Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. R = F (t, u) where t = t (w, x, y, z), u – u (w, x, y, z)
> Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.
> Find three positive numbers whose sum is 100 and whose product is a maximum.
> Find the points on the surface y2 = 9 + xz that are closest to the origin.
> Find the points on the cone z2 = x2 + y2 that are closest to the point (4, 2, 0).
> Find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 1).
> Find the shortest distance from the point (2, 0, -3) to the plane x + y + z = 1.
> If a function of one variable is continuous on an interval and has only one critical number, then a local maximum has to be an absolute maximum. But this is not true for functions of two variables. Show that the function has exactly one critical point,
> Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your pre
> For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the function has only two critical points, but has local maxima a
> Find the absolute maximum and minimum values of f on the set D. f (x, y) = x3 - 3x - y3 + 12y, D is the quadrilateral whose vertices are (-2, 3), (2, 3), (2, 2), and (-2, -2)
> Find the absolute maximum and minimum values of f on the set D. f(x, y) = 2x³ + y*, D= {(x, y) | x² + y² < 1}
> Find the absolute maximum and minimum values of f on the set D. f(x, y) = xy², D= {(x, y) | x > 0, y > 0, x² + y? < 3}
> Find the absolute maximum and minimum values of f on the set D. f(x, y) = x² + 2y² D = {(x, y) | 0 < x< 2,0 < y< 3} — 2х — 4у + 1,
> Find the absolute maximum and minimum values of f on the set D. f(x, y) = x² + xy + y? – 6y, D = {(x, y) | –3 <I< 3,0 < y< 5}
> Find the absolute maximum and minimum values of f on the set D. f(x, y) = x² + y² + x*y + 4, D = {(r, y) | |x| < 1, |y| < 1}
> Find the absolute maximum and minimum values of f on the set D. f (x, y) = x + y - xy, D is the closed triangular region with vertices (0, 0), (0, 2), and (4, 0)
> Find the absolute maximum and minimum values of f on the set D. f (x, y) = x2 + y2 - 2x, D is the closed triangular region with vertices (2, 0), (0, 2), and (0, -2)
> Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or l
> Use the level curves in the figure to predict the location of the critical points of f and whether f has a saddle point or a local maximum or minimum at each critical point. Explain your reasoning. Then use the Second Derivatives Test to confirm your pre
> Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or l
> Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or l
> Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or l
> Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. f(x, y) = sin x + sin y + cos(x + y), 0 < x< T/4, 0 < y</4
> Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. f(x, y) = sin x + sin y + sin(x + y), 0 <x< 27, 0 <y< 27
> Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. f(x, y) = (x – y)e²-y²
> Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. f(x, y) = x² + y² +x-?y-2
> Show that f (x, y) = x2ye-x2-y2 has maximum values at (±1, 1/ 2 ) and minimum values at (±1, -1/ 2 ). Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddl
> Show that f (x, y) = x2 + 4y2 - 4xy + 2 has an infinite number of critical points and that D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point.
> 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
> The contour map shows the average maximum temperature for November 2004 (in 0C). Estimate the value of the directional derivative of this temperature function at Dubbo, New South Wales, in the direction of Sydney. What are the units? 100 200 300 (Di
> Let p(t) = f (t(t), h(t)), where f is differentiable, t(2) = 4, g'(2) = -3, h(2) = 5, h'(2) = 6, fx (4, 5) = 2, fy (4, 5) = 8. Find p'(2).
> Find the directional derivative of the function at the given point in the direction of the vector v. t (u, v) = u2e-v, (3, 0), v = 3i + 4 j
> Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y) = (1, 2), v = (3, 5) .2 + y
> (a). Find the gradient of f. (b). Evaluate the gradient at the point P. (c). Find the rate of change of f at P in the direction of the vector u. f(x, y, 2) 3 у'е"»,, Р(), 1, —1), %3D u= 13. 13 13
> Use the Chain Rule to find dz/dt or dw/dt. w = In /x² + y? + z?, x= sin t, y = cos t, z = tan t %3D
> Use the figure to estimate Du f (2, 2). yA (2, 2) u Vf(2, 2)
> Assume that all the given functions have continuous second-order partial derivatives. If u = f (x, y), where x = es cos t and y = es sin t, show that u ar au -2s e ду?
> Use the Chain Rule to find dz/dt or dw/dt. w = xey/z, x = t2, y = 1 - t, z = 1 + 2t
> Assume that all the given functions have continuous second-order partial derivatives. Show that any function of the form is a solution of the wave equation. z = f (x + at) + t (x – at) az a'z at? [Hint: Let u = x + at, v = x – at.
> The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance R is slowly increasing as the resistor heats up. Use Ohm’s Law, V = IR, to find how the current I is changing at the moment when R = 400 Ω, I = 0.0
> The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 1 m and w = h = 2 m, and ℓ, and w are increasing at a rate of 2 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at whi
> 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) = xy + ey
> Use Equation 6 to find dy/dx. Equation 6: cos (xy) = 1 + sin y aF dy F. dx F,
> Use the Chain Rule to find the indicated partial derivatives. u = xe", x= a°B, y = B'y, t= ya; au du du da' aß' ay when a — -1, B — 2, у %3D 1
> Use the Chain Rule to find the indicated partial derivatives. p + q N p= u + vw, q- v + uw,r= w + uv; %3D p +r' aN aN aN when u = 2, v = 3, w = 4 ne dv' dw
> Use the Chain Rule to find the indicated partial derivatives. P = Vu? + v? + w², u= xe', v= ye", w = e*"; aP ap when x = 0, y = 2 ax ay
> Use the Chain Rule to find the indicated partial derivatives. = xy + yz + zx, x=r cos 0, y = r sin 0, z = = re; dw dw when r= 2,0 = 7/2 ar' a0
> Use the Chain Rule to find the indicated partial derivatives. T = 2и + u = pq r, v= pvg r; aT aT aT when p = 2, q = 1, r = 4 ap' ag' ar
> Use the Chain Rule to find the indicated partial derivatives. z = x* + x*y, x= s + 2t – u, y = stu"; zę ze zę as' at au when s = 4, t = 2, u = 1
> Suppose f is a differentiable function of x and y, and t (r, s) = f (2r - s, s2 - 4r). Use the table of values in Exercise 15 to calculate tr (1, 2) and ts (1, 2). Exercise 15: Suppose f is a differentiable function of x and y, and t (u, v) = f (eu + s
> Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. T = F (p, q, r), where p = p (x, y, z), q = q (x, y, z), r = r (x, y, z)
> Show that every normal line to the sphere x2 + y2 + z2 = r2 passes through the center of the sphere.
> If f is homogeneous of degree n, show that a'f + 2ху a'f a'f + y?. ду? — n(n — 1),f(х, у) дх ду
> A function f is called homogeneous of degree n if it satisfies the equation f (tx, ty) = tn f (x, y) for all t, where n is a positive integer and f has continuous second-order partial derivatives. (a). Verify that f (x, y) = x2y + 2xy2 + 5y3 is homogeneo
> Show that the equation of the tangent plane to the elliptic paraboloid z/c = x2/a2 + y2/b2 at the point (x0, y0, z0) can be written as 2xxo 2yyo + z + z0 a? b?
> The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.
> Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.
> Suppose f is a differentiable function of x and y, and t (u, v) = f (eu + sin v, eu + cos v). Use the table of values to calculate gu (0, 0) and tv (0, 0). f f. fy (0, 0) 6 8. (1, 2) 3 5 4. 2. 3.
> Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.
> Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6.
> Assume that all the given functions are differentiable. If z =[f(ax + y) + g(ax – y)], show that y a? a у? ду
> Assume that all the given functions are differentiable. If z =-[f(x – y) + g(x + y)], show that () - az x²- dy
> Assume that all the given functions are differentiable. If u = f (x, y), where x = es cos t and y = es sin t, show that 2 2 ди du ди + ду no ди no no ax ds at
> A sound with frequency fs is produced by a source traveling along a line with speed vs. If an observer is traveling with speed vo along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is
> A manufacturer has modeled its yearly production function P (the value of its entire production, in millions of dollars) as a Cobb-Douglas function P (L, K) = 1.47L0.65K0.35 where L is the number of labor hours (in thousands) and K is the invested capi
> The pressure of 1 mole of an ideal gas is increasing at a rate of 0.05 kPa/s and the temperature is increasing at a rate of 0.15 K/s. Use the equation PV = 8.31T in Example 2 to find the rate of change of the volume when the pressure is 20 kPa and the te
> Let R(s, t) = G(u((, t), v(s, t)), where G, u, and v are differentiable, u(1, 2) = 5, us(1, 2) = 4, ut(1, 2) = -3, v(1, 2) = 7, vs(1, 2) = 2, vt(1, 2) = 6, Gu(5, 7) = 9, Gv (5, 7) = -2. Find Rs (1, 2) and Rt (1, 2).
> The second directional derivative of f (x, y) is Dåf(x, y) = Da[Du f(x, y)] (. ), calculate If f(x, y) = x³ + 5x²y + y³ and u = Då f(2, 1).
> The radius of a right circular cone is increasing at a rate of 1.8 in/s while its height is decreasing at a rate of 2.5 in/s. At what rate is the volume of the cone changing when the radius is 120 in. and the height is 140 in.?
> Show that the operation of taking the gradient of a function has the given property. Assume that u and v are differentiable functions of x and y and that a, b are constants. (a). ∆ (au + bv) = a ∆u + b âˆ
> Suppose that over a certain region of space the electrical potential V is given by V (x, y, z) − 5x2 - 3xy + xyz. (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 cha
> The temperature at a point (x, y, z) is given by where T is measured in 0C and x, y, z in meters. (a). Find the rate of change of temperature at the point P (2, -1, 2) in the direction toward the point (3, -3, 3). (b). In which direction does the tempe
> The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 1200. (a). Find the rate of change of T at (1, 2, 2) in the direction towar
> 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) = x* – 2x?
> Use Equation 6 to find dy/dx. Equation 6: ey sin x = x + xy aF dy F. dx F,
> Use Equation 6 to find dy/dx. Equation 6: tan-1(x2y) = x + xy2 aF dy F. dx F,
> Find the directions in which the directional derivative of f (x, y) = x2 + xy3 at the point (2, 1) has the value 2.
> Use Equation 6 to find dy/dx. Equation 6: y cos x = x2 + y2 aF dy F. dx F,
> Find the maximum rate of change of f at the given point and the direction in which it occurs. f (p, q, r) = arctan (pqr), (1, 2, 1)
> Find the maximum rate of change of f at the given point and the direction in which it occurs. f (x, y, z) = x/ (y + z), (8, 1, 3)
> Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y, z) = x In(yz), (1,2, )
> Find the maximum rate of change of f at the given point and the direction in which it occurs. f (x, y) = sin(xy), (1, 0)
> Find the maximum rate of change of f at the given point and the direction in which it occurs. f (s, t) = test, (0, 2)
> 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) = x³ + y³ –
> Find the directional derivative of f (x, y, z) = xy2z3 at P (2, 1, 1) in the direction of Q (0, -3, 5).
> 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) = y² – 2y
> 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) = (x² + y)e
> Find the directional derivative of the function at the given point in the direction of the vector v. h (r, s, t) = ln (3r + 6s + 9t), (1, 1, 1), v = 4i + 12j + 6k
> 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) = xye-&²+y³
> 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) = ex cos y
> 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) = y cos x
> Find the directional derivative of the function at the given point in the direction of the vector v. g(s, t) = s Jī, (2,4), v= 2i – j
> Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y) — е'sin y, (0, п/3), v%3 (-6, 8)
> (a). Find the gradient of f. (b). Evaluate the gradient at the point P. (c). Find the rate of change of f at P in the direction of the vector u. f(x, y) = x² In y, P(3, 1), u = -i + Bj
> 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) — х3 — Зх +
> (a). Find the gradient of f . (b). Evaluate the gradient at the point P. (c). Find the rate of change of f at P in the direction of the vector u. f(x, y) %3D х/у, Р(2, 1), u = +
> 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) = x² + xy +
> A table of values for the wind-chill index W = f (T, v) is given in Exercise 14.3.3 on page 923. Use the table to estimate the value of Du f (-20, 30), where u = (i + j)/ 2 . Table from Exercise 14.3.3: Wind speed (km/h) | 20 30 40 50 60 70 T -10 -
> Suppose (0, 2) is a critical point of a function t with continuous second derivatives. In each case, what can you say about t? (a) g:(0, 2) = –1, 9zy(0, 2) = 6, „(0, 2) = 1 (b) gr(0, 2) = –1, gry(0, 2) = 2, gry(0, 2) = –8 %3D (c) gx(0, 2) = 4, 9zy(0
> Use the Chain Rule to find dz/dt or dw/dt. z = xy3 - x2y, x = t2 + 1, y = t2 - 1
