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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 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) = xy2 − 6x2 − 3y2
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(−2.1, 3.5), (−1.3, 2.7), (1.5, 1.3), (2.7, −1.5)
Evaluate the double integrals in Exercises 1 through 18. -y Jo Jy-1 (2x + y) dx dy
In Exercises 1 through 20, compute all first-order partial derivatives of the given function. f(x, y) e² -x 1₂2
In Exercises 5 through 12, use the formulas to find the corresponding least-squares line.(−1.73, −4.33), (0.03, −2.19), (0.93, 0.15), (3.82, 1.61)
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.z = xexy
For each of the functions in Exercises 11 through 14, compute the second-order partial derivatives fxx, fyy, fxy, and fyx.f(x, y) = ex2+y2
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) = ln(xy2) subject to 2x2 + 3y2 = 8 for x > 0 and y > 0.
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 − 6xy − 2y3
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = xyex
For each of the functions in Exercises 11 through 14, compute the second-order partial derivatives fxx, fyy, fxy, and fyx.f(x, y) = x2 + y3 − 2xy2
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, z) = xyz subject to x + 2y + 3z = 24.
In Exercises 1 through 16, compute the indicated functional value.f(x, y, z) = xyz; f(1, 2, 3), f(3, 2, 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) = (x2 + 2y2)e1−x2−y2
For each of the functions in Exercises 11 through 14, compute the second-order partial derivatives fxx, fyy, fxy, and fyx.f(x, y) = x ln y
Evaluate the double integrals in Exercises 1 through 18. [S² 2xy 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 and minimum values of f(x, y, z) = x + 3y − z subject to z = 2x2 + y2.
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) = e−(x2+y2−6y)
In Exercises 1 through 16, compute the indicated functional value.g(x, y, z) = (x + y)eyz; g(1, 0, −1), g(1, 1, 2)
For each of the functions in Exercises 11 through 14, compute the second-order partial derivatives fxx, fyy, fxy, and fyx.f(x, y) = (5x2 − y)3
In Exercises 1 through 20, compute all first-order partial derivatives of the given function.f(x, y) = xex+2y
Evaluate the double integrals in Exercises 1 through 18. (4 0 Vxy dy dx
In Exercises 1 through 16, compute the indicated functional value. F(r, s, t) In(r + t) r+s+t F(1, 1, 1), F(0, e², 3e²)
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 − 4xy + y3
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, z) = x + 2y + 3z. Find the maximum and minimum values of f(x, y, z) subject to the constraint x2 + y2 + z2 = 16.
In Exercises 1 through 20, compute all first-order partial derivatives of the given function. f(x, y) = 2x + 3y y - x
During his annual medical checkup, Byron is advised by his doctor to adopt a regimen of exercise, diet, and medication to lower his blood cholesterol level to 220 milligrams per deciliter (mg/dL). Suppose Byron finds that his cholesterol level t days after beginning the regimen isa. What is
When she is 30, Sue starts making annual deposits of $2,000 into a bond fund that pays 8% annual interest compounded continuously. Assuming that her deposits are made as a continuous income flow, how much money will be in her account if she retires at age 55?
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. *5 [²12² J2 (2 + 2t + 31²) dt
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. X 2x + 1 dx
Environmentalists estimate that the population of a certain endangered species is currently 3,000. The population is expected to be growing at the rate of R(t) = 10e0.01t individuals per year t years from now, and the fraction that survive t years is given by S(t) = e−0.07t. What will the
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 2 S² f₁ (2x - 4)² dx
In Exercises 31 through 34, solve the given initial value problem for y = f(x).dy/dx = e−x where y = 3 when x = 0
A manufacturer supplies S(p) = 0.5p2 + 3p + 7 hundred units of a certain commodity to the market when the price is p dollars per unit. Find the average supply as the price varies from p = $2 to p = $5.
In Exercises 1 through 20, find the indicated indefinite integral. S √3x + 1 dx
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. S (²-3/) dx
A patient is injected with a drug, and t hours later, the concentration of the drug remaining in the patient’s bloodstream is given byWhat is the average concentration of the drug during the first 3 hours after the injection? C(t) = 0.3t (t² + 16) ¹/2 mg/cm³
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. 2²-1 2xet dx
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. 1(1² (1² + 1)5 dt +
In Exercises 1 through 20, find the indicated indefinite integral. [(3. (3x + 1)√3x² + 2x + 5 dx
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the curve y = x2 + 2 from x = −1 to x = 3.
In Exercises 7 through 10, p = D(q) is the price (dollars per unit) at which q units of a particular commodity will be demanded by the market (that is, all q units will be sold at this price), and q0 is a specified level of production. In each case, find the price p0 = D(q0) at which q0 units will
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. (31² V5t + 2) dt
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the curve y = 1/x2, and the lines y = x and y = x/8.
In Exercises 9 through 14, estimate the area under the graph of y = f(x) over the interval 0 ≤ x ≤ 4 by computing a Riemann sum of f over the interval with 8 subintervals, using left endpoints. Then find the actual area using the fundamental theorem of calculus.f(x) = 4 − x
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the curvefrom x = − 2 to x = 2. y = √4 = x²
In Exercises 9 through 14, estimate the area under the graph of y = f(x) over the interval 0 ≤ x ≤ 4 by computing a Riemann sum of f over the interval with 8 subintervals, using left endpoints. Then find the actual area using the fundamental theorem of calculus. I + X X f(x)
In Exercises 1 through 20, find the indicated indefinite integral. fox (x + 2)(x² + 4x + 2)5 dx
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. [3₁\ 3t√² + 8 dt
In Exercises 7 through 10, p = D(q) is the price (dollars per unit) at which q units of a particular commodity will be demanded by the market (that is, all q units will be sold at this price), and q0 is a specified level of production. In each case, find the price p0 = D(q0) at which q0 units will
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. (x1/3 3x-2/3 + 6) dx
In Exercises 1 through 20, find the indicated indefinite integral. x + 2 x² + 4x + 2 dx
In Exercises 9 through 14, estimate the area under the graph of y = f(x) over the interval 0 ≤ x ≤ 4 by computing a Riemann sum of f over the interval with 8 subintervals, using left endpoints. Then find the actual area using the fundamental theorem of calculus.f(x) = x2 + 1
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the curve y = 4 − x2 from x = − 2 to x = 2.
In Exercises 9 through 14, estimate the area under the graph of y = f(x) over the interval 0 ≤ x ≤ 4 by computing a Riemann sum of f over the interval with 8 subintervals, using left endpoints. Then find the actual area using the fundamental theorem of calculus. f(x) 1 5-x
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region bounded by the curves y = x2 − 2x and y = −x2 + 4.
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the curvefrom x = 1 to x = e2. y 1 Vx
In Exercises 1 through 20, find the indicated indefinite integral. 3x + 6 (2x² + 8x + 3)² - dx
In Exercises 11 through 14, p = S(q) is the price (dollars per unit) at which q units of a particular commodity will be supplied to the market by producers, and q0 is a specified level of production. In each case, find the price p0 = S(q0) at which q0 units will be supplied and compute the
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. x²(x + 1)³/4 dx
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. Jovy- √y - 2y-³) dy
In Exercises 15 through 19, the demand and supply functions, D(q) and S(q), for a particular commodity are given. Specifically, q units of the commodity will be demanded (sold) at a price of p = D(q) dollars per unit, while q units will be supplied by producers when the price is p = S(q) dollars
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. xel-x² dx Xx
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. J 2y 1 2 3 + Vy dy
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region between the curves y = x3 − 3x2 and y = x2 + 5x.
In Exercises 1 through 20, find the indicated indefinite integral. fa- 5)¹2 dt
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the curve y = 1/x from x = 1 to x = 10.
In Exercises 11 through 14, p = S(q) is the price (dollars per unit) at which q units of a particular commodity will be supplied to the market by producers, and q0 is a specified level of production. In each case, find the price p0 = S(q0) at which q0 units will be supplied and compute the
In Exercises 5 through 18, sketch the given region R and then find its area.R is the region between the curve y = x3 and the line y = 9x, for x ≥ 0.
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. J(² + xVx) dx 2
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 2 15 -1 5 dx
In Exercises 1 through 20, find the indicated indefinite integral. Sve 12 v(v - 5)¹2 dv
In Exercises 15 through 19, the demand and supply functions, D(q) and S(q), for a particular commodity are given. Specifically, q units of the commodity will be demanded (sold) at a price of p = D(q) dollars per unit, while q units will be supplied by producers when the price is p = S(q) dollars
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. 2y4 ST dy + 1
In Exercises 11 through 14, p = S(q) is the price (dollars per unit) at which q units of a particular commodity will be supplied to the market by producers, and q0 is a specified level of production. In each case, find the price p0 = S(q0) at which q0 units will be supplied and compute the
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. y² (³ + 5)2 dy
In Exercises 9 through 14, estimate the area under the graph of y = f(x) over the interval 0 ≤ x ≤ 4 by computing a Riemann sum of f over the interval with 8 subintervals, using left endpoints. Then find the actual area using the fundamental theorem of calculus.f(x) = x(4 − x)
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 1 π dx -2
In Exercises 7 through 14, find the volume of the solid formed by rotating the region R about the x axis.R is the region under the curve y = e−0.1x from x = 0 to x = 10.
In Exercises 11 through 14, p = S(q) is the price (dollars per unit) at which q units of a particular commodity will be supplied to the market by producers, and q0 is a specified level of production. In each case, find the price p0 = S(q0) at which q0 units will be supplied and compute the
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. 1 [(VF 2√7¹ V) dx + 2
In Exercises 5 through 18, sketch the given region R and then find its area.R is the triangle bounded by the line y = 4 − 3x and the coordinate axes.
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. 5 fo 10 (3x + 2) dx
In Exercises 5 through 18, sketch the given region R and then find its area.R is the triangle with vertices (−4, 0), (2, 0), and (2, 6).
In Exercises 1 through 20, find the indicated indefinite integral. In(3x) X - dx
It is projected that t years from now the population of a certain country will be changing at the rate of e0.02t million per year. If the current population is 50 million, what will be the population 10 years from now?
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. fox. (x + 1)(x² + 2x + 5)¹² dx
In Exercises 1 through 20, find the indicated indefinite integral. 5xe-x²2 dx
In Exercises 15 through 19, the demand and supply functions, D(q) and S(q), for a particular commodity are given. Specifically, q units of the commodity will be demanded (sold) at a price of p = D(q) dollars per unit, while q units will be supplied by producers when the price is p = S(q) dollars
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. [u' (3-1) du 3u
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. [(³3r² (3x² - 1)et¹. 1et¹-x dx
In Exercises 15 through 19, the demand and supply functions, D(q) and S(q), for a particular commodity are given. Specifically, q units of the commodity will be demanded (sold) at a price of p = D(q) dollars per unit, while q units will be supplied by producers when the price is p = S(q) dollars
In Exercises 1 through 30, find the indicated integral. Check your answers by differentiation. (2 2e" + la 6 U + In 2) du
In Exercises 5 through 18, sketch the given region R and then find its area.R is the rectangle with vertices (1, 0), (−2, 0), (−2, 5), and (1, 5).
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. 5 3x² + 12x³ + 6 + 5x + 10x + 12 dx
In Exercises 1 through 20, find the indicated indefinite integral. X 16(₁ = 4 x - dx
In Exercises 15 through 44, evaluate the given definite integral using the fundamental theorem of calculus. ₁ (5 (5 - 2t) dt
A study indicates that x months from now, the population of a certain town will be increasing at the rate of 10 + 2√x people per month. By how much will the population increase over the next 9 months?
In Exercises 15 through 19, the demand and supply functions, D(q) and S(q), for a particular commodity are given. Specifically, q units of the commodity will be demanded (sold) at a price of p = D(q) dollars per unit, while q units will be supplied by producers when the price is p = S(q) dollars
In Exercises 3 through 36, find the indicated integral and check your answer by differentiation. би - 3 4u² 4u + 1 du
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