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mathematics
precalculus

Calculus Of A Single Variable 11th Edition Ron Larson, Bruce H. Edwards - Solutions

In Exercises 63–68, find fx, fy, and fz, and evaluate each at the given point. f(x, y, z) = 3x² + y² = 2z², (1, -2, 1)
In Exercises 67–70, discuss the continuity of the composite function f ° g. f(1) = ²/ g(x, y) = 2х - Зу
In Exercises 71–76, find each limit. (a) lim Δε 0 (b) lim Δy Ο f(x + Δx, y) - f(x,y) Δι f(x,y + Δy) - f(x,y) Δν
In Exercises 67–70, discuss the continuity of the composite function f ° g. f(1) = ²/ t g(x, y) = x² + y²
A principal of $5000 is deposited in a savings account that earns interest at a rate of r (written as a decimal), compounded continuously. The amount A(r, t) after t years is A(r, t) = 5000ert.Use this function of two variables to complete the table. 0.02 Rate 5 0.03 0.04 Number of Years 0.05 5 10
In Exercises 71–76, find each limit. (a) lim Δε 0 (b) lim Δy Ο f(x + Δx, y) - f(x,y) Δι f(x,y + Δy) - f(x,y) Δν
In Exercises 67–70, discuss the continuity of the composite function f ° g. ƒ(1) = 1 1² / ₁ f(t) -t g(x, y) = x² + y²
In Exercises 71–76, find each limit.f(x, y) = x2 − 4y (a) lim Δε 0 (b) lim Δy Ο f(x + Δx, y) - f(x,y) Δι f(x,y + Δy) - f(x,y) Δν
In Exercises 71–76, find each limit.f(x, y) = 3x2 + y2 (a) lim Δε 0 (b) lim Δy Ο f(x + Δx, y) - f(x,y) Δι f(x,y + Δy) - f(x,y) Δν
The average length of time that a customer waits in line for service iswhere y is the average arrival rate, written as the number of customers per unit of time, and x is the average service rate, written in the same units. Evaluate each of the following.(a) W(15, 9)(b) W(15, 13)(c) W(12, 7)(d) W(5,
The table shows the net sales x (in billions of dollars), the total assets y (in billions of dollars), and the shareholder’s equity z (in billions of dollars) for Walmart for the years 2010 through 2015.A model for the data is z = f(x, y) = 0.428x - 0.653y + 8.172.(a) Complete a fourth row in the
The table shows the national health expenditures (in billions of dollars) for the Department of Veterans Affairs x, workers’ compensation y, and Medicaid z from 2009 through 2014.A model for the data is given by z = -0.120x2 + 0.657y2 + 17.70x - 51.53y + 842.5.(a) Find (b) Determine the
In Exercises 105 and 106, show that the functions u and v satisfy the Cauchy-Riemann equationsu = ex cos y, v = ex sin y ди ду ах dy and ди ду ду ах
In Exercises 105 and 106, show that the functions u and v satisfy the Cauchy-Riemann equationsu = x2 - y2, v = 2xy ди ду ах dy and ди ду ду ах
In Exercises 69–76, find all values of x and y such that fx(x, y) = 0 and fy(x, y) = 0 simultaneously.f(x, y) = ln(x2 + y2 + 1)
In Exercises 91–94, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If f(x0, y0) = f(x1, y1), then x0 = x1 and y0 = y1.
In Exercises 81 and 82, use the Cobb-Douglas production function to find the production level when x = 600 units of labor and y = 350 units of capital.f(x, y) = 80x0.5y0.5
In Exercises 77–86, find the four second partial derivatives. Observe that the second mixed partials are equal.z = √x2 + y2
In Exercises 95–98, show that the function satisfies Laplace’s equation ∂2z /∂x2 + ∂2z/∂y2 = 0.z = 5xy
Consider the function defined by(a) Find fx(x, y) and fy(x, y) for (x, y) ≠ (0, 0).(b) Use the definition of partial derivatives to find fx(0, 0) and fy(0, 0).(c) Use the definition of partial derivatives to find fxy(0, 0) and fyx(0, 0).(d) Using Theorem 13.3 and the result of part (c), what can
Consider the function f(x, y) = (x2 + y2)2/3.Show that f(x, y)=3(x² 0, 4x + y²)1/3² (x, y) = (0, 0) (x, y) = (0, 0)
In Exercises 21–32, find the domain and range of the function. g(x, y) = y X
In Exercises 11–24, find the limit and discuss the continuity of the function. lim (x, y, z) (1, 3, 4) √x + y + z
In Exercises 11–24, find the limit and discuss the continuity of the function. lim (x, y, z) (-2, 1, 0) xez
In Exercises 11–24, find the limit and discuss the continuity of the function. arccos(x/y) lim (x, y) (0.1) 1 + xy
In Exercises 11–24, find the limit and discuss the continuity of the function. lim (x, y) (1,-4) sin X y
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. f(x, y) = 3x2 – 21 f(x + Δx, y) - f(x,y) · (2) Δε (b) f(x, y + Δy) - f(x,y) Δν
In Exercises 11–24, find the limit and discuss the continuity of the function. arcsin xy lim (x, y) (0, 1) 1 xy
In Exercises 11–24, find the limit and discuss the continuity of the function. lim (x, y) (n/3, 2) y cos xy
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. f(x, y) = 2x + y2 (a) f(x + Δx, y) - f(x, y) Δε (b) f(x, y + Δy) - f(x, y) Δν
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. g(x, y) = [ (2t (2t - 3) dt (a) g(4,0) (b) g(4, 1) (c) g(4,3) (d) g(ૐ, 0)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. V(r, h) = r²h (a) V(3, 10) (b) V(5,2) (c) V(4,8) (d) V(6, 7)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. f(x, y) = x sin y (a) f(2, π/4) (b) f(3, 1) (c) f(-3,0) (d) f(4, π/2)
In Exercises 11–24, find the limit and discuss the continuity of the function. lim (x, y) (-1, 1) (x + 4y² + 5)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. = [² / dr dt g(x, y) = (a) g(4,1) (b) g(6, 3) (c) g(2,5) (d) g(1,7)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. h(x, y, z) = xy N (a) h(-1, 3, -1) (b) h(2,2,2) (c) h(4,4t, 1²) (d) h(-3,2,5)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. f(x, y, z) = √√√x + y + z (a) f(2, 2, 5) (b) f(0, 6, -2) (c) f(8, -7, 2) (d) f(0, 1, -1)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. g(x, y) = In|x + y (a) g(1,0) (b) g(0, -1²) (c) g(e, 0) (d) g(e, e)
In Exercises 21–32, find the domain and range of the function.g(x, y) = x√y
In Exercises 21–32, find the domain and range of the function.f(x, y) = 3x2 - y
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. f(x, y) = 4x² - 4y² (a) f(0, 0) (c) f(2, 3) (e) f(x, 0) (b) f(0, 1) (d) f(1, y) (f) f(t, 1)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. f(x, y) = 2x - y + 3 (a) f(0, 2) (b) f(-1,0) (c) f(5, 30) (d) f(3, y) (e) f(x, 4) (f) ƒ(5, 1)
In Exercises 11–24, find the limit and discuss the continuity of the function. lim (x, y) (3, 1) (x²-2y)
In Exercises 9–20,evaluate the function at the given values of the independent variables. Simplify the results. f(x, y) = xex (a) f(-1,0) (b) f(0, 2) (c) f(x, 3) (d) f(t, -y)
A vertical force of 40 pounds acts on a wrench, as shown in the figure. Find the torque at P. P 9 in. 60° F = 40 lb
Explain how to sketch a contour map of a function of x and y.
Explain how examining limits along different paths might show that a limit does not exist. Does this type of examination show that a limit does exist? Explain.
Explain why z2 = x + 3y is not a function of x and y.
Find a set of parametric equations of the line with the given characteristics.The line passes through the point (-6, -8, 2) and is perpendicular to the xz-plane.
(a) Find the projection of u onto v(b) Find the vector component of u orthogonal to v.u = 〈6, 7〉, v = 〈1,4〉
(a) Find the projection of u onto v,(b) Find the vector component of u orthogonal to v.u = 〈1, -1, 1〉, v = 〈2, 0, 2〉
(a) Find the projection of u onto v,(b) Find the vector component of u orthogonal to v.u = 4i + 2j, v = 3i + 4j
Determine whether u and v are orthogonal, parallel, or neither.u = 〈-3, 0, 9〉v = 〈1, 0, -3〉
Find the distance between the points.(-2, 1, -5), (4, -1, -1)
Describe and sketch the surface.y = 1/2z
Convert the point from rectangular coordinates to(a) Cylindrical coordinates (b) Spherical coordinates.(-√3, 3, -5)
Convert the point from rectangular coordinates to(a) Cylindrical coordinates(b) Spherical coordinates.(8, 8, 1)
Convert the point from cylindrical coordinates to rectangular coordinates.(5, π, 1)
Find an equation in rectangular coordinates for the surface represented by the cylindrical equation, and sketch its graph.z = r2 sin2 θ + 3r cos θ
Find an equation in rectangular coordinates for the surface represented by the cylindrical equation, and sketch its graph.r = -5z
Find an equation in rectangular coordinates for the surface represented by the spherical equation, and sketch its graph.∅ = π/4
Find an equation in rectangular coordinates for the surface represented by the spherical equation, and sketch its graph.ρ = 9 sec θ
Consider points P and Q on a curve. What does it mean for the curvature at P to be less than the curvature at Q?
Determine whether u and v are orthogonal, parallel, or neither.u = 〈7, -2, 3〉v = 〈-1, 4, 5〉
Find a unit vector in the direction of u = 〈2, 3, 5〉.
Find the standard equation of the sphere with the given characteristics.Center: (3, -2, 6); Radius: 4
Find the distance between the points.(1, 6, 3), (-2, 3, 5)
In Exercises 1–4,(a) Find the domain of r,(b) Determine the interval(s) on which the function is continuous. r(t) = fi + نها - -j + k k
An object moves along a curve in the plane. What information do you gain about the motion of the object from the velocity vector to the curve at time t?
In Exercises 1–4,(a) Find the domain of r,(b) Determine the interval(s) on which the function is continuous. r(t): = √²9i- j + In(t − 1)k
In Exercises 3 –10, find r'(t), r(t0), and r'(t0) for the given value of t0. Then sketch the curve represented by the vector-valued function and sketch the vectors r(t0), and r'(t0).r(t) = (1 - t2)i + tj, t0 = 3
For each scenario, describe the direction of the acceleration vectors. Explain your reasoning.(a) A comet traveling through our solar system in a parabolic path(b) An object thrown on Earth’s surface
You want to toss an object to a friend who is riding a Ferris wheel (see figure). The following parametric equations give the path of the friend r1(t) and the path of the object r2(t). Distance is measured in meters, and time is measured in seconds.(a) Locate your friend’s position on the Ferris
Find the domain of the vector-valued function.r(t) = √4 - t2 i + t2j - 6tk
Find the domain of the vector-valued function.r(t) = ln ti - etj - tk
In Exercises 9–14, find the unit tangent vector T(t) and a set of parametric equations for the tangent line to the space curve at point P.r(t) = cos ti + 3 sin t j + (3t - 4)k, P(1, 0, -4)
In Exercises 11–18, find r'(t).r(t) = t4i - 5tj
In Exercises 11–18, find r'(t).r(t) = √ti + (1 - t3) j
Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations.P(0, 0, 0), Q(5, 2, 2)
Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]r(t) = ti + ln tj + 2t/3k, 0.1 ≤ t ≤ 5 (a) (c) x -2 Z 2 Z 2 y (b) (d) X -2 X 4 2 4 2 Z 2 2 4 y
In Exercises 11–18, find r'(t).r(t) = 4√ti + t2 √t j + ln t2k
Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations.P(-3, -6, -1), Q(-1, -9, -8)
In Exercises 19–22, find (a) r'(t), (b) r''(t), and (c) r'(t) · r''(t).r(t) = t3i + 1/2 t2j
In Exercises 19 and 20, find (a) r'(t), (b) r''(t), and (c) r'(t) · r''(t).r(t) = (t2 + 4t) i - 3t2j
In Exercises 19–22, find (a) r'(t), (b) r''(t), and (c) r'(t) · r''(t).r(t) = (t2 + t)i + (t2 - t)j
In Exercises 19 and 20, find (a) r'(t), (b) r''(t), and (c) r'(t) · r''(t).r(t) = 5 cos ti + 2 sin tj
In Exercises 21 and 22, find (a) r'(t), (b) r''(t), (c) r'(t) · r''(t), and (d) r'(t) x r''(t).r(t) = 2t3i + 4tj - t2k
In Exercises 23 and 24, find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) t : _—__²__i + tj + √1 + tk t-2
In Exercises 19–22, find (a) r'(t), (b) r''(t), and (c) r'(t) · r''(t).r(t) = 4 cos ti + 4 sin tj
In Exercises 19–22, find the curvature of the curve, where s is the arc length parameter.r(s) = cos 1/2si + √3/2sj + sin 1/2sk
In Exercises 19–22, find the curvature of the curve, where s is the arc length parameter.r(s) = cos si + sin sj + 5k
Sketch the plane curve represented by the vector-valued function and give the orientation of the curve.r(t) = t/4i + (t - 1)j
In Exercises 23 and 24, find the open interval(s) on which the curve given by the vector-valued function is smooth.r(t) = (t - 1)3i + (t - 1)4j
In Exercises 27–30, find the indefinite integral. (Pi + 5tj + 81³k) dt
In Exercises 23–26, find (a) r'(t), (b) r''(t), (c) r'(t) · r''(t), and (d) r'(t) x r''(t).r(t) = 1/2 t2i - t j + 1/6 t3k
In Exercises 23–28, find the curvature of the plane curve at the given value of the parameter. r(t) =
In Exercises 23–28, find the curvature of the plane curve at the given value of the parameter.r(t) = t2i + j, t = 2
Sketch the plane curve represented by the vector-valued function and give the orientation of the curve.r(t) = t3i + t2j
Sketch the plane curve represented by the vector-valued function and give the orientation of the curve.r(t) = (5 - t)i + √tj
In Exercises 23–26, find (a) r'(t), (b) r''(t), (c) r'(t) · r''(t), and (d) r'(t) x r''(t).r(t) = 〈cos t + t sin t, sin t - t cos t, t〉

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