- A satellite TV station is to be located at P(x, y) so that the sum of the squares of the distances from P to the three towns A, B, and C is a minimum (see the figure). Find the coordinates of P, the
- (A) Show that (0, 0) is a critical point of the function g(x, y) = exy2 + x2y3 + 2, but that the second derivative test for local extrema fails.(B) Use cross sections, as in Example 2, to decide
- A mailing service states that a rectangular package shall have the sum of its length and girth not to exceed 120 inches (see the figure on page 838). What are the dimensions of the largest (in
- Consider the problem of minimizingf(x, y) = x2 + 2y2subject to the constraint g(x, y) = yex2 - 1 = 0.(A) Solve the constraint equation for y, and then substitute into f(x, y) to obtain a function
- A heavy industrial plant located in the center of a small town emits particulate matter into the atmosphere. Suppose that the concentration of fine particulate matter (in parts per million) at a
- In Problem use Theorem 2 to find the local extrema.f(x, y) = 8 - x2 + 12x - y2 - 2y THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0
- Use Lagrange multipliers to maximize f(x, y) = xy subject to 2x + 3y = 24.
- THEOREM 1 Local Extrema and Partial DerivativesIn Problem find fx(x, y) and fy(x, y), and explain, using Theorem 1, why f(x, y) has no local extrema.f(x, y) = yex - 3x + 4y THEOREM 1 Local Extrema
- The marginal revenue for a company that manufactures and sells x graphing calculators per week is given bywhere R(x) is the revenue in dollars. Find the revenue function and the number of calculators
- The rate of change of the monthly sales of a new basketball game is given byS′(t) = 350 ln (t + 1) S(0) = 0where t is the number of months since the game was released and S(t) is
- Find the producers’ surplus at a price level of P̅ = $55 for the price–supply equation P = S(x) 15+ 0.1x + 0.003x² =
- Find the producers’ surplus at a price level of P̅ = $20 for the price–supply equation P = S(x) Р || 10x 300 - x
- In Problem find each antiderivative. Then use the antiderivative to evaluate the definite integral. (A) J X Vy + x² dy (B) S₁²= X Vy √y + x² dy
- In Problem use absolute value on a graphing calculator to find the area between the curve and the x axis over the given interval. Find answers to two decimal places.y = x3 ln x; 0.1 ≤ x ≤ 3.1
- Use the method of Lagrange multipliers in Problem.Maximize f(x, y) = 2xysubject to x + y = 6
- In Problem find the indicated values of the functionsg(3, -3) f(x, y) 2x + 7y - 5 and g(x, y) 88 x² + 3y
- Find all critical points and test for extrema forf(x, y) = x3 - 12x + y2 - 6y
- Use the method of Lagrange multipliers in Problem.Maximize the product of two numbers if their sum must be 10.
- In Problem find each antiderivative. Then use the antiderivative to evaluate the definite integral. (A) | 3y2ary dy 2 ) 32 0 (B) 3y2ex+y dy
- In Problem find the least squares line and use it to estimate y for the indicated value of x. Round answers to two decimal places.Estimate y when x = 2. ت 1 1 3 نرا y 00 9 الا 5
- In Problem give a verbal description of the region R and determine whether R is a regular x region, a regular y region, both, or neither. R = {(x, y) ||x| ≤ 2, y ≤ 3}
- In Problem use Theorem 2 to find the local extrema.f(x, y) = x2 + 8x + y2 + 25 THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a
- In Problem find the indicated first-order partial derivative for each function z = f(x, y).fx (x, y) if f(x, y) = 4x - 3y + 6
- In Problem evaluate each iterated integral.See Problem 7 -1. 12x²y³ dy dx
- In Problem use Theorem 2 to find the local extrema.f(x, y) = 15x2 - y2 - 10y THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a critical
- Use Lagrange multipliers to minimize f(x, y, z) = x2 Step: 1+ y2 + z2 subject to 2x + y + 2z = 9.
- In Problem find the indicated values off(0, 0, 0) f(x, y, z) = 2x - 3y² + 5z³ - 1
- Use the method of Lagrange multipliers in Problem.Minimize f(x, y, z) = x2 + y2 + z2subject to x + y + 3z = 55
- Find the least squares line for the data in the following table. X 10 20 30 40 50 y 50 45 50 55 65 X 60 70 80 90 100 y 80 85 90 90 110
- In Problem find the indicated first-order partial derivative for each function z = f(x, y).fx (x, y) if f(x, y) = 7x + 8y - 2
- In Problem find the indicated values off(0, 0, 2) f(x, y, z) = 2x - 3y² + 5z³ - 1
- In Problem find the least squares line and use it to estimate y for the indicated value of x. Round answers to two decimal places.Estimate y when x = 8. X 0.5 2 3.5 5 6.5 X 9.5 25 22 21 21
- Use the method of Lagrange multipliers in Problem.Maximize f(x, y, z) = 300 - x2 - 2y2 - z2subject to 4x + y + z = 70
- In Problem give a verbal description of the region R and determine whether R is a regular x region, a regular y region, both, or neither. R = {(x, y) x² + y² ≥ 1, x ≤ 2, 0 ≤ y ≤ 2}
- In Problem evaluate each iterated integral.See Problem 9 3 It's -2 (4x + 6y + 5) dx dy
- In Problem find the indicated first-order partial derivative for each function z = f(x, y).fy(x, y) if f(x, y) = x2 - 3xy + 2y2
- In Problem use Theorem 2 to find the local extrema.f(x, y) = x2 + y2 + 6x - 8y + 10 THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a
- In Problem find the indicated values off(6, -5, 0) f(x, y, z) = 2x - 3y² + 5z³ - 1
- Find the average value of f(x, y) = x2/3 y1/3 over the rectangleR = {(x, y) | -8 ≤ x ≤ 8, 0 ≤ y ≤ 27}
- Use the method of Lagrange multipliers in Problem.Maximize f(x, y, z) = 900 - 5x2 - y2 - 2z2subject to x + y + z = 34
- In Problem use Theorem 2 to find the local extrema.f(x, y) = 100 + 6xy - 4x2 - 3y2 THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a
- In Problem find the indicated first-order partial derivative for each function z = f(x, y).fy(x, y) if f(x, y) = 3x2 + 2xy - 7y2
- In Problem find the indicated first-order partial derivative for each function z = f(x, y). az ay if z = (5x + 2y) 10
- Find the volume of the solid under the graph of z = 3x2 + 3y2 over the rectangleR = {(x, y) | 0 ≤ x ≤ 1, -1 ≤ y ≤ 1}
- In Problem evaluate each iterated integral.See Problem 13 re² In x ху -dy dx
- Use the method of Lagrange multipliers in Problem.Minimize f(x, y, z) = x2 + 4y2 + 2z2subject to x + 2y + z = 10
- In Problem find the indicated first-order partial derivative for each function z = f(x, y). 肛 ax if z= (2x - 3y) 8
- In Problem use Theorem 2 to find the local extrema.f(x, y) = 5x2 - y2 + 2y + 6 THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a
- Use the method of Lagrange multipliers in Problem.Maximize and minimize f(x, y, z) = x - 2y + zsubject to x2 + y2 + z2 = 24
- In Problem find the indicated value of the given function.V(4, 12) for V(R, h) = πR2h
- In Problem use Theorem 2 to find the local extrema.f(x, y) = x2 + xy + y2 - 7x + 4y + 9 THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is
- In Problem use the description of the region R to evaluate the indicated integral. ffe etty dA; R R = {(x, y)-x ≤ y ≤x, 0≤x≤ 2}
- Find the volume of the solid under the graph of F(x, y) = 60x2y over the region R bounded by the graph of x + y = 1 and the coordinate axes.
- Use the method of Lagrange multipliers in Problem.Maximize and minimize f(x, y, z) = x - y - 3zsubject to x2 + y2 + z2 = 99
- In Problem evaluate each iterated integral.See Problem 15 [S 3y²ex+y³ dx dy 0 0
- In Problem use Theorem 2 to find the local extrema.f(x, y) = -x2 + 2xy - 2y2 - 20x + 34y + 40 THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a,
- A company produces x units of product A and y units of product B (both in hundreds per month). The monthly profit equation (in thousands of dollars) is given byP(x, y) = -4x2 + 4xy - 3y2 + 4x + 10y +
- In Problem find the indicated value of the given function. S(3, 10) for S(R, h) = TRVR² + h²
- In Problem graph the region R bounded by the graphs of the indicated equations. Describe R in set notation with double inequalities, and evaluate the indicated integral. : J R y = x + 1, y = 0, x =
- In Problem use Theorem 2 to find the local extrema.f(x, y) = exy THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a critical point] 3.
- In Problem find the indicated value.fx (4, 1) if f(x, y) = x2y2 - 5xy3
- A company’s annual profits (in millions of dollars) over a 5-year period are given in the following table. Use the least squares line to estimate the profit for the sixth year.
- Use both orders of iteration to evaluate each double integral in Problem. [[ xy dA; R = {(x, y) |0 ≤ x ≤ 2, 0≤ y ≤ 4} R
- In Problem find the indicated value.fy(1, 0) if f(x, y) = 3xey
- In Problem graph the region R bounded by the graphs of the indicated equations. Describe R in set notation with double inequalities, and evaluate the indicated integral. y = x², y = √x; [[₁ 12xy
- In Problem find the indicated value of the given function.A(100, 0.06, 3) for A(P, r, t) = P + Prt
- In Problem use Theorem 2 to find the local extrema.f(x, y) = x3 + y3 - 3xy THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a critical
- In Problem find the indicated value.fy(2, 4) if f(x, y) = x4 ln y
- In Problem graph the region R bounded by the graphs of the indicated equations. Describe R in set notation with double inequalities, and evaluate the indicated integral. y = 4x = x², y = 0; J R Vy +
- When diving using scuba gear, the function used for timing the duration of the dive iswhere T is the time of the dive in minutes, V is the volume of air (in cubic feet, at sea-level pressure)
- Use both orders of iteration to evaluate each double integral in Problem. [[ (x + y) ³ dA; R = {(x, y) | − 1 ≤x≤1, 1≤y≤2} R
- The Cobb–Douglas production function for a product isN(x, y) = 10x0.8y0.2where x is the number of units of labor and y is the number of units of capital required to produce N units of the
- In Problem find the indicated value.fy(2, 1) if f(x, y) = ex2 - 4y
- In Problem find the indicated value of the given function. P(4, 2) for P(r, T) - [x x" dx
- In Problem find the indicated value. f(1,-1) if f(x, y) 2xy 1 + x²y² 2
- In Problem graph the region R bounded by the graphs of the indicated equations. Describe R in set notation with double inequalities, and evaluate the indicated integral. x = 1 + 3y, x = 1 - y, y =
- Consider the problem of maximizing f(x, y) = e-(x2 + y2) subject to the constraint g(x, y) = x2 + y - 1 = 0.(A) Solve the constraint equation for y, and then substitute into f(x, y) to obtain a
- In Problem find the average value of each function over the given rectangle. f(x, y) = (x + y)²; R = {(x, y) |1 ≤x≤5, -1≤ y ≤ 1}
- In Problem graph the region R bounded by the graphs of the indicated equations. Describe R in set notation with double inequalities, and evaluate the indicated integral. y = 1 - √x, y = 1 + √x, x
- In Problem use Theorem 2 to find the local extrema.f(x, y) = 2x4 + y2 - 12xy THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a critical
- Data on U.S. property crimes (in number of crimes per 100,000 population) are given in the table for the years 2001 through 2015.(A) Find the least squares line for the data, using x = 0 for 2000.(B)
- In Problem find the indicated value.fy(3, 3) if f(x, y) = e3x - y2
- In Problem find the indicated value. fx(-1,2) if f(x, y) x² - y² 1 + x²
- In Problem graph the region R bounded by the graphs of the indicated equations. Describe R in set notation with double inequalities, and evaluate the indicated integral. y = x, y = 6 x, y =
- In Problem find the average value of each function over the given rectangle. f(x, y) = x² + y2²; R = {(x, y) |-1 ≤ x ≤ 2, 1 ≤ y ≤ 4}
- Data for U.S. honey production are given in the table for the years 1990 through 2015.(A) Find the least squares line for the data, using x = 0 for 1990.(B) Use the least squares line to predict U.S.
- A manufacturing company produces two models of an HDTV per week, x units of model A and y units of model B at a cost (in dollars) ofC(x, y) = 6x2 + 12y2If it is necessary (because of shipping
- At the beginning of the semester, students in a foreign language course take a proficiency exam. The same exam is given at the end of the semester. The results for 5 students are shown in the
- In Problem find the indicated function f of a single variable.f(x) = G(x, 0) for G(x, y) = x2 + 3xy + y2 - 7
- In Problem use Theorem 2 to find the local extrema.f(x, y) = x3 - 3xy2 + 6y2 THEOREM 2 Second-Derivative Test for Local Extrema If 1. z = f(x, y) 2. fx(a, b) = 0 and f(a, b) = 0 [(a, b) is a critical
- In Problem evaluate each integral. Graph the region of integration, reverse the order of integration, and then evaluate the integral with the order reversed. 3-x f(x + 2y) dy dx
- The market research department for a drugstore chain chose two summer resort areas to test-market a new sunscreen lotion packaged in 4-ounce plastic bottles. After a summer of varying the selling
- The following table gives the U.S. population per square mile for the years 1960–2010:(A) Find the least squares line for the data, using x = 0 for 1960.(B) Use the least squares line to estimate
- In Problem M(x, y) = 68 + 0.3x - 0.8y gives the mileage (in mpg) of a new car as a function of tire pressure x (in psi) and speed (in mph). Find the indicated quantity (include the appropriate units)
- In Problem find the average value of each function over the given rectangle. f(x, y) = x/y; R = {(x, y) |1 ≤ x ≤ 4, 2 ≤ y ≤7}
- The following table gives life expectancies for males and females in a sample of Central and South American countries:(A) Find the least squares line for the data.(B) Use the least squares line to
- In Problem evaluate each integral. Graph the region of integration, reverse the order of integration, and then evaluate the integral with the order reversed. f(y-x) dx dy 4
- In Problem find the average value of each function over the given rectangle. f(x, y) = x²y³; R = {(x,y) | -1 ≤ x ≤ 1, 0≤ y ≤ 2}
- A consulting firm for a manufacturing company arrived at the following Cobb–Douglas production function for a particular product:N(x, y) = 50x0.8y0.2In this equation, x is the number of units of
- The table gives the winning heights in the pole vault in the Olympic Games from 1980 to 2016.(A) Use a graphing calculator to find the least squares line for the data, letting x = 0 for 1980.(B)