Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112280
Author: James Stewart
Publisher: Cengage Learning
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Chapter 11, Problem 12RCC
If z is defined implicitly as a function of x and y by an equation of the form F(x, y, z) = 0, how do you find ∂z/∂x and ∂z/∂y?
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Chapter 11 Solutions
Essential Calculus: Early Transcendentals
Ch. 11.1 - Let g(x, y) = cos(x + 2y). (a) Evaluate g(2, 1)....Ch. 11.1 - Let F(x,y)=1+4y2. (a) Evaluate F(3,1). (b) Find...Ch. 11.1 - Let f(x,y,z)=x+y+z+ln(4x2y2z2). (a) Evaluate f(1,...Ch. 11.1 - Let g(x,y,z)=x3y2z10xyz. (a) Evaluate g(1, 2, 3)....Ch. 11.1 - Find and sketch the domain of the function....Ch. 11.1 - Find and sketch the domain of the function....Ch. 11.1 - Find and sketch the domain of the function. 15....Ch. 11.1 - Find and sketch the domain of the function....Ch. 11.1 - Find and sketch the domain of the function. 19....Ch. 11.1 - Find and sketch the domain of the function. f(x,...
Ch. 11.1 - Find and sketch the domain of the function....Ch. 11.1 - Find and sketch the domain of the function. 22....Ch. 11.1 - Prob. 14ECh. 11.1 - Prob. 13ECh. 11.1 - Prob. 15ECh. 11.1 - Sketch the graph of the function. f(x, y) = eyCh. 11.1 - 1320 Sketch the graph of the function. 17. f(x, y)...Ch. 11.1 - Prob. 18ECh. 11.1 - Prob. 20ECh. 11.1 - Prob. 19ECh. 11.1 - A contour map for a function f is shown. Use it to...Ch. 11.1 - Two contour maps are shown. One is for a function...Ch. 11.1 - Locate the points A and B on the map of Lonesome...Ch. 11.1 - Make a rough sketch of a contour map for the...Ch. 11.1 - Prob. 25ECh. 11.1 - Draw a contour map of the function showing several...Ch. 11.1 - Draw a contour map of the function showing several...Ch. 11.1 - Prob. 28ECh. 11.1 - Draw a contour map of the function showing several...Ch. 11.1 - Prob. 30ECh. 11.1 - Draw a contour map of the function showing several...Ch. 11.1 - Draw a contour map of the function showing several...Ch. 11.1 - Sketch both a contour map and a graph of the...Ch. 11.1 - Prob. 34ECh. 11.1 - Prob. 35ECh. 11.1 - Prob. 36ECh. 11.1 - Prob. 37ECh. 11.1 - Use a computer to graph the function using various...Ch. 11.1 - Prob. 39ECh. 11.1 - Prob. 40ECh. 11.1 - Prob. 41ECh. 11.1 - Prob. 42ECh. 11.1 - Prob. 43ECh. 11.1 - Prob. 44ECh. 11.1 - Prob. 45ECh. 11.1 - Prob. 46ECh. 11.1 - Prob. 47ECh. 11.1 - Prob. 48ECh. 11.1 - Prob. 49ECh. 11.1 - Prob. 50ECh. 11.1 - Prob. 51ECh. 11.1 - Prob. 52ECh. 11.1 - Prob. 53ECh. 11.1 - Prob. 55ECh. 11.1 - Prob. 54ECh. 11.1 - Prob. 56ECh. 11.2 - Prob. 1ECh. 11.2 - Prob. 2ECh. 11.2 - Prob. 3ECh. 11.2 - Prob. 4ECh. 11.2 - Prob. 5ECh. 11.2 - Prob. 6ECh. 11.2 - Prob. 7ECh. 11.2 - Prob. 8ECh. 11.2 - Prob. 9ECh. 11.2 - Prob. 10ECh. 11.2 - Prob. 11ECh. 11.2 - Prob. 12ECh. 11.2 - Prob. 13ECh. 11.2 - Prob. 14ECh. 11.2 - Prob. 15ECh. 11.2 - Prob. 16ECh. 11.2 - Prob. 17ECh. 11.2 - Prob. 18ECh. 11.2 - Prob. 19ECh. 11.2 - Prob. 20ECh. 11.2 - Determine the set of points at which the function...Ch. 11.2 - Determine the set of points at which the function...Ch. 11.2 - Determine the set of points at which the function...Ch. 11.2 - Determine the set of points at which the function...Ch. 11.2 - Determine the set of points at which the function...Ch. 11.2 - Prob. 26ECh. 11.2 - Prob. 27ECh. 11.2 - Prob. 28ECh. 11.2 - Prob. 29ECh. 11.2 - Prob. 30ECh. 11.2 - Prob. 31ECh. 11.2 - Prob. 32ECh. 11.2 - Prob. 33ECh. 11.2 - Prob. 34ECh. 11.3 - Prob. 1ECh. 11.3 - Prob. 3ECh. 11.3 - Prob. 4ECh. 11.3 - Prob. 2ECh. 11.3 - Prob. 5ECh. 11.3 - Prob. 6ECh. 11.3 - Prob. 7ECh. 11.3 - Prob. 8ECh. 11.3 - Prob. 9ECh. 11.3 - Prob. 10ECh. 11.3 - Prob. 11ECh. 11.3 - Prob. 12ECh. 11.3 - Prob. 13ECh. 11.3 - Prob. 14ECh. 11.3 - Prob. 15ECh. 11.3 - Prob. 16ECh. 11.3 - Prob. 17ECh. 11.3 - Prob. 18ECh. 11.3 - Prob. 19ECh. 11.3 - Prob. 20ECh. 11.3 - Prob. 21ECh. 11.3 - Prob. 22ECh. 11.3 - Prob. 23ECh. 11.3 - Find the first partial derivatives of the...Ch. 11.3 - Prob. 25ECh. 11.3 - Prob. 26ECh. 11.3 - Prob. 27ECh. 11.3 - Prob. 28ECh. 11.3 - Prob. 29ECh. 11.3 - Prob. 30ECh. 11.3 - Prob. 31ECh. 11.3 - Find the indicated partial derivative. 32. f(x, y)...Ch. 11.3 - Prob. 33ECh. 11.3 - Prob. 34ECh. 11.3 - Prob. 35ECh. 11.3 - Prob. 36ECh. 11.3 - 3738 Find fx and fy and graph f, fx, and fy with...Ch. 11.3 - Prob. 38ECh. 11.3 - Prob. 39ECh. 11.3 - Use implicit differentiation to find z/x and z/y....Ch. 11.3 - Use implicit differentiation to find z/x and z/y....Ch. 11.3 - Prob. 42ECh. 11.3 - Prob. 43ECh. 11.3 - Find z/x and z/y. 52. (a) z = f(x)g(y) (b) z =...Ch. 11.3 - Prob. 45ECh. 11.3 - Find all the second partial derivatives. 54. f(x,...Ch. 11.3 - Find all the second partial derivatives. 55....Ch. 11.3 - Prob. 48ECh. 11.3 - Prob. 49ECh. 11.3 - Prob. 50ECh. 11.3 - Prob. 51ECh. 11.3 - Verify that the conclusion of Clairauts Theorem...Ch. 11.3 - Prob. 53ECh. 11.3 - Prob. 54ECh. 11.3 - Prob. 55ECh. 11.3 - Prob. 56ECh. 11.3 - Prob. 57ECh. 11.3 - Find the indicated partial derivative(s). 70. u =...Ch. 11.3 - Prob. 59ECh. 11.3 - Prob. 60ECh. 11.3 - Prob. 61ECh. 11.3 - Determine whether each of the following functions...Ch. 11.3 - Verify that the function u=1/x2+y2+z2 is a...Ch. 11.3 - Prob. 64ECh. 11.3 - Prob. 65ECh. 11.3 - Prob. 66ECh. 11.3 - Show that the function z = xey + yex is a solution...Ch. 11.3 - Prob. 68ECh. 11.3 - Prob. 69ECh. 11.3 - Prob. 71ECh. 11.3 - Prob. 70ECh. 11.3 - The wind-chill index is modeled by the function W...Ch. 11.3 - Prob. 73ECh. 11.3 - Prob. 74ECh. 11.3 - Prob. 75ECh. 11.3 - Prob. 76ECh. 11.3 - Prob. 77ECh. 11.3 - Prob. 78ECh. 11.3 - Prob. 79ECh. 11.3 - Prob. 80ECh. 11.3 - Prob. 81ECh. 11.3 - Prob. 82ECh. 11.4 - Find an equation of the tangent plane to the given...Ch. 11.4 - Find an equation of the tangent plane to the given...Ch. 11.4 - Prob. 3ECh. 11.4 - Find an equation of the tangent plane to the given...Ch. 11.4 - Find an equation of the tangent plane to the given...Ch. 11.4 - Prob. 6ECh. 11.4 - Prob. 7ECh. 11.4 - Prob. 8ECh. 11.4 - Prob. 9ECh. 11.4 - Prob. 11ECh. 11.4 - Explain why the function is differentiable at the...Ch. 11.4 - Prob. 14ECh. 11.4 - Prob. 12ECh. 11.4 - Prob. 15ECh. 11.4 - Verify the linear approximation at (0, 0). 16....Ch. 11.4 - Prob. 17ECh. 11.4 - Prob. 18ECh. 11.4 - Prob. 19ECh. 11.4 - Find the differential of the function. 26....Ch. 11.4 - Prob. 21ECh. 11.4 - Prob. 22ECh. 11.4 - Prob. 23ECh. 11.4 - Prob. 24ECh. 11.4 - Prob. 25ECh. 11.4 - If z = x2 xy + 3y2 and (x, y) changes from (3, 1)...Ch. 11.4 - The length and width of a rectangle are measured...Ch. 11.4 - Use differentials to estimate the amount of metal...Ch. 11.4 - Prob. 29ECh. 11.4 - Prob. 32ECh. 11.4 - Prob. 30ECh. 11.4 - Prob. 33ECh. 11.4 - Prob. 31ECh. 11.4 - Suppose you need to know an equation of the...Ch. 11.4 - Show that the function is differentiable by...Ch. 11.4 - Show that the function is differentiable by...Ch. 11.4 - Prob. 37ECh. 11.4 - (a) The function...Ch. 11.5 - Use the Chain Rule to find dz/dt or dw/dt. 1....Ch. 11.5 - Use the Chain Rule to find dz/dt or dw/dt. 2....Ch. 11.5 - Use the Chain Rule to find dz/dt or dw/dt. 5. w =...Ch. 11.5 - Use the Chain Rule to find dz/dt or dw/dt. 6. w =...Ch. 11.5 - Use the Chain Rule to find z/s and z/t. 5....Ch. 11.5 - Use the Chain Rule to find z/s and z/t. 6....Ch. 11.5 - Use the Chain Rule to find z/s and z/t. 11. z = er...Ch. 11.5 - Prob. 8ECh. 11.5 - If z = f(x, y), where f is differentiable, and...Ch. 11.5 - Let W(s,t)=F(u(s,t),(s,t)), where F, u and are...Ch. 11.5 - Suppose f is a differentiable function of x and y,...Ch. 11.5 - Suppose f is a differentiable function of x and y,...Ch. 11.5 - Prob. 13ECh. 11.5 - Prob. 14ECh. 11.5 - Prob. 15ECh. 11.5 - Prob. 16ECh. 11.5 - Use the Chain Rule to find the indicated partial...Ch. 11.5 - Use the Chain Rule to find the indicated partial...Ch. 11.5 - Use the Chain Rule to find the indicated partial...Ch. 11.5 - Use the Chain Rule to find the indicated partial...Ch. 11.5 - Use the Chain Rule to find the indicated partial...Ch. 11.5 - Use Equation 6 to find dy/dx. 28. cos(xy) = 1 +...Ch. 11.5 - Use Equation 6 to find dy/dx. 29. tan1(x2y) = x +...Ch. 11.5 - Use Equation 6 to find dy/dx. 30. ey sin x = x +...Ch. 11.5 - Use Equations 7 to find z/x and z/y. 31. x2 + 2y2...Ch. 11.5 - Use Equations 7 to find z/y and z/y. 26....Ch. 11.5 - Use Equations 7 to find z/x and z/y. 33. ez = xyzCh. 11.5 - Use Equations 7 to find z/x and z/y. 34. yz + x ln...Ch. 11.5 - The temperature at a point (x, y) is T(x, y),...Ch. 11.5 - Prob. 30ECh. 11.5 - The speed of sound traveling through ocean water...Ch. 11.5 - Prob. 32ECh. 11.5 - Prob. 33ECh. 11.5 - The voltage V in a simple electrical circuit is...Ch. 11.5 - The pressure of 1 mole of an ideal gas is...Ch. 11.5 - A sound with frequency fs, is produced by a source...Ch. 11.5 - Assume that all the given functions are...Ch. 11.5 - Assume that all the given functions are...Ch. 11.5 - Assume that all the given functions are...Ch. 11.5 - Prob. 40ECh. 11.5 - Prob. 41ECh. 11.5 - Prob. 42ECh. 11.5 - Prob. 43ECh. 11.5 - Prob. 44ECh. 11.5 - Prob. 45ECh. 11.5 - Prob. 46ECh. 11.5 - Prob. 47ECh. 11.5 - Prob. 48ECh. 11.6 - Find the directional derivative of f at the given...Ch. 11.6 - Find the directional derivative of f at the given...Ch. 11.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 11.6 - Prob. 4ECh. 11.6 - Prob. 5ECh. 11.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 11.6 - Prob. 7ECh. 11.6 - Prob. 8ECh. 11.6 - Prob. 9ECh. 11.6 - Prob. 10ECh. 11.6 - Find the directional derivative of the function at...Ch. 11.6 - Use the figure to estimate Du, f(2, 2).Ch. 11.6 - Prob. 13ECh. 11.6 - Prob. 14ECh. 11.6 - Prob. 16ECh. 11.6 - Prob. 15ECh. 11.6 - Find the maximum rate of change of f at the given...Ch. 11.6 - Find the maximum rate of change of f at the given...Ch. 11.6 - (a) Show that a differentiable function f...Ch. 11.6 - Find the directions in which the directional...Ch. 11.6 - Find all points at which the direction of fastest...Ch. 11.6 - Near a buoy, the depth of a lake at the point with...Ch. 11.6 - The temperature T in a metal ball is inversely...Ch. 11.6 - Prob. 24ECh. 11.6 - Suppose that over a certain region of space the...Ch. 11.6 - Suppose you are climbing a hill whose shape is...Ch. 11.6 - Prob. 27ECh. 11.6 - Shown is a topographic map of Blue River Pine...Ch. 11.6 - Show that the operation of taking the gradient of...Ch. 11.6 - Prob. 30ECh. 11.6 - Prob. 31ECh. 11.6 - Prob. 32ECh. 11.6 - Find equations of (a) the tangent plane and (b)...Ch. 11.6 - Find equations of (a) the tangent plane and (b)...Ch. 11.6 - Find equations of (a) the tangent plane and (b)...Ch. 11.6 - Find equations of (a) the tangent plane and (b)...Ch. 11.6 - Use a computer to graph the surface, the tangent...Ch. 11.6 - Use a computer to graph the surface, the tangent...Ch. 11.6 - If f(x, y) = xy, find the gradient vector f(3, 2)...Ch. 11.6 - If g(x, y) = x2 + y2 4x, find the gradient vector...Ch. 11.6 - Show that the equation of the tangent plane to the...Ch. 11.6 - At what point on the paraboloid y=x2+z2 is the...Ch. 11.6 - Are there any points on the hyperboloid x2 y2 z2...Ch. 11.6 - Show that the ellipsoid 3x2 + 2y2 + z2 = 9 and the...Ch. 11.6 - Where does the normal line to the paraboloid z =...Ch. 11.6 - Prob. 46ECh. 11.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 11.6 - Prob. 48ECh. 11.6 - Find parametric equations tor the tangent line to...Ch. 11.6 - Prob. 50ECh. 11.6 - Prob. 51ECh. 11.6 - Prob. 52ECh. 11.6 - Suppose that the directional derivatives of f(x,...Ch. 11.6 - Prob. 54ECh. 11.7 - Suppose (1, 1) is a critical point of a function f...Ch. 11.7 - Use the level curves in the figure to predict the...Ch. 11.7 - Find the local maximum and minimum values and...Ch. 11.7 - Find the local maximum and minimum values and...Ch. 11.7 - Find the local maximum and minimum values and...Ch. 11.7 - Find the local maximum and minimum values and...Ch. 11.7 - Find the local maximum and minimum values and...Ch. 11.7 - Find the local maximum and minimum values and...Ch. 11.7 - Prob. 9ECh. 11.7 - Prob. 10ECh. 11.7 - Prob. 11ECh. 11.7 - Prob. 12ECh. 11.7 - Prob. 13ECh. 11.7 - Prob. 14ECh. 11.7 - Prob. 15ECh. 11.7 - Prob. 16ECh. 11.7 - Prob. 17ECh. 11.7 - Use a graph or level curves or both to estimate...Ch. 11.7 - Prob. 19ECh. 11.7 - Prob. 20ECh. 11.7 - Use a graphing device as in Example 4 (or Newtons...Ch. 11.7 - Use a graphing device as in Example 4 (or Newtons...Ch. 11.7 - Find the absolute maximum and minimum values of f...Ch. 11.7 - Find the absolute maximum and minimum values of f...Ch. 11.7 - Find the absolute maximum and minimum values of f...Ch. 11.7 - Find the absolute maximum and minimum values of f...Ch. 11.7 - Find the absolute maximum and minimum values of f...Ch. 11.7 - Find the absolute maximum and minimum values of f...Ch. 11.7 - Prob. 29ECh. 11.7 - Prob. 30ECh. 11.7 - Prob. 31ECh. 11.7 - Prob. 32ECh. 11.7 - Prob. 33ECh. 11.7 - Prob. 34ECh. 11.7 - Prob. 35ECh. 11.7 - Prob. 36ECh. 11.7 - Prob. 37ECh. 11.7 - Prob. 38ECh. 11.7 - Prob. 39ECh. 11.7 - Prob. 40ECh. 11.7 - Prob. 41ECh. 11.7 - The base of an aquarium with given volume V is...Ch. 11.7 - Prob. 43ECh. 11.7 - Prob. 44ECh. 11.7 - Prob. 45ECh. 11.7 - Prob. 46ECh. 11.7 - Prob. 47ECh. 11.7 - Prob. 48ECh. 11.8 - Pictured are a contour map of f and a curve with...Ch. 11.8 - Prob. 21ECh. 11.8 - Use Lagrange multipliers to find the maximum and...Ch. 11.8 - Each of these extreme value problems has a...Ch. 11.8 - Use Lagrange multipliers to find the maximum and...Ch. 11.8 - Prob. 4ECh. 11.8 - Each of these extreme value problems has a...Ch. 11.8 - Prob. 6ECh. 11.8 - Use Lagrange multipliers to find the maximum and...Ch. 11.8 - Prob. 8ECh. 11.8 - Prob. 9ECh. 11.8 - Prob. 10ECh. 11.8 - Prob. 11ECh. 11.8 - Prob. 12ECh. 11.8 - Find the extreme values of f subject to both...Ch. 11.8 - Prob. 14ECh. 11.8 - Prob. 15ECh. 11.8 - Prob. 16ECh. 11.8 - Find the extreme values of f on the region...Ch. 11.8 - Prob. 18ECh. 11.8 - Prob. 19ECh. 11.8 - Prob. 22ECh. 11.8 - Prob. 23ECh. 11.8 - Prob. 25ECh. 11.8 - Prob. 26ECh. 11.8 - Use Lagrange multipliers to prove that the...Ch. 11.8 - Use Lagrange multipliers to prove that the...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Use Lagrange multipliers to give an alternate...Ch. 11.8 - Prob. 37ECh. 11.8 - Prob. 38ECh. 11.8 - Prob. 39ECh. 11.8 - Prob. 40ECh. 11.8 - Prob. 41ECh. 11.8 - Prob. 42ECh. 11.8 - The plane x + y + 2z = 2 intersects the paraboloid...Ch. 11.8 - Prob. 44ECh. 11.8 - (a) Find the maximum value of...Ch. 11.8 - Prob. 48ECh. 11 - Prob. 1RCCCh. 11 - What is a function of three variables? How can you...Ch. 11 - Prob. 3RCCCh. 11 - (a) What does it mean to say that f is continuous...Ch. 11 - Prob. 5RCCCh. 11 - What does Clairauts Theorem say?Ch. 11 - Prob. 7RCCCh. 11 - Define the linearization of f at (a, b). What is...Ch. 11 - Prob. 9RCCCh. 11 - If z = f(x, y), what arc the differentials dx, dy,...Ch. 11 - State the Chain Rule for the case where z = f(x,...Ch. 11 - If z is defined implicitly as a function of x and...Ch. 11 - Prob. 13RCCCh. 11 - Prob. 14RCCCh. 11 - Prob. 15RCCCh. 11 - Prob. 16RCCCh. 11 - State the Second Derivatives Test.Ch. 11 - (a) What is a closed set in 2? What is a bounded...Ch. 11 - Explain how the method of Lagrange multipliers...Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Prob. 2RQCh. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Prob. 8RQCh. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Determine whether the statement is true or false....Ch. 11 - Find and sketch the domain of the function. 1....Ch. 11 - Find and sketch the domain of the function. 2....Ch. 11 - Sketch the graph of the function. 3. f(x, y) = 1 ...Ch. 11 - Sketch the graph of the function. 4. f(x, y) = x2...Ch. 11 - Sketch several level curves of the function. 5....Ch. 11 - Sketch several level curves of the function. 6....Ch. 11 - Make a rough sketch of a contour map for the...Ch. 11 - The contour map of a function f is shown, (a)...Ch. 11 - Prob. 9RECh. 11 - Prob. 10RECh. 11 - Prob. 11RECh. 11 - Prob. 12RECh. 11 - Prob. 13RECh. 11 - Prob. 14RECh. 11 - Prob. 15RECh. 11 - The speed of sound traveling through ocean water...Ch. 11 - Prob. 17RECh. 11 - Prob. 18RECh. 11 - Prob. 19RECh. 11 - Prob. 20RECh. 11 - If z = xy + xey/x show that xzx+yzy=xy+z.Ch. 11 - Prob. 22RECh. 11 - Prob. 23RECh. 11 - Prob. 24RECh. 11 - Prob. 25RECh. 11 - Prob. 26RECh. 11 - Prob. 27RECh. 11 - Prob. 28RECh. 11 - Prob. 29RECh. 11 - Find du if u = ln(1 + se2t).Ch. 11 - Prob. 31RECh. 11 - Prob. 32RECh. 11 - Prob. 33RECh. 11 - If v = x2sin y + yexy, where x = s + 2t and y =...Ch. 11 - Prob. 35RECh. 11 - Prob. 36RECh. 11 - Prob. 37RECh. 11 - The length x of a side of a triangle is increasing...Ch. 11 - Prob. 39RECh. 11 - If cos(xyz) = 1 + .x2y2 + z2, find zx and zy.Ch. 11 - Prob. 41RECh. 11 - Prob. 42RECh. 11 - Prob. 43RECh. 11 - Find the directional derivative of f at the given...Ch. 11 - Find the maximum rate of change of f(x,y)=x2y+y at...Ch. 11 - Find parametric equations of the tangent line at...Ch. 11 - Find the local maximum and minimum values and...Ch. 11 - Find the local maximum and minimum values and...Ch. 11 - Find the local maximum and minimum values and...Ch. 11 - Find the local maximum and minimum values and...Ch. 11 - Find the absolute maximum and minimum values of f...Ch. 11 - Find the absolute maximum and minimum values of f...Ch. 11 - Use a graph or level curves or both to estimate...Ch. 11 - Use a graphing calculator or computer (or Newtons...Ch. 11 - Use Lagrange multipliers to find the maximum and...Ch. 11 - Use Lagrange multipliers to find the maximum and...Ch. 11 - Use Lagrange multipliers to find the maximum and...Ch. 11 - Use Lagrange multipliers to find the maximum and...Ch. 11 - Prob. 59RECh. 11 - A package in the shape of a rectangular box can be...Ch. 11 - A pentagon is formed by placing an isosceles...Ch. 11 - Prob. 62RE
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- 2.6 Applications: Growth and Decay; Mathematics of Finances 1. A couple wants to have $50,000 in 5 years for a down payment on a new house. (a) How much should they deposit today, at 6.2% compounded quarterly, to have the required amount in 5 years? (b) How much interest will be earned? (c) If they can deposit only $30,000 now, how much more will they need to complete the $50,000 after 5 years? Note, this is not 50,000-P3.arrow_forwardThe graph of f(x) is given below. Select each true statement about the continuity of f(x) at x = 1. Select all that apply: ☐ f(x) is not continuous at x = 1 because it is not defined at x = 1. ☐ f(x) is not continuous at x = 1 because lim f(x) does not exist. x+1 ☐ f(x) is not continuous at x = 1 because lim f(x) ‡ f(1). x+→1 ☐ f(x) is continuous at x = 1.arrow_forwarda is done please show barrow_forward
- A homeware company has been approached to manufacture a cake tin in the shape of a "ghost" from the Pac-Man video game to celebrate the 45th Anniversary of the games launch. The base of the cake tin has a characteristic dimension / and is illustrated in Figure 1 below, you should assume the top and bottom of the shape can be represented by semi-circles. The vertical sides of the cake tin have a height of h. As the company's resident mathematician, you need to find the values of r and h that minimise the internal surface area of the cake tin given that the volume of the tin is Vfixed- 2r Figure 1 - Plan view of the "ghost" cake tin base. (a) Show that the Volume (V) of the cake tin as a function of r and his 2(+1)²h V = 2arrow_forward15. Please solve this and show each and every step please. PLEASE no chatgpt can I have a real person solve it please!! I am stuck. I am doing pratice problems and I do not even know where to start with this. The question is Please compute the indicated functional value.arrow_forwardUse a graph of f to estimate lim f(x) or to show that the limit does not exist. Evaluate f(x) near x = a to support your conjecture. Complete parts (a) and (b). x-a f(x)= 1 - cos (4x-4) 3(x-1)² ; a = 1 a. Use a graphing utility to graph f. Select the correct graph below.. A. W → ✓ Each graph is displayed in a [- 1,3] by [0,5] window. B. in ✓ ○ C. und ☑ Use the graphing utility to estimate lim f(x). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. x-1 ○ A. The limit appears to be approximately ☐ . (Round to the nearest tenth as needed.) B. The limit does not exist. b. Evaluate f(x) for values of x near 1 to support your conjecture. X 0.9 0.99 0.999 1.001 1.01 1.1 f(x) ○ D. + ☑ (Round to six decimal places as needed.) Does the table from the previous step support your conjecture? A. No, it does not. The function f(x) approaches a different value in the table of values than in the graph, after the approached values are rounded to the…arrow_forward
- x²-19x+90 Let f(x) = . Complete parts (a) through (c) below. x-a a. For what values of a, if any, does lim f(x) equal a finite number? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. x→a+ ○ A. a= (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no values of a for which the limit equals a finite number. b. For what values of a, if any, does lim f(x) = ∞o? Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. x→a+ A. (Type integers or simplified fractions) C. There are no values of a that satisfy lim f(x) = ∞. + x-a c. For what values of a, if any, does lim f(x) = -∞0? Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. x→a+ A. Either a (Type integers or simplified fractions) B.arrow_forwardSketch a possible graph of a function f, together with vertical asymptotes, that satisfies all of the following conditions. f(2)=0 f(4) is undefined lim f(x)=1 X-6 lim f(x) = -∞ x-0+ lim f(x) = ∞ lim f(x) = ∞ x-4 _8arrow_forwardDetermine the following limit. lim 35w² +8w+4 w→∞ √49w+w³ 3 Select the correct choice below, and, if necessary, fill in the answer box to complete your choice. ○ A. lim W→∞ 35w² +8w+4 49w+w3 (Simplify your answer.) B. The limit does not exist and is neither ∞ nor - ∞.arrow_forwardCalculate the limit lim X-a x-a 5 using the following factorization formula where n is a positive integer and x-➡a a is a real number. x-a = (x-a) (x1+x-2a+x lim x-a X - a x-a 5 = n- + xa an-2 + an−1)arrow_forwardThe function s(t) represents the position of an object at time t moving along a line. Suppose s(1) = 116 and s(5)=228. Find the average velocity of the object over the interval of time [1,5]. The average velocity over the interval [1,5] is Vav = (Simplify your answer.)arrow_forwardFor the position function s(t) = - 16t² + 105t, complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = 1. Time Interval Average Velocity [1,2] Complete the following table. Time Interval Average Velocity [1, 1.5] [1, 1.1] [1, 1.01] [1, 1.001] [1,2] [1, 1.5] [1, 1.1] [1, 1.01] [1, 1.001] ப (Type exact answers. Type integers or decimals.) The value of the instantaneous velocity at t = 1 is (Round to the nearest integer as needed.)arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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