Questions from Calculus

Q: Verify that the conclusion of Clairaut’s Theorem holds, that is,

Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx. u = cos (x2y)

Q: Find the indicated partial derivative(s). f (x

Find the indicated partial derivative(s). f (x, y) = sin (2x + 5y); fyxy

Q: Use the table of values of f (x, y)

Use the table of values of f (x, y) to estimate the values of fx (3, 2), fx (3, 2.2), and fxy (3, 2).

Q: Level curves are shown for a function f. Determine whether the

Level curves are shown for a function f. Determine whether the following partial derivatives are positive or negative at the point P. (a). fx (b). fy (c). fxx (d). fxy (e). fyy

Q: Show that each of the following functions is a solution of the

Show that each of the following functions is a solution of the wave equation utt = a2uxx. (a). u = sin (kx) sin (akt) (b). u = t/ (a2t2 - x2) (c). u = (x – at)6 + (x + at)6 (d). u = sin (x – at) + ln...

Q: The temperature at a point (x, y) on a

The temperature at a point (x, y) on a flat metal plate is given by T (x, y) = 60/ (1 + x2 + y2), where T is measured in 8C and x, y in meters. Find the rate of change of temperature with respect to d...

Q: / (a). Evaluate t (1, 2

(a). Evaluate t (1, 2, 3). (b). Find and describe the domain of t.

Q: Cobb and Douglas used the equation P (L, K)

Cobb and Douglas used the equation P (L, K) = 1.01L0.75K0.25 to model the American economy from 1899 to 1922, where L is the amount of labor and K is the amount of capital. (See Example 14.1.3.) (a)....

Q: At the beginning of this section we discussed the function I =

At the beginning of this section we discussed the function I = f (T, H), where I is the heat index, T is the temperature, and H is the relative humidity. Use Table 1 to estimate fT (92, 60) and fH (92...

Q: Determine the signs of the partial derivatives for the function f whose

Determine the signs of the partial derivatives for the function f whose graph is shown. (a). fxx (21, 2) (b). fyy (21, 2)