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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

Define the derivative of the function z = ƒ(x, y) in the direction u = cos θi + sin θj.
Show that if ƒ(x, y) ishomogeneous of degree n, then xf(x, y) + yf(x, y) = nf (x, y).
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values. 111 f(x, y) = exy/2, c = 2, 3, 4, 2, 3, 4
The figure shows the level curves for an unknown function ƒ(x, y). What, if any, information can be given about ƒ at the points A, B, C, and D? Explain your reasoning. y B A H X
(a) Find the gradient of the function at P(b) Find a unit normal vector to the level curve ƒ(x, y) = c at P(c) Find the tangent line to the level curve ƒ(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x² + 4y² c = 40,
Discuss the continuity of the function. Z f(x, y, z) = x² + y² - 4
Discuss the relationship between the tangent plane to a surface and approximation by differentials.
Demonstrate the result of Exercise 53 for the functionsData from in Exercise 53Given the functions u(x, y) and v(x, y), verify that the Cauchy-Riemann differential equationscan be written in polar coordinate form as u = In√√x² + y2 and v = v = arctan X
Find the first partial derivatives with respect to x, y, and z.ƒ(x, y, z)= 3x²y -5xyz + 10yz²
Discuss the continuity of the function. f(x, y, z) = 1 √x² + y² + z²
(a) Find the gradient of the function at P(b) Find a unit normal vector to the level curve ƒ(x, y) = c at P(c) Find the tangent line to the level curve ƒ(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 3x²2y² c = 1, P(1, 1)
For some surfaces, the normal lines at any point pass through the same geometric object. What is the common geometric object for a sphere? What is the common geometric object for a right circular cylinder? Explain.
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values. f(x, y) = √√√9 — x² - y², c = 0, 1, 2, 3
Given the functions u(x, y) and v(x, y), verify that the Cauchy-Riemann differential equationscan be written in polar coordinate form as ди ax ду ду and ди ду ду ax
Find thefirst partial derivatives with respect to x, y, and z.H(x, y, z) = sin(x + 2y + 3z)
Find the slopes of the surface in the x- and y-directions at the given point. h(x, y) = x² - y² (-2, 1, 3) X 7 6 5 4 3 2 Z 3
Consider the functionsƒ(x, y) = x² - y² and g(x, y) = x² + y².(a) Show that both functions have a critical point at (0, 0). (b) Explain how ƒ and g behave differently at this critical point.
Sketch the graph of an arbitrary function ƒ satisfying the given conditions. State whether the function has any extrema or saddle points. (There are many correct answers.) f(0, 0) = 0, f(0, 0) = 0 < 0, x < 0 y < 0 > 0, x > 0' y > 0 fxx (x, y) > 0, fyy (x, y) < 0, and f(x, y) = 0 for all (x,
Give the standard form of the equation of the tangent plane to a surface given by F(x, y, z) = 0 at (x0, Y0, Z0).
(a) Find the gradient of the function at P(b) Find a unit normal vector to the level curve ƒ(x, y) = c at P(c) Find the tangent line to the level curve ƒ(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = x - y² c = 3, P(4, -1)
Use polar coordinates and L’Hôpital’s Rule to find the limit. lim (x, y)→(0, (x² + y²)In (x² + y²) 0)
Find a point on the hyperboloid x² + 4y² - z² = 1 where the tangent plane is parallel to the plane x + 4y - z = 0.
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values.z = 6 - 2x - 3y, c = 0, 2, 4, 6, 8, 10
The two radii of the frustum of a right circular cone are increasing at a rate of 4 centimeters per minute, and the height is increasing at a rate of 12 centimeters per minute (see figure). Find the rates at which the volume and surface area are changing when the two radii are 15 centimeters and 25
Show thatfor w = ƒ(x, y), x = u - v, and y = v - u. aw ди дw + = 0 Əv
Find a point on the ellipsoid x² + 4y² + z² = 9 where the tangent plane is perpendicular to the line with parametric equations
(a) Find the gradient of the function at P(b) Find a unit normal vector to the level curve ƒ(x, y) = c at P(c) Find the tangent line to the level curve ƒ(x, y) = c at P, and (d) sketch the level curve, the unit normalvector, and the tangent line in the xy-plane. f(x, y) = 4x² - y c = 6, P(2, 10)
Use polar coordinates and L’Hôpital’s Rule to find the limit. lim (x, y)→(0, 0) - 1 − cos(x² + y²) x² + y²
Sketch the graph of an arbitrary function ƒ satisfying the given conditions. State whether the function has any extrema or saddle points. (There are many correct answers.)ƒ(x, y) > 0 and ƒy(x, y) < 0 for all (x, y).
Find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results. 4xy f(x, y) (x² + 1)(y² + 1) R = {(x, y): 0 ≤ x ≤ 1,0 ≤ y ≤ 1}
Evaluate ƒx and ƒy at the given point. f(x, y) = 2xy 4x² + 5y²² 2 (1, 1)
Sketch thegraph of an arbitrary function ƒ satisfying the givenconditions. State whether the function has any extrema orsaddle points. (There are many correct answers.)All of the first and second partial derivatives of ƒ are 0.
Find a normal vector to the level curve ƒ(x, y) = c at P. f(x, y) = X x² + y² c=½, P(1, 1) с
Use polar coordinates and L’Hôpital’s Rule to find the limit. sin(x² + y²) lim (x, y) (0,0) x² + y²
(a) Show that the surfaces intersect at the given point(b) Show that the surfaces have perpendicular tangent planes at this point. x² + y² + z² + 2x - 4y - 4z - 12 = 0, 4x² + y² + 16z² = 24, (1, -2, 1)
An annular cylinder has an inside radius of r₁ and an outside radius of r2 (see figure). Its moment of inertia is I = 1/2m(r²1+ r22), where m is the mass. The tworadii are increasing at a rate of 2 centimeters per second. Findthe rate at which I is changing at the instant the radii are6
Evaluate ƒx and ƒy at the given point. f(x, y) = xy x - y (2,-2)
Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y X
Use polar coordinates and L’Hôpital’s Rule to find the limit. lim (x, y)→(0, 0) sin√x² + y² /x² + y²
Find a normal vector to the level curve ƒ(x, y) = c at P. f(x,y) = xy C c = -3, P(-1, 3)
Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given -values.z = x + y, c = -1, 0, 2, 4
Find a normal vector to the level curve ƒ(x, y) = c at P. f(x, y) = x² + y² c=25, P(3, 4)
Define each of the following for a function of two variables.(a) Relative minimum (b) Relative maximum(c) Critical point (d) Saddle point
Show that the surfaces are tangent to each other at the given point by showing that the surfaces have the same tangent plane at this point. x² + y² + z² - 8x x² + y² + 2z = 7, 12y + 4z + 42 = 0, (2, 3, -3)
Use polar coordinates to find the limit. Let x = r cos θ and y = r sin θ, and note that (x,y) → (0, 0) implies r → 0. lim (x, y)→(0, 0) sinx² + y²
(a) Show that the surfaces intersect at the given point(b) Show that the surfaces have perpendicular tangent planes at this point.z = 2xy², 8x² - 5y² - 8z = -13, (1, 1, 2)
Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y X
Evaluate ƒx and ƒy at the given point.ƒ(x, y) = arccos xy, (1, 1)
Evaluate ƒx and ƒy at the given point. f(x, y) = arctan X (2,-2)
Show that the surfaces are tangent to each other at the given point by showing that the surfaces have the same tangent plane at this point. x² + 2y² + 3z² = 3, x² + y² + z² + 6x 10y + 14 = 0, (-1, 1, 0)
Use polar coordinates to find the limit. Let x = r cos θ and y = r sin θ, and note that (x,y) → (0, 0) implies r → 0. lim (x, y)→(0, 0) cos(x² + y²)
The Ideal Gas Law is pV = mRT, where p is the pressure, V is the volume, m is the constant mass, R is a constant, T is the temperature, and p and V are functions of time. Find dT/dt, the rate at which the temperature changes with respect to time.
Find a normal vector to the level curve ƒ(x, y) = c at P. f(x, y) = 6 - 2x - Зу c = 6, P(0, 0)
The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. What are the rates of change of the volume and surface area when the radius is 12 inches and the height is 36 inches?
Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y X
Evaluate ƒx and ƒy at the given point. f(x, y) = sin xy, 2, F|4
Find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results.ƒ(x, y) = x² + 2xy + y², R = {(x, y): [x] ≤ 2, |y| ≤ 1}
Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).](a)(b)(c)(d) y X
Use polar coordinates to find the limit. Let x = r cos θ and y = r sin θ, and note that (x,y) → (0, 0) implies r → 0. x² - y² lim (x, y) (0,0) √√√x² + y²
Find the point(s) on the surface at which the tangent plane is horizontal. < X + = + Áx = 2 Ax I
The graph of the function w = ƒ(x, y) is shown below.(a) Assume that x and y are functions of a single variable r. Give the chain rule for finding dw/dr.(b) Assume that x and y are functions of two variables r and θ. Give the chain rule for finding ∂w/∂r and∂w/∂θ. -2 X 2 2
Use polar coordinates to find the limit. Let x = r cos θ and y = r sin θ, and note that (x,y) → (0, 0) implies r → 0. x²y² lim (x, y) (0,0) x² + y²
Evaluate ƒx and ƒy at the given point. f(x,y) = cos(2x – y), ㅠㅠ 4' 3
Find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results.ƒ(x, y) = 2x - 2xy + y²R: The region in the xy-plane bounded by the graphs of y = x²and y = 1
Find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results.ƒ(x, y) = 3x² + 2y² - 4yR: The region in the xy-plane bounded by the graphs of y = x²and y = 4
Find the point(s) on the surface at which the tangent plane is horizontal.z = 5xy
For ƒ(x, y) = 0, give therule for finding dy/dx implicitly. For ƒ(x, y, z) = 0, givethe rule for finding ∂z/∂x and ∂z/∂y implicitly.
Use a computer algebra system to graph the function.ƒ(x, y) = x sin y
Evaluate ƒx and ƒy at the given point.ƒ(x, y) = e-x cos y, (0, 0)
Find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results.ƒ(x, y) = (2x - y)²R: The triangular region in the xy-plane with vertices (2, 0),(0, 1), and (1, 2)
Use polar coordinates to find the limit. Let x = r cos θ and y = r sin θ, and note that (x,y) → (0, 0) implies r → 0. x3 x² + y² lim (x, y) (0,0) x² + y²
Use the function to show that ƒx(0, 0) and ƒy(0, 0) both exist, but that ƒ is not differentiable at (0, 0). f(x, y) = = 5x²y x³ +³ 0, (x, y) = (0, 0) (x, y) = (0, 0)
Find the point(s) on the surface at which the tangent plane is horizontal.z = 4x² + 4xy - 2y² + 8x - 5y - 4
(a) Use the graph to estimate the components of the vector in the direction of the maximum rate of increase in the function at the given point.(b) Find the gradient at the point and compare it with yourestimate in part (a). (c) In what direction would the function be decreasing at the greatest
Let w = ƒ(x, y) be a function in which x and y are functions of two variables s and t. Give the Chain Rule for finding ∂w/∂s and ∂w/∂t.
Use a computer algebra system to graph the function.ƒ(x, y) = x²e(-xy/2)
Find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results.ƒ(x, y) = 12 - 3x - 2yR: The triangular region in the xy-plane with vertices (2,0),(0, 1), and (1, 2)
Use polar coordinates to find the limit. Let x = r cos θ and y = r sin θ, and note that (x,y) → (0, 0) impliesr → 0. lim xy² (x, y) (0,0) x² + y²
Use the function to show that ƒx(0, 0) and ƒy(0, 0) both exist, but that ƒ is notdifferentiable at (0, 0). f(x, y) 3x²y x² + y² 0, (x, y) = (0, 0) (x, y) = (0,0)
(a) Use the graph to estimate the components of the vector in the direction of the maximum rate of increase in the function at the given point.(b) Find the gradient at the point and compare it with your estimate in part (a). (c) In what direction would the function be decreasing at the
Let w = ƒ(x, y), x = g(s, t), andy = h(s, t), where ƒ, g, and h are differentiable. Use theappropriate Chain Rule to find ws(1, 2) and wt(1, 2), given thefollowing table of values. g(1, 2) 4 g,(1, 2) -2 h(1, 2) g,(1, 2) 3 -3 h,(1, 2) 8 f(4,3) -5 h,(1, 2) 5 f,(4,3) 7
Find the point(s) on the surface at which the tangent plane is horizontal.z = x² - xy + y² - 2x - 2y
Use the limit definition of partial derivatives to find ƒx(x, y) and ƒy(x, y). f(x, y) = 1 x + y
Let w = ƒ(x, y) be a function in whichx and y are functions of a single variable t. Give the ChainRule for finding dw/dt.
Use a computer algebra system to graph the function. 2 1/2√144- - 16x²-9y²
Discuss the continuity of the functions ƒ and g. Explain any differences. f(x, y) g(x, y) = + 2xy² + y², (x, y) = (0, 0) x² + y² (x, y) = (0,0) 0, + 2xy² + y², (x, y) ‡ (0, 0) x² + y² (x, y) = (0, 0) 1,
Find the absolute extrema of the function over the region R. (In each case, R contains the boundaries.) Use a computer algebra system to confirm your results.ƒ(x, y) = x² + xy, R = {(x, y): |x| ≤ 2, |y| ≤ 1}
Show that the function is differentiable by finding values of ε1 and ε2 as designated in the definition of differentiability, and verify that both ε1 and ε2 approach 0 as (Δx, Δy) → (0, 0).ƒ(x, y) = 5x - 10y + y³-
Find the point(s) on the surface at which the tangent plane is horizontal.z = 3x² + 2y² - 3x + 4y - 5
Consider the functionFind a unit vector u orthogonal to Δƒ(3, 2) and calculateDuƒ(3, 2). Discuss the geometric meaning of the result. f(x, y) = 3 X -- 3 2
Consider the functionFind the maximum value of the directional derivative at (3, 2). f(x, y) = 3 X -- 3 2
Use the Second Partials Test to verify that the formulas for a and b given in Theorem 13.18 yield a minimum.Data from in Theorem 13.18 THEOREM 13.18 Least Squares Regression Line The least squares regression line for {(x₁, y₁), (x₂, y2),. ... (x, yn)} is given by f(x) = ax + b,
Let w = ƒ(x, y), x = g(t), andy = h(t), where ƒ, g, and h are differentiable. Use the appropriateChain Rule to find dw/dt when t = 2, given the following tableof values. g(2) 4 h(2) g'(2) h'(2) (4,3),(4,3) 3 − 1 6 -5 7
Use a computer algebra system to graph the function.z + y2 - x2 + 1
Find theabsolute extrema of the function over the region R. (In eachcase, R contains the boundaries.) Use a computer algebra systemto confirm your results. f(x, y) = x² - 4xy + 5 R = {(x, y): 1 ≤ x ≤ 4,0 ≤ y ≤ 2}
Use the limit definition of partial derivatives to find ƒx(x, y) and ƒy(x, y).ƒ(x, y) = √x + y
Discuss the continuity of the functions ƒ and g. Explain any differences. f(x, y) = g(x, y) = [x4-y4 x² + y² (0, [x² - y²) x² + y² (1, करे (x, y) = (0,0) (x, y) = (0, 0) (x, y) = (0, 0) (x, y) = (0, 0)
Show that the function is differentiable by finding values of ε1 and ε2 as designated in the definition of differentiability, and verify that both ε1 and ε2 approach 0 as (Δx, Δy) → (0, 0).ƒ(x, y) = x²y
Find the point(s) on the surface at which the tangent plane is horizontal.z = 3 - x² - y² + 6y
Consider the functionFind Δƒ(x, y). f(x, y) = 3 X -- 3 2

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