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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
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 of a random variable that has an exponential distribution with parameter λ.
In her role as editor of a new national book club, Denise Briggs has compiled statistics suggesting that the fraction of its members who are still active t months after joining is given by the function S(t) = e−0.02t. The club currently has 5,000 members, and Denise expects new members to join at
Use integration by parts to verify reduction Formula 27: (In u)" du u(In u)" - n (In u)"-¹ du -nfan 1
The population P(t) (thousands) of a bacterial colony t hours after the introduction of a toxin is changing at the rate P´(t) = (1 − 0.5t)e0.5t thousand bacteria per hour. By how much does the population change during the fourth hour?
The concentration of a drug t hours after injection into a patient’s bloodstream is C(t) = 4te(2−0.3t) mg/mL. What is the average concentration of drug in the patient’s bloodstream over the first 6 hours after the injection?
To study the degradation of certain hazardous wastes with a high toxic content, biological researchers sometimes use the Haldane equationwhere a, b, and c are positive constants and S(t) is the concentration of substrate (the substance acted on by bacteria in the waste material).* Find the general
Use the integral table to findThen storeinto Y1 andinto Y2 using the bold graphing style. Graph both functions using the modified window [−3.7, 5.7]1 by [−2, 2]1. Verify that F´(x) = f (x) for x = (−2, 1). 1 x(3x - 6) dx.
After t weeks, contributions in response to a local fund-raising campaign were coming in at the rate of 2,000te−0.2t dollars per week. How much money was raised during the first 5 weeks?
A new virus has just been declared an epidemic by health officials. Currently, 10,000 people have the disease and it is estimated that t days from now new cases will be reported at the rate of R(t) = 10te−0.1t people per day. If the fraction of victims who still have the virus t days after first
In each case, first describe the domain of the given function and then find the partial derivatives fx, fy, fxx, and fyx. a. f(x, y) = x² + 2xy² - 3y4 2x + y b. f(x, y) = x - y c. f(x, y) = ²xy + In (y² - 2x)
Use integration by parts to verify reduction Formula 26: [ureau du - - - - 1 -uneau a a J leau du
Use 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 of the region bounded by the curves.
After t seconds, an object is moving with velocity te−t/2 meters per second. Express the position of the object as a function of time.
Repeat Exercise 72 for the curvesy = e2x + 4 and y = 5exData 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
Repeat Exercise 72 for the curvesy = ln x and y = x2 − 5x + 4Data 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
Evaluate the double integrals in Exercises 1 through 18. Jo J1 xy dx dy
Evaluate the double integrals in Exercises 1 through 18. ff. 2²y dy dx J1 Jo
In Exercises 1 through 4, use the method of Example 7.4.1 to find the corresponding least-squares line.(0, 1), (2, 3), (4, 2)Data from Example 7.4.1.Use the least-squares criterion to find the equation of the line that is closest to the three points (1, 1), (2, 3), and (4, 3).
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Find the maximum value of f(x, y) = xy subject to the constraint x + y = 1.
In Exercises 1 through 10, find the partial derivatives fx and fy. f(x, y) = 2x³y + 3xy² + y X
In Exercises 1 through 4, use the method of Example 7.4.1 to find the corresponding least-squares line.(1, 1), (2, 2), (6, 0)Data from Example 7.4.1.Use the least-squares criterion to find the equation of the line that is closest to the three points (1, 1), (2, 3), and (4, 3).
In Exercises 1 through 16, compute the indicated functional value.f(x, y) = 5x + 3y; f(−1, 2), f(3, 0)
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = 7x − 3y + 4
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) = 5 − x2 − y2
Evaluate the double integrals in Exercises 1 through 18. In 20 10 -1 2xe dx dy
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Find the maximum and minimum values of the function f(x, y) = xy subject to the constraint x2 + y2 = 1.
In Exercises 1 through 16, compute the indicated functional value.f(x, y) = x2 + x − 4y; f(1, 3), f(2, −1)
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = x − xy + 3
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Let f(x, y) = x2 + y2. Find the minimum value of f(x, y) subject to the constraint xy = 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) = 2x2 − 3y2
In Exercises 1 through 10, find the partial derivatives fx and fy.f(x, y) = (xy2 + 1)5
In Exercises 1 through 16, compute the indicated functional value.g(x, y) = x(y − x3); g(1, 1), g(−1, 4)
Describe the level curves of each of these functions:a. f(x, y) = x2 + y2b. f(x, y) = x + y2
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = 4x3 − 3x2y + 5x
In Exercises 1 through 4, use the method of Example 7.4.1 to find the corresponding least-squares line.(1, 2), (2, 4), (4, 4), (5, 2)Data from Example 7.4.1.Use the least-squares criterion to find the equation of the line that is closest to the three points (1, 1), (2, 3), and (4, 3).
In each case, find all critical points of the given function f(x, y) and use the second partials test to classify each as a relative maximum, a relative minimum, or a saddle point.a. f(x, y) = 4x3 + y3 − 6x2 − 6y2 + 5b. f(x, y) = x2 − 4xy + 3y2 + 2x − 4yc. f(x, y) = xy 1 y 1 x
Evaluate the double integrals in Exercises 1 through 18. *1 [f 2 نرا (x + 2y) dy dx
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) = xy
In Exercises 1 through 10, find the partial derivatives fx and fy.f(x, y) = √x(x − y2)
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Let f(x, y) = x2 + 2y2 − xy. Find the minimum value of f(x, y) subject to the constraint 2x + y = 22.
In Exercises 1 through 16, compute the indicated functional value.g(x, y) = xy − x(y + 1); g(1, 0), g(−2, 3)
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = 2x( y − 3x) − 4y
Evaluate each of these double integrals: 3 af fry dedy a. y c. ff. ² dx dy C. 1 bffre b. xey dx dy d. ff*xe- 0 xe dy dx
In Exercises 1 through 4, use the method of Example 7.4.1 to find the corresponding least-squares line.(1, 5), (2, 4), (3, 2), (6, 0)Data from Example 7.4.1.Use the least-squares criterion to find the equation of the line that is closest to the three points (1, 1), (2, 3), and (4, 3).
Use the method of Lagrange multipliers to find these constrained extrema:a. The smallest value of f(x, y) = x2 + y2 subject to x + 2y = 4.b. The largest and the smallest values of the function f(x, y) = xy2 subject to 2x2 + y2 = 6.
Evaluate the double integrals in Exercises 1 through 18. SC 1 Jo 2xy x²2² +1 dx dy
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) = 16 X 6 + = + x² - 3y² y
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(1, 2), (2, 2), (2, 3), (5, 5)
Evaluate the double integrals in Exercises 1 through 18. Jo Jo xe dy dx
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) = x2 + 2y2 − xy + 14y
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Let f(x, y) = 8x2 − 24xy + y2. Find the maximum and minimum values of the function f(x, y) subject to the constraint 8x2 + y2 = 1.
In Exercises 1 through 10, find the partial derivatives fx and fy. = √² + √² y f(x, y) =
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. ×100 8 f(x, y) = xy + = + = X y
In Exercises 1 through 16, compute the indicated functional value. f(x, y) 3x + 2y 2x + 3y ; f(1, 2), f(-4, 6)
In Exercises 1 through 10, find the partial derivatives fx and fy.f(x, y) = xe−y + ye−x
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) = x2 − y2 subject to the constraint x2 + y2 = 4.
In Exercises 1 through 16, compute the indicated functional value.f(x, y) = (x − 1)2 + 2xy3; f(2, −1), f(1, 2)
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = 2xy5 + 3x2y + x2
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(−4, −1), (−3, 0), (−1, 0), (0, 1), (1, 2)
Evaluate the double integrals in Exercises 1 through 18. xp kp & x* JJ
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Let f(x, y) = x2 − y2 − 2y. Find the maximum and minimum values of the function f(x, y) subject to the constraint x2 + y2 = 1.
In Exercises 1 through 16, compute the indicated functional value. 2 g(x, y) = √y² – x²; g(4, 5), g(−1, 2)
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(−2, 5), (0, 4), (2, 3), (4, 2), (6, 1)
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.z = 5x2y + 2xy3 + 3y2
A company will produce Q(K, L) = 120K3/4L1/4 hundred units of a particular commodity when the capital expenditure is K thousand dollars and the size of the workforce is L worker-hours. Find the marginal productivity of capital QK and the marginal productivity of labor QL when the capital
In Exercises 1 through 10, find the partial derivatives fx and fy.f(x, y) = x ln(x2 − y) + y ln( y − 2x)
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) = 2x3 + y3 + 3x2 − 3y − 12x − 4
Everett has just received $500 as a birthday gift and has decided to spend it on DVDs and video games. He has determined that the utility (satisfaction) derived from the purchase of x DVDs and y video games is U(x, y) = ln (x2√y) If each DVD costs $20 and each video game costs $50, how many DVDs
In Exercises 1 through 10, find the partial derivatives fx and fy. f(x, y) +3 x + y ху
Evaluate the double integrals in Exercises 1 through 18. (5 y SJ J1 0/ √/1 - y² dx dy
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(−6, 2), (−3, 1), (0, 0), (0, −3), (1, −1), (3, −2)
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.z = (3x + 2y)5
Evaluate the double integrals in Exercises 1 through 18. J2 ²x + y ху dy dx
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Find the maximum value of f(x, y) = xy2 subject to the constraint x + y2 = 1.
A company’s annual profit (in millions of dollars) for the first 5 years of operation is shown in this table:a. Plot these data on a graph.b. Find the equation of the least-squares line through the data.c. Use the least-squares line to predict the company’s 6th year profit. Year Profit
In Exercises 1 through 16, compute the indicated functional value. f(r, s) = S In r' f(e², 3), f(ln 9, e³)
In Exercises 1 through 16, compute the indicated functional value.g(u, v) = 10u1/2v2/3; g(16, 27), g(4, −1,331)
Evaluate the double integrals in Exercises 1 through 18. TT (+3) 1 J2 dy dx
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 − 1)2 + y3 − 3y2 − 9y + 5
In Exercises 1 through 10, find the partial derivatives fx and fy. f(x, y) = x2² - y² 2x + y
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = (x + xy + y)3
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Let f(x, y) = 2x2 + 4y2 − 3xy − 2x − 23y + 3. Find the minimum value of the function f(x, y) subject to the constraint x + y = 15.
A certain disease can be treated by administering at least 70 units of drug C, but that level of medication sometimes results in serious side effects. Looking for a safer approach, a physician decides instead to use drugs A and B, which result in no side effects as long as their combined dosage is
In Exercises 1 through 10, find the partial derivatives fx and fy.f(x, y) = xyexy
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) = x3 + y2 − 6xy + 9x + 5y + 2
A flat metal plate lying in the xy plane is heated in such a way that the temperature at the point (x, y) is T (ºC), where T(x, y) = 10ye−xy Find the average temperature over a rectangular portion of the plate for which 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1.
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(0, 1), (1, 1.6), (2.2, 3), (3.1, 3.9), (4, 5)
In Exercises 1 through 16, compute the indicated functional value. g(x, y) = X + g(1, 2), g(2, -3) X y
In Exercises 1 through 10, find the partial derivatives fx and fy. f(x, y) = In ху x + 3у
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(s, t) = 3t/2s
Evaluate the double integrals in Exercises 1 through 18. Vx ST Jo Jo xy dy dx
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(3, 5.72), (4, 5.31), (6.2, 5.12), (7.52, 5.32), (8.03, 5.67)
Evaluate the double integrals in Exercises 1 through 18. [fy o J1 xy VI - y²2 dx dy
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Let f(x, y) = 2x2 + y2 + 2xy + 4x + 2y + 7. Find the minimum value of the function f(x, y) subject to the constraint 4x2 + 4xy = 1
In Exercises 1 through 16, compute the indicated functional value.f(x, y) = xyexy; f(1, ln 2), f(ln 3, ln 4)
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) = −x4 − 32x + y3 − 12y + 7
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.(1, 15.6), (3, 17), (5, 18.3), (7, 20), (10, 22.4)Data from Example 7.4.4.A manufacturer gathers these data relating the level of
In Exercises 1 through 16, use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.Find the maximum and minimum values of f(x, y) = exy subject to x2 + y2 = 4.
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.z = t2/s3
In Exercises 1 through 16, compute the indicated functional value. f(s, t) = est est(1, 0), f(ln 2, 2) 2-
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