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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

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
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
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
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
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
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
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) =
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 +
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
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.
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
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
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
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
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
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
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
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 +
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)
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
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,
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
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
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
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
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
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
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
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
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
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
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
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,
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
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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