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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

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 =
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
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
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
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).
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
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
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 =
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
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
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.
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.
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
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.
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.
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 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)
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 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)
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 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.
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 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
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)
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)
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
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)
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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