater within 10 minutes of (before or after) the start of the film.
Money is transferred into an account at the rate of R(t) = 1,000te−0.3t dollars per year for 5 years. If the account pays 4% interest compounded continuously, how much will accumulate in the account over a 5-year period?
The accompanying table gives the personal consumption expenditure and the corresponding disposable income (in billions of dollars) for the United States in the period 2003–2008:a. Plot these data on a graph, with disposable income on the x axis and consumption expenditure on the y axis.b. Find
In Exercises 17 through 22, describe the domain of the given function.f(x, y) = ln(x + y −4)
In Exercises 19 through 24, use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the rectangle with vertices (−1, 1), (2, 1), (2, 2), and (−1, 2).
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0). f(x, y) = x + X y - 3x at (1, 1)
The accompanying table gives the Dow Jones Industrial Average (DJIA) at the close of the first trading day of the year shown:a. Plot these data on a graph, with the number of years after 2001 on the x axis and the DJIA on the y axis.b. Find the equation of the least-squares line for the data.c.
In Exercises 21 through 28, evaluate the partial derivatives fx(x, y) and fy(x, y) at the given point (x0, y0).f(x, y) = x3y − 2(x + y) at (1, 0)
Use the table of integrals (Table 6.1) to find the integrals in Exercises 27 through 38. dx x(2 + 3x)
In Exercises 21 through 34, either evaluate the given improper integral or show that it diverges. ∞0+! 1 J2 t(In t)2 dt
A quarter circle of radius 1 has the equationfor 0 ≤ x ≤ 1 and has area π/4. Thus,Use this formula to estimate π by applying:a. The trapezoidal rule.b. Simpson’s rule. In each case, use n = 8 subintervals. y= V1-x² √1-2²
In Exercises 21 through 26, determine how many subintervals are required to guarantee accuracy to within 0.00005 in the approximation of the given integral by (a) The trapezoidal rule (b) Simpson’s rule. ၂ ၃ to dx
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. L e¹-x dx
In Exercises 1 through 26, use integration by parts to find the given integral. 1³ 2 √x² + 1 dx
Use the table of integrals (Table 6.1) to find the integrals in Exercises 27 through 38. x dx 3 – 5x
A manufacturer supplies refrigerators to two stores, A and B. The manager estimates that if x units are delivered to store A and y units to store B each month, the monthly profit will be P(x, y) hundred dollars, whereEach month, the company can produce exactly 700 refrigerators. How many
In Exercises 21 through 34, either evaluate the given improper integral or show that it diverges. Jo +00 x²e-2x dx
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. -1 [ 2 -∞0 X dx
In Exercises 27 through 32, find the expected value for the random variable with the density function given in the indicated problem.a. P(2 ≤ X ≤ 5)b. P(3 ≤ X ≤ 4)c. P(X ≥ 4) f(x) = 3 if 2 ≤ x ≤ 5 (0 otherwise
In Exercises 27 through 32, find the expected value for the random variable with the density function given in the indicated problem.a. P(0 ≤ X ≤ 2)b. P(1 ≤ X ≤ 2)c. P( X ≤ 1) X f(x) = {2 0 if 0 ≤ x ≤2 otherwise
In Exercises 19 through 26, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities.a. P(0 ≤ X b. P(2 ≤ X ≤ 4)c. P(X ≥ 6) f(x) xe x if x ≥ 0 if x < 0
In Exercises 21 through 26, determine how many subintervals are required to guarantee accuracy to within 0.00005 in the approximation of the given integral by (a) The trapezoidal rule (b) Simpson’s rule. 2.4 J1.2 et dx
In Exercises 21 through 34, either evaluate the given improper integral or show that it diverges. ∞0+. 2x dx
In Exercises 19 through 26, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities.a. P(X ≥ 0)b. P(1 ≤ X ≤ 2)c. P(X ≤ 2) f(x) = [2xe-x² 0 if x ≥ 0 if x < 0
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. 3e4x dx 8 ·00
In Exercises 1 through 26, use integration by parts to find the given integral. fre dx
In Exercises 21 through 26, determine how many subintervals are required to guarantee accuracy to within 0.00005 in the approximation of the given integral by (a) The trapezoidal rule (b) Simpson’s rule. 3 - dx 1 X
In Exercises 17 through 20, solve the given initial value problem using either integration by parts or a formula from Table 6.1. Note that Exercises 19 and 20 involve separable differential equations. dy dx 4 x² + 2x - 3 where y = 1 when x = 0
In Exercises 21 through 26, determine how many subintervals are required to guarantee accuracy to within 0.00005 in the approximation of the given integral by (a) The trapezoidal rule (b) Simpson’s rule. JI In(1+x) dx
In Exercises 19 through 26, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities.a. P(0 ≤ X b. P(X ≤ 2)c. P(X ≥ 5) f(x) = 10 0 -x/10 if x ≥ 0 if x < 0
In Exercises 21 through 26, determine how many subintervals are required to guarantee accuracy to within 0.00005 in the approximation of the given integral by (a) The trapezoidal rule (b) Simpson’s rule. 뭐 2 X dx
In Exercises 19 through 26, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities.a. P(1 ≤ X b. P(1 ≤ X ≤ 2)c. P(X ≥ 2) 3 f(x) = {x² 0 if x ≥ 1 if x < 1
In Exercises 21 through 34, either evaluate the given improper integral or show that it diverges. •+∞o -2x xe dx
In Exercises 1 through 26, use integration by parts to find the given integral. x(In x)² dx
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. +00 xex dx
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. *+∞ 1/x e J1 zł xp.
In Exercises 19 through 26, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities.a. P(0 ≤ X ≤ 4)b. P(1 ≤ X ≤ 2)c. P(X ≤ 1) 3 (4x - x²) if 0≤x≤ 4 otherwise f(x)=32 0
In Exercises 21 through 34, either evaluate the given improper integral or show that it diverges. *+∞o 3e-5x dx
In Exercises 19 through 26, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities.a. P(2 ≤ X ≤ 5)b. P(3 ≤ X ≤ 4)c. P(X ≥ 4) f(x) = 3 if 2 ≤ x ≤ 5 (0 otherwise
In Exercises 1 through 26, use integration by parts to find the given integral. In x 2 x dx
In Exercises 1 through 26, use integration by parts to find the given integral. (t-1)e¹-¹ dt
In Exercises 21 through 34, either evaluate the given improper integral or show that it diverges. *+∞ 3t 2² +1 dt
In Exercises 19 through 26, f(x) is a probability density function for a particular random variable X. Use integration to find the indicated probabilities.a. P(0 ≤ X ≤ 2)b. P(1 ≤ X ≤ 2)c. P( X ≤ 1) X f(x) = {2 0 if 0 ≤ x ≤2 otherwise
In Exercises 1 through 30, either evaluate the given improper integral or show that it diverges. 12 +00 1 x ln x - dx
In Exercises 15 through 20, approximate the given integral and estimate the error |En| using(a) The trapezoidal rule(b) Simpson’s rule with the specified number of subintervals. J1 In x dx; n = 4
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