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mathematics
precalculus

Calculus Early Transcendentals 8th edition James Stewart - Solutions

Find the maximum rate of change of f (x, y) = x2y + √y at the point (2, 1). In which direction does it occur?
Find the directional derivative of f at the given point in the indicated direction.f(x, y) = x2e-y, (-2, 0), in the direction toward the point (2, -3)
(a) When is the directional derivative of f a maximum?(b) When is it a minimum?(c) When is it 0?(d) When is it half of its maximum value?
Find the gradient of the function f (x, y, z) = x2eYZ2.
If cos(x/z) = 1 + x2y2 + z2, find ∂z/∂x and ∂z/∂y.
If z = f (u, v), where u = xy, v = y/x, and f has continuous second partial derivatives, show that д?2 - y². ду? д?z a²z -4uv дz + 2v до дх? ди ду
The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate of 2 in/s, and the contained angle is increasing at a rate of 0.05 radian/s. How fast is the area of the triangle changing when x = 40 in, y = 50 in, and θ = π/6?
If z = y + f (x2 - y2), where f is differentiable, show that y ∂z/∂x + x ∂z/∂y = x
Use a tree diagram to write out the Chain Rule for the case where w = f (t, u, v), t = t( p, q, r, s), u = u( p, q, r, s), and v = v( p, q, r, s) are all differentiable functions.
If v = x2 sin y + yexy, where x = s + 2t and y = st, use the Chain Rule to find ∂v/∂s and ∂v/∂t when s = 0 and t = 1.
If u = x2y3 + z4, where x = p + 3p2, y = pep, and z = p sin p, use the Chain Rule to find du/dp.
The two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum error in the calculated value of (a) the area of the triangle and(b) the length of the hypotenuse.
Find the linear approximation of the function f (x, y, z) = x3√y2 + z2 at the point (2, 3, 4) and use it to estimate the number (1.98)3√(3.01)2 + (3.97)2.
Find du if u = ln(1 + se2t).
Find the points on the hyperboloid x2 + 4y2 - z2 = 4 where the tangent plane is parallel to the plane 2x + 2y + z = 5.
Use a computer to graph the surface z = x2 + y4 and its tangent plane and normal line at (1, 1, 2) on the same screen. Choose the domain and viewpoint so that you get a good view of all three objects.
Find equations of (a) The tangent plane and (b) The normal line to the given surface at the specified point.sin(xyz) = x + 2y + 3z, (2, -1, 0)
Find equations of (a) The tangent plane and (b) The normal line to the given surface at the specified point.xy + yz + zx = 3, (1, 1, 1)
Find equations of (a) The tangent plane and (b) The normal line to the given surface at the specified point.x2 + 2y2 - 3z2 = 3, (2, 21, 1)
Find equations of (a) The tangent plane and (b) The normal line to the given surface at the specified point.z = ex cos y, (0, 0, 1)
Find equations of (a) The tangent plane and (b) The normal line to the given surface at the specified point.z = 3x2 - y2 + 2x, (1, -2, 1)
If z = sin(x + sin t), show that∂z/∂x ∂2z/∂x ∂t = ∂z/∂t ∂2z/∂x2
If z = xy + xey/x, show that x ∂z/∂x + y ∂z/∂y = xy + z.
Find all second partial derivatives of f .v = r cos(s + 2t)
Find all second partial derivatives of f .f(x, y, z) = xkylzm
Find all second partial derivatives of f .z = xe-2y
Find all second partial derivatives of f .f(x, y) = 4x3 - xy3
The speed of sound traveling through ocean water is a function of temperature, salinity, and pressure. It has been modeled by the functionC = 1449.2 + 4.6T - 0.055T2 + 0.00029T3 + (1.34 - 0.01T)(S - 35) + 0.016)where C is the speed of sound (in meters per second), T is the temperature (in degrees
Find the first partial derivatives.S(u, v, w) = u arctan(v√w)
Find the first partial derivatives.F(a, B) = a2 ln(a2 + 2)
Find the first partial derivatives.g(u, v) = u + 2v/u2 + v2
Find the first partial derivatives.f(x, y) = (5y3 + 2x2y)8
Find a linear approximation to the temperature function T(x, y) in Exercise 11 near the point (6, 4). Then use it to estimate the temperature at the point (5, 3.8).
Evaluate the limit or show that it does not exist. 2xy lim (x, y)→(0, 0) x² + 2y?
Evaluate the limit or show that it does not exist. 2ху (х, у) —- (1, 1) х2 + 2y? lim
The contour map of a function f is shown.(a) Estimate the value of f (3, 2).(b) Is fx (3, 2) positive or negative? Explain.(c) Which is greater, fy(2, 1) or fy(2, 2)? Explain. УА 5 4 80 70 3 40 20 х 4. 2.
Make a rough sketch of a contour map for the function whose graph is shown. ZA y
Sketch several level curves of the function.f(x, y) = ex + y
Sketch several level curves of the function.f (x, y) = √4x2 + y2
Sketch the graph of the function.f (x, y) = x2 + (y - 2)2
Sketch the graph of the function.f (x, y) = 1 - y2
Find and sketch the domain of the function.f (x, y) = √4 - x2 - y2 + √1 - x2
Find and sketch the domain of the function.f (x, y) = ln(x + y + 1)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f (x, y) has two local maxima, then f must have a local minimum.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f (x, y) = sin x + sin y, then -√2 < Duf (x, y) < √2.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f (x, y) = ln y, then ∇f (x, y) = 1/y.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f is a function, then lim (х, у) — (2, 5) f(x, y) = f(2, 5)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f has a local minimum at (a, b) and f is differentiable at (a, b), then ∇f (a, b) = 0.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If fx(a, b) and fy(a, b) both exist, then f is differentiable at (a, b).
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f(x, y) → L as (x, y) → (a, b) along every straight line through (a, b), then lim(x, y)l(a, b) f (x, y) = L.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Dk f(x, y, z) = f(x, y, z)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.fxy = ∂2f/∂x∂y
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.There exists a function f with continuous second-order partial derivatives such that fx(x, y) = x + y2 and f/(x, y) = x - y2.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. f(a, y) – f(a, b) f,(a, b) = lim y – b
Referring to Exercise 27, we now suppose that the production is fixed at bLaK1-a = Q, where Q is a constant. What values of L and K minimize the cost functionC(L, K) = mL + nK?
Consider the problem of minimizing the function f (x, y) = x on the curve y2 + x4 - x3 = 0 (a piriform).(a) Try using Lagrange multipliers to solve the problem.(b) Show that the minimum value is f (0, 0) = 0 but the Lagrange condition ∇f (0, 0) = λ∇g(0, 0) is not satisfied for any value of
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.f (x, y) = sin x sin y, -π < x < π, -π < y < π
The helix r(t) = (cos(πt/2), sin(πt/2), t) intersects the sphere x2 + y2 + z2 = 2 in two points. Find the angle of intersection at each point.
Where does the helix r(t) = kcos πt, sin πt, t) intersect the paraboloid z = x2 + y2? What is the angle of intersection between the helix and the paraboloid? (This is the angle between the tangent vector to the curve and the tangent plane to the paraboloid.)
Suppose that over a certain region of space the electrical potential V is given by V(x, y, z) = 5x2 - 3xy + x/z.(a) Find the rate of change of the potential at P(3, 4, 5) in the direction of the vector v = i + j - k.(b) In which direction does V change most rapidly at P?(c) What is the maximum rate
Find the directional derivative of the function at the given point in the direction of the vector v.f (x, y) = ex sin y, (0, π/3), v = (-6, 8)
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f (x, y) = √2x + 3y , (3, 1), θ = -π/6
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f (x, y) = y cos (xy), (0, 1), θ = π/4
Find the directional derivative of f at the given point in the direction indicated by the angle θ.f (x, y) = xy3 - x3, (1, 2), = π/3
Assume that all the given functions have continuous second-order partial derivatives.If z = f (x, y), where x = r cos θ and y = r sin θ, show that 1 3z + dr? 1 əz r år
Assume that all the given functions are differentiable.If z = f (x, y), where x = r cos θ and y = r sin θ,(a) find ∂z/∂r and ∂z/∂θ and (b) show that е-Ө'-Ө-не r? дх ду ar
Suppose f is a differentiable function of x and y, and g(u, v) = f (eu + sin v, eu + cos v). Use the table of values to calculate gu(0, 0) and gv(0, 0). fx f 3 fy (0, 0) (1, 2) 4 5 2.
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = tan(uyv), u = 2s + 3t, v = 3s - 2t
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.z = er cos θ, r = st, θ = √s2 + t2
Use the Chain Rule to find dz/dt or dw/dt.z = x - y/x + 2y, x = eπt, y = e-πt-
Show that the function is differentiable by finding values єof є1 and є2 that satisfy Definition 7.f (x, y) = xy - 5y2
Find the differential of the function.R = aB2 cos у
Find the differential of the function.z = e-2x cos 2πt
Find the linear approximation of the function f(x, y) = 1 - xy cos πy at (1, 1) and use it to approximate f(1.02, 0.97). Illustrate by graphing f and the tangent plane.
Show that the Cobb-Douglas production function satisfies P(L, K0) = C1(K0)Lby solving the differential equationdP/dL = a P/L
Show that the Cobb-Douglas production function P = bLaK bsatisfies the equation ӘР ӘР ( α +β )Ρ L - + K Әк
Find the first partial derivatives of the function.f (x, t) = t2e-x
If c E Vn, show that the function f given by f (x) = c • x is continuous on Rn.
Use a computer to investigate the family of surfaces z = x2 + y2 + cxy. In particular, you should determine the transitional values of c for which the surface changes from one type of quadric surface to another.
Use a computer to investigate the family of surfacesz = (ax2 + by2)e-x2-y2How does the shape of the graph depend on the numbers a and b?
Investigate the family of functions f (x, y) = ecx2+y2. How does the shape of the graph depend on c?
Graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both x and y become large? What happens as (x, y) approaches the origin?f (x, y) = xy/x2 + y2
Graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both x and y become large? What happens as (x, y) approaches the origin?f (x, y) = x + y/x2 + y2
Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the “peaks and valleys.” Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be “local maximum points”? What
Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the “peaks and valleys.” Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be “local maximum points”? What
Describe how the graph of t is obtained from the graph of f .(a) g(x, y) = f (x - 2, y)(b) g(x, y) = f (x, y + 2)(c) g(x, y) = f (x + 3, y - 4)
Describe how the graph of t is obtained from the graph of f .(a) g(x, y) = f (x, y) + 2(b) g(x, y) = 2f (x, y)(c) g(x, y) = -f (x, y)(d) g(x, y) = 2 - f (x, y)
Describe the level surfaces of the function.f (x, y, z) = x2 - y2 - z2
Describe the level surfaces of the function.f (x, y, z) = y2 + z2
Describe the level surfaces of the function.f (x, y, z) = x2 + 3y2 + 5z2
Describe the level surfaces of the function.f (x, y, z) = x + 3y + 5z
Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices.z = x - y/1 + x2 + y2 B D II III IV VI
Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices.z = (1 - x2)(1 - y2) B D II III IV VI
Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices.z = sin x - sin y B D II III IV VI
Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices.z = sin(x - y) B D II III IV VI
Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices.z = ex cos y B D II III IV VI
Match the function (a) with its graph (labeled A–F below) and (b) with its contour map (labeled I–VI). Give reasons for your choices.z = sin(xy) B D II III IV VI
Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.f (x, y) = cos x cos y B D II III IV
Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.f (x, y) = e-(x2+y2)/3(sin(x2) + cos(
Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.f (x, y) = xy3 - yx3 (dog saddle)

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