Question: At the beginning of this section we

> 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)

> Determine the set of points at which the function is continuous. F(x, y) 1+ x² + y? 1– x² – y2

> Determine the set of points at which the function is continuous. F(x, y) = cos /1 + x – y

> Find h (x, y) = g (f (x, y)) and the set of points at which h is continuous. g(t) = t² + JT, ƒ(x, y) = 2x + 3y – 6

> A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle 508 above the horizontal. Is it a home run? (In other words, does the ba

> Use a computer graph of the function to explain why the limit does not exist. lim (x, y) (0, 0) 2.x? + 3xy + 4y² 3x? + 5y?

> Find the limit, if it exists, or show that the limit does not exist. x?y?z? lim (3, x 2(0, 0, 0) x? + y? + z?

> Find the limit, if it exists, or show that the limit does not exist. ху + yz lim (x, y, z-(0,0,0) x² +: y? + z?

> Find the limit, if it exists, or show that the limit does not exist. lim (x, y, 2)-(m, 0, 1/3) e tan(xz)

> Find the limit, if it exists, or show that the limit does not exist. x² + y? lim (x, y)-(0, 0) x² + y² + I – 1

> Find the limit, if it exists, or show that the limit does not exist. ху lim (x, y)-(0, 0) x* + y4

> Find the limit, if it exists, or show that the limit does not exist. xy cos y lim + y* .2 (x, y) (0, 0) x

> 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

> A rifle is fired with angle of elevation 36°. What is the muzzle speed if the maximum height of the bullet is 1600 ft?

> 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. B D E II III IV VI yA (OKO C O O O O).O O O) O O KO C O OD O x- y 1+ x² + y²

> Draw a contour map of the function showing several level curves. f(x, y) = xy

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f.

> 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 E II III IV VI yA (OKO C O O O O).O O O

> 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 E II III IV VI yA (OKO C O O O O).O O O) O O KO C O OD O

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f. y.

> 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

> 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

> 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

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f. 14, * 13 12

> A company makes three sizes of cardboard boxes: small, medium, and large. It costs $2.50 to make a small box, $4.00 for a medium box, and $4.50 for a large box. Fixed costs are $8000. (a). Express the cost of making x small boxes, y medium boxes, and z l

> A thin metal plate, located in the xy-plane, has temperature T (x, y) at the point (x, y). Sketch some level curves (isothermals) if the temperature function is given by 100 T(x, y) = 1+ x² + 2y²

> Make a rough sketch of a contour map for the function whose graph is shown.

> Locate the points A and B on the map of Lonesome Mountain (Figure 12). How would you describe the terrain near A? Near B? Figure 12: 90 90 180 40 60 60- -60 70 50 60 80 70 60 70 80 70 30- -30 -100 80 0- -- 70 80 30- 60 -30 50 40 60- -60 30 20 10 90

> Draw a contour map of the function showing several level curves. f (x, y) = y/ (x2 + y2)

> Draw a contour map of the function showing several level curves. f(x, y) = x² + y?

> Draw a contour map of the function showing several level curves. f (x, y) = y - arctan x

> Draw a contour map of the function showing several level curves. f(x, y) = ye²

> Draw a contour map of the function showing several level curves. f(x, y) = In(x² + 4y³)

> Draw a contour map of the function showing several level curves. f(x, y) = VF + y

> A ball is thrown at an angle of 45° to the ground. If the ball lands 90 m away, what was the initial speed of the ball?

> A rocket burning its onboard fuel while moving through space has velocity v(t) and mass m(t) at time t. If the exhaust gases escape with velocity ve relative to the rocket, it can be deduced from Newton’s Second Law of Motion that (b

> Draw a contour map of the function showing several level curves. f(x, y) = x² – y?

> A contour map of a function is shown. Use it to make a rough sketch of the graph of f. 5 3 2 15 3 2

> Find the tangential and normal components of the acceleration vector at the given point. -k, (1, 1, 1)

> Find the tangential and normal components of the acceleration vector. r(t) = t i + 2et j + e2t k

> Find the tangential and normal components of the acceleration vector. r(t) = cos t i + sin t j + t k

> Find the tangential and normal components of the acceleration vector. r(t) = 2t2 i + ( 2 3 t3 - 2t) j

> Find the tangential and normal components of the acceleration vector. r(t) = (t2 + 1) i + t3 j, t > 0

> Rework Exercise 23 if the projectile is fired from a position 100 m above the ground. Exercise 23: A projectile is fired with an initial speed of 200 m/s and angle of elevation 60°. Find (a) the range of the projectile, (b) the maximum height reached,

> Determine the set of points at which the function is continuous. f(x, y, z) = Vy – x² In z

> Find the limit, if it exists, or show that the limit does not exist. x³ - y3 lim (x, y)-0, 0) x? + xy + y? .2 2

> Find the limit, if it exists, or show that the limit does not exist. ху — у lim (x, 3)(1, 0) (x – 1)² + y?

> Determine the set of points at which the function is continuous. G(x, y) = Vx + /T- x² – y²

> Sketch the graph of the function. f (x, y) = cos y

> Sketch the graph of the function. f(x, y) = V4 – 4x² – y²