Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx. u = ln (x + 2y)
> Determine the set of points at which the function is continuous. e* + e H(x, y) = eу — 1
> 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 the function and observe where it is discontinuous. Then use the formula to explain what you have observed. 1 f(x, y) = 1 – x² – y? ,2
> Graph the function and observe where it is discontinuous. Then use the formula to explain what you have observed. f (x, y) = e1/(x-y)
> Sketch both a contour map and a graph of the function and compare them. f(x, y) = V36 – 9x² – 4y²
> Sketch both a contour map and a graph of the function and compare them. f (x, y) = x2 + 9y2
> Find h (x, y) = g (f (x, y)) and the set of points at which h is continuous. g(t) = t + In t, f(x, y) : 1 - ху 1+ x?y? .2,2
> Use a computer graph of the function to explain why the limit does not exist. хуз lim (x, y)(0, 0) x? + y6
> Find the limit, if it exists, or show that the limit does not exist. xy4 lim (x, y)- (0, 0) x + y® ,2 8
> The temperature-humidity index I (or humidex, for short) is the perceived air temperature when the actual temperature is T and the relative humidity is h, so we can write I = f (T, h). The following table of values of I is an excerpt from a table compile
> The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h = f (v, t) are recorded in feet in Table 4. Table 4: (a). What is the value of f (40,
> Two contour maps are shown. One is for a function f whose graph is a cone. The other is for a function t whose graph is a paraboloid. Which is which, and why?
> Level curves (isothermals) are shown for the typical water temperature (in 0C) in Long Lake (Minnesota) as a function of depth and time of year. Estimate the temperature in the lake on June 9 (day 160) at a depth of 10 m and on June 29 (day 180) at a dep
> Shown is a contour map of atmospheric pressure in North America on August 12, 2008. On the level curves (called isobars) the pressure is indicated in millibars (mb). (a). Estimate the pressure at C (Chicago), N (Nashville), S (San Francisco), and V (Va
> The position function of a spaceship is and the coordinates of a space station are (6, 4, 9). The captain wants the spaceship to coast into the space station. When should the engines be turned off? 4 r(t) = (3 + f)i + (2 + In f) j + (7- k t2 + 1,
> A contour map for a function f is shown. Use it to estimate the values of f (-3, 3) and f (3, -2). What can you say about the shape of the graph? 70 60 50 40 -1 30 20 10
> The magnitude of the acceleration vector a is 10 cm/s2. Use the figure to estimate the tangential and normal components of a. a
> Sketch the graph of the function. f (x, y) = x2
> The body mass index is defined in Exercise 39. Draw the level curve of this function corresponding to someone who is 200 cm tall and weighs 80 kg. Find the weights and heights of two other people with that same level curve. Exercise 39: The body mass i
> In Example 2 we considered the function W = f (T, v), where W is the wind-chill index, T is the actual temperature, and v is the wind speed. A numerical representation is given in Table 1 on page 889. Table: (a). What is the value of f (-15, 40)? What
> The body mass index (BMI) of a person is defined by where m is the person’s mass (in kilograms) and h is the height (in meters). Draw the level curves B (m, h) = 18.5, B (m, h) = 25, B (m, h) = 30, and B (m, h) = 40. A rough guideline i
> Find and sketch the domain of the function. f(x, y, 2) = /4 – x² + v9 – y² + VT –
> A manufacturer has modeled its yearly production function P (the monetary 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 inves
> The wind-chill index W discussed in Example 2 has been modeled by the following function: Check to see how closely this model agrees with the values in Table 1 for a few values of T and v. W(T, v) 13.12 + 0.6215T – 11.37v0.16 + 0.3965TV0.16
> What is the connection between vector functions and space curves?
> What is a vector function? How do you find its derivative and its integral?
> If z = 5x2 + y2 and (x, y) changes from (1, 2) to (1.05, 2.1), compare the values of ∆z and dz.
> Find the differential of the function. L = xze-y 2-z2
> Find and sketch the domain of the function. In(2 — х) g(x, y) = 1- x² – y?
> Find the differential of the function. T = 1+ uvw
> Find the differential of the function. m = p5q3.
> Find the differential of the function. u = Jx? + 3y2 %3D
> Find the differential of the function. z = e-2x cos 2 πt.
> Verify the linear approximation at (0, 0). ex cos (xy) ≈ x + 1
> Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point. f (x, y) = 4 arctan (xy), (1, 1)
> Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point. f(x, y) =+Y, 1+ x (1, 3)
> Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point. fu, y) — Vху, (1,4)
> (a). If a particle moves along a straight line, what can you say about its acceleration vector? (b). If a particle moves with constant speed along a curve, what can you say about its acceleration vector?
> Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point. f (x, y) = 1 + x ln (xy – 5), (2, 3)
> The diffusion equation where D is a positive constant, describes the diffusion of heat through a solid, or the concentration of a pollutant at time t at a distance x from the source of the pollution, or the invasion of alien species into a new habitat.
> If u = e4+4; ,++4,, where af + až + . + a; = 1, show that Fu ax? + = u ax? ax?
> If f and t are twice differentiable functions of a single variable, show that the function u (x, t) = f (x + at) + t (x – at) is a solution of the wave equation given in Exercise 78. Exercise 78: Show that each of the following functions is a solution
> A particle has position function r(t). If r'(t) = c × r(t), where c is a constant vector, describe the path of the particle.
> Find the indicated partial derivative(s). u = x*y*z*; ax dy² az³ .3
> Find the indicated partial derivative(s). a³w aw у+ 22 дг ду дх" дх? ду
> Find the indicated partial derivative(s). V = In(r + s² + t); ar as at
> Find the indicated partial derivative(s). W = Ju + v?; 2 au dv
> Find the indicated partial derivative(s). g (r, s, t) = er sin (st); grst
> Find the indicated partial derivative(s). f (x, y) = x4y2 - x3y; fxxx, fxyx
> Determine the signs of the partial derivatives for the function f whose graph is shown. (a). fx (21, 2) (b). fy (21, 2)
> Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx. u = x4y3 - y4
> Find all the second partial derivatives. w = V1 + uv? ,2
> Find all the second partial derivatives. v = sin (s2 - t2)
> Find all the second partial derivatives. y 2х + Зу
> Find all the second partial derivatives. f (x, y) = ln (ax + by)
> Determine the signs of the partial derivatives for the function f whose graph is shown. (a). fx (1, 2) (b). fy (1, 2)
> Use the definition of partial derivatives as limits (4) to find fx (x, y) and fy (x, y). f(x, y) = x + y?
> Use the definition of partial derivatives as limits (4) to find fx (x, y) and fy (x, y). f (x, y) = xy2 - x3y
> Find the indicated partial derivative. f (x, y, z) = xyz; fz (e, 1, 0)
> Find the indicated partial derivative. 1- f(x, y, z) = In Vx? + y² + z? 1+ Jx? + y? + z² f,(1, 2, 2)
> Find the indicated partial derivative. f (x, y) = y sin-1 (xy); fy (1, 1 2 )
> Find the indicated partial derivative. R (s, t) = tes/t; Rt (0, 1)
> Find the first partial derivatives of the function. u = sin (x1 + 2x2 + ∙ ∙ ∙ + nxn)
> Find the first partial derivatives of the function. Vx} + x} + .… + x금 u =
> Find the first partial derivatives of the function. ах + By? yz + 8t? Ф(х, у, г, 1) -
> A ball with mass 0.8 kg is thrown southward into the air with a speed of 30 m/s at an angle of 30° to the ground. A west wind applies a steady force of 4 N to the ball in an easterly direction. Where does the ball land and with what speed?
> Find the first partial derivatives of the function. u = xy/z
> Find the first partial derivatives of the function. p = Vt4 + u? cos v
> Find the first partial derivatives of the function. w = y tan (x + 2z)
> Find the first partial derivatives of the function. w = ln (x + 2y + 3z)
> Find the first partial derivatives of the function. f (x, y, z) = xy2e-xz
> Find the first partial derivatives of the function. f (x, y, z) = x3yz2 + 2yz
> Find the first partial derivatives of the function. F(a, ß) = (" V3 + 1 dt [ :
> Find the first partial derivatives of the function. F(x, y) = [" cos(e') dt
> Find the first partial derivatives of the function. f (x, y) = xy
> Find the first partial derivatives of the function. R (p, q) = tan-1(pq2)
> A ball is thrown eastward into the air from the origin (in the direction of the positive x-axis). The initial velocity is 50 i + 80 k, with speed measured in feet per second. The spin of the ball results in a southward acceleration of 4 ft/s2, so the acc
> Find the first partial derivatives of the function. g (u, v) = (u2v - v3)5
> Find the first partial derivatives of the function. e" w и + u + v? 2
> Find the first partial derivatives of the function. ах + by f(x, y) = сх + dy Cx
> Use Clairaut’s Theorem to show that if the third-order partial derivatives of f are continuous, then fxyy = fyxy = fyyx
> Find the first partial derivatives of the function. f(x, y) (x + y)?
> Find the first partial derivatives of the function. f(x, y) = y
> Find the first partial derivatives of the function. z = x sin (xy)
> Find the first partial derivatives of the function. z = ln (x + t2)
> Show that a projectile reaches three-quarters of its maximum height in half the time needed to reach its maximum height.
> The table gives coordinates of a particle moving through space along a smooth curve. (a). Find the average velocities over the time intervals [0, 1], [0.5, 1], [1, 2], and [1, 1.5]. (b). Estimate the velocity and speed of the particle at t = 1. t y
> The kinetic energy of a body with mass m and velocity v is K = 1 2 mv2. Show that aK dK = K am dv?
> Find the first partial derivatives of the function. f(x, t) = /3x + 4t
> Find the first partial derivatives of the function. f (x, t) = t2e-x
> Find the first partial derivatives of the function. f (x, y) = x2y - 3y4
> Find the first partial derivatives of the function. f (x, y) = x4 + 5xy3
> The wind-chill index is modeled by the function W = 13.12 + 0.6215T - 11.37v0.16 + 0.3965Tv 0.16 where T is the temperature (°C) and v is the wind speed (km/h). When T − 215°C and v − 30 km/h, by how much would you expect the apparent temperature W to
> For the ideal gas of Exercise 88, show that Exercise 88: The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. Show that ,aP av = mR aT ƏT aP av əT -1 av aT əP
> The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. Show that aP av əT -1 av aT əP
> A medieval city has the shape of a square and is protected by walls with length 500 m and height 15 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m. Your plan is to set fire to the city by catapulting heated
> Find the limit, if it exists, or show that the limit does not exist. x²y + xy² lim x² – y² (x, y)(2, -1) x – y?
> Find the limit, if it exists, or show that the limit does not exist. (x²y³ – 4y²) (х, у)— (3, 2)
> At the beginning of this section we considered the function and guessed on the basis of numerical evidence that f (x, y) → 1 as (x, y) → (0, 0). Use polar coordinates to confirm the value of the limit. Then graph the f
> Determine the set of points at which the function is continuous. F(x, y) 1+ x² + y? 1– x² – y2
