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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
Find the gradient of the function and the maximum value of the directional derivative at the given point. Function f(x, y) x + y y + 1 Point (0, 1)
The height and radius of a right circular cone are measured as h = 16 meters and r = 6 meters. In the process of measuring, errors Δr and Δh aremade. Let S be the lateral surface area of the cone. Completethe table above to show the relationship between ΔS and dS forthe indicated errors.
The possible error involved in measuring each dimension of a rectangular box is ±0.02 inch. The dimensions of the box are 8 inches by 5 inches by 12 inches. Approximate the propagated error and the relative error in the calculated volume of the box.
Find both first partial derivatives.z = cos xy
Find the limit (if it exists). If the limit does not exist, explain why. x-y-1 lim (x, y) (2, 1)√√x - y - 1
Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. Use the graphing utility to plot the points and graph the regression line.(6, 4), (1, 2), (3, 3), (8, 6), (11, 8), (13, 8)
Find the gradient of the function and the maximum value of the directional derivative at the given point. Function f(x, y) = x² + 2xy Point (1, 0)
Find the domain and range of the function.ƒ(x, y)=arcsin(y/x)
Differentiate implicitly to find the first partial derivatives of z.x + sin(y + z) = 0
Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function ƒ(x, y) at the critical point (x0, y0).ƒxx(X0, Y0) = -3, ƒyy(X0, Y0) = -8, ƒxy(X0, Y0) = 2
Find both first partial derivatives.ƒ(x, y) = √2x + y³
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.z = ye2xy, (0, 2, 2)
Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. Use the graphing utility to plot the points and graph the regression line.(0, 6), (4, 3), (5, 0), (8, -4), (10,-5)
Find the limit (if it exists). If the limit does not exist, explain why. x-y lim (x, y)→(0, 0) √√x √y
A right circular cone of height h = 8 and radius r = 4 is constructed, and in the process, errors Δr and Δh are made in the radius and height, respectively.Complete the table to show the relationship between ΔV anddV for the indicated errors. 0.1 0.1 0.001 -0.0001 Ah 0.1 -0.1 0.002 0.0002 dV AV
Find the domain and range of the function.ƒ(x, y) = arccos(x + y)
Differentiate implicitly to find the first partial derivatives of z.x² + 2yz + z² = 1
Examine the function for extrema without using the derivative tests, and use a computer algebra system to graph the surface. Z = (x² - y²)² x² + y²
Find the domain and range of the function. f(x, y) = √4x² - 4y² - 2
Find the limit (if it exists). If the limit does not exist, explain why. lim (x, y)→(0,0) 1 x²y²
Determine whetherthere is a relative maximum, a relative minimum, a saddlepoint, or insufficient information to determine the nature of thefunction ƒ(x, y) at the critical point (x0, y0).ƒxx(X0, Y0) = 9, ƒyy(X0, Y0) = 4, ƒxy (X0, Y0) = 6
The volume of the red right circular cylinder in the figure is V = πr²h. The possible errors in the radius andthe height are Δr and Δh, respectively. Find dV and identify thesolids in the figure whose volumes are given by the terms ofdV. What solid represents the difference between ΔV and dV?
Find both first partial derivatives.ƒ(x, y) = √x² + y²
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.xyz = 10, (1, 2, 5)
Find the domain and range of the function. f(x, y) = √√√4x² — y²
Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. Use the graphing utility to plot the points and graph the regression line.(1, 0), (3, 3), (5, 6)
Differentiate implicitly to find the first partial derivatives of z.xz + yz + xy = 0
Use the gradient to find the directional derivative of the function at P in the direction of Q.h(x, y, z) = In(x + y + z), P(1, 0, 0), Q(4, 3, 1)
Examine the function for extrema without using the derivative tests, and use a computer algebra system to graph the surface. Z || (x - y)4 x² + y²
Find the limit (if it exists). If the limit does not exist, explain why. 1 lim (x, y) (0,0) x + y
Find both first partial derivatives.g(x, y) = In √x² + y²
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.xy - z = 0, (-2, -3, 6)
(a) Find the least squares regression line(b) Calculate S, the sum of the squared errors. Use the regression capabilities of a graphing utility to verify your results. 2 y (1, 0) (3, 0) 1 2 3 (2,0) T (5,2) (4,2) (3, 1) (4,1) (6,2) || X 4 5 6
Find the domain and range of the function. Z = ху х-у y
The area of the shaded rectangle in the figure is A = lh. The possible errors in the length and height are Δl andΔh, respectively. Find dA and identify the regions in the figurewhose areas are given by the terms of dA. What regionrepresents the difference between ΔA and dA? Δη h ι ΔΙ
Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. Use the graphing utility to plot the points and graph the regression line.(0, 0), (1, 1), (3, 4), (4, 2), (5, 5)
Differentiate implicitly to find dy/dx. X x² + y² - y² = 6
Differentiate implicitly to find the first partialderivatives of z.x² + y² + z² = 1
Use the gradient to find the directional derivative of the function at P in the direction of Q.g(x, y, z) = xyez, P(2, 4, 0), Q(0, 0, 0)
Find both first partial derivatives. Z = xy x² + y² 2
Which point has a greater differential, (2, 2) or (1/2, 1/2)? Explain. (Assume that dx and dy are the same for both points.) X 3 31 3 У
Find both first partial derivatives.h(x, y) = e-(x² + y²)
Find the limit (if it exists). If the limit does not exist, explain why. x²y lim (x, y) (1,-1) 1 + xy²
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.z = x² - y², (3, 2, 5)
Find the domain and range of the function. 2= x + y ху
Use the gradient to find the directional derivative of the function at P in the direction of Q.ƒ(x, y) = 3x² - y² + 4, P(-1, 4), Q(3, 6)
(a) Find the least squares regression line(b) Calculate S, the sum of the squared errors. Use the regression capabilities of a graphing utility to verify your results. 4 3 2 1 y (0,4) (1,3) (1, 1) + 1 (2,0) 3 2 4 ➤X
Use a computer algebra system to graph the surface and locate any relative extrema and saddle points.z = exy
Find the limit (if it exists). If the limit does not exist, explain why. xy - 1 lim (x, y) (1.1) 1 + xy
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.z = 16 - x² - y², (2, 2, 8)
Find the domain and range of the function. g(x, y) y n X
Use the gradient to find the directionalderivative of the function at P in the direction of Q.g(x, y) = x² + y² + 1, P(1, 2), Q(2, 3)
(a) Find the least squares regression line(b) Calculate S, the sum of the squared errors. Use the regression capabilities of a graphing utility to verify your results. 4 3 2 (-1, 1) (-3,0) 1 -3-2-1 -2 y (3,2) (1, 1) | | | x 1 2 3
Differentiate implicitly to find dy/dx.In√x² + y² + x + y = 4
Use a computer algebra system to graph the surface and locate any relative extrema and saddle points.z = (x² + 4y²) e¹-x²-y²
Find the limit and discuss the continuity of the function. lim (x, y, z) (-2, 1, 0) xeyz
Use the gradient to find the directional derivative of the function at P in the direction of v. f(x, y, z) = xy + yz + xz, P(1, 2, -1), v= 2i+j - k
Find both first partial derivatives.z = ln(x² + y²)
When using differentials, what is meant by the terms propagated error and relative error?
(a) Find the least squares regression line(b) Calculate S, the sum of the squared errors. Use theregression capabilities of a graphing utility to verify your results. (-2, 0) -2 -1 3 2 1 -1 y (0, 1) 1 (2,3) 2 X
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.x² + y² + z = 9, (1, 2, 4)
Find both first partial derivatives. x + y x - y z = In:
Use a computer algebra system to graph the surface and locate any relative extrema and saddle points. Z || - 4x +2 x² + y² + 1
Differentiate implicitly to find dy/dx.sec xy + tan xy + 5 = 0
Find the limit and discuss the continuity of the function. lim (x, y, z)→(1, 3, 4) √x + y + z
Use a computer algebra system to graph the surface and locate any relative extrema and saddle points.ƒ(x, y) = y³ -3yx2 - 3y² - 3x² + 1
Use the gradient to find the directional derivative of the function at P in the direction of v. 3 f(x, y, z) = x² + y² + z², P(1, 1, 1), V = 3 (i- j + k)
What is meant by a linear approximation of z = ƒ(x, y) at the point P(x0 y0)?
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.x² + y² + z² = 9, (1, 2, 2)
Find the domain and range of the function.g(x, y) = x√y
Differentiate implicitly to find dy/dx.x² - xy + y² - x + y = 0
Find z = ƒ(x, y) and use the total differential to approximate the quantity. sin[(1.05)² + (0.95)²] - sin(1²+1²)
Examine the function for relative extrema and saddle points. 2 = ( ²2 - x² + y²) ¹-2²-² X N Z y
Find both first partial derivatives.z = In(x² + y²)
Find the limit and discuss the continuity of the function. arccos(x/y) lim (x, y) (0, 1) 1 + xy
Use the gradient to find the directional derivative of the function at P in the direction of v. h(x, y) P(1.). 2 = ex sin y, P1, V = -i
Describe the change in accuracy of dz as an approximation of Δz as Δx and Δy increase.
Find an equation of the tangent plane and find a set of symmetric equations for the normal line to the surface at the given point.x + y + z = 9, (3, 3, 3)
Find z = ƒ(x, y) and use the total differential to approximate the quantity. (5.05)² + (3.1)² — √5² + 3² -
In your own words, describe the method of least squares for finding mathematical models.
Find the domain and range of the function.ƒ(x, y)= exy
Find an equation of the tangent plane to the surface at the given point. z e*(sin y + 1), 0,2
Usethe gradient to find the directional derivative of the function atP in the direction of v. f(x, y) = xy, P(0, -2), v = (i + √√√3j)
Find aw/as and ∂w/∂t (a) By using the appropriate Chain Rule (b) By converting w to a function of s and t before differentiating.w = x cos yz, x = s², y = t², z = s - 2t
Examine the function for relative extrema and saddle points. z = e* sin y X 00 6 + 2 LLL 3π y
Find the limit and discuss the continuity of the function. arcsin xy lim (x, y) (0, 1) 1 xy
Find both first partial derivatives. z = ln In X y
Find both first partial derivatives.z = In√ху
In your own words, state the problem-solving strategy for applied minimum and maximum problems.
Find the domain and range of the function.ƒ(x, y) = x² + y²
Find z = ƒ(x, y) and use the total differential to approximate the quantity. 1 - (3.05)² (5.95)² 1 - 3² 6²
Find aw/as and ∂w/∂t (a) By using the appropriate Chain Rule (b) By converting w to a function of s and t before differentiating.w = zexy,x = s - t, y = s + t, z = st
A trough with trapezoidal cross sections is formed by turning up the edges of a 30-inch-wide sheet of aluminum (see figure). Find the cross section of maximum area. X 30 - 2x X 0
Examine the function for relative extrema and saddle points. f(x, y) = 2xy – (x + y) + 1 - + Z -2 2 3 X
Find the limit and discuss the continuity of the function. lim (x, y)→(2m, X sin 4) y
Find and simplify the function values.ƒ(x, y) = 3x² - 2y(a)(b) f(x + Δx, y) - f(x, y) Δε
Find an equation of the tangent plane to the surface at the given point.xy² + 3x - z² = 8, (1, -3, 2)
Find the gradient of the function at the given point.w = x tan(y + z), (4, 3, -1)
Find aw/as and ∂w/∂t (a) By using the appropriate Chain Rule (b) By converting w to a function of s and t before differentiating.w = x² + y² + z², x = t sin s, y = t cos s, z = st²
Find both first partial derivatives.z = yey/x
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