Find the directional derivative of f(x, y, z) = xy + yz + zx at P(1, -1, 3) in the direction of Q)2, 4, 5).

Find the Limit, if it exists, or show that the limit does not exist. lim x2y + xy2 / x2 - y2 (x,y)-->(2, -1)

Consider lim_(x, y)→(0, 0). x²y / x^4 + y². (a) Determine (if possible) the limit along any line of the form y = ax.

Find the Limit, if it exists, or show that the limit does not exist. lim xy2 cos y / x2 + y2 (x,y)-->(0,0)

Find the limit, if it exists, or show that the limit does not exist. lim (x,y,z) to (0,0,0) xy+yz/x^2+y^2+z^2

Find the limit, if it exists, or show that the limit does not exist. $\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y+y z^{2}+x z^{2}}{x^{2}+y^{2}+z^{4}}$

Consider the function f(x, y) = 4xy / x²+y², (x, y) ≠ (0, 0) {0, (x, y) = (0, 0) and the unit vector u = 1/√2 (i + j). Does the directional derivative of f at P(0, 0) in the direction of u exist? If f(0, 0) were defined as 2 instead of 0, would the directional derivative exist?

Use the level curves in the figure to predict the location of the critical points of $f(x, y) = 3x - x^3 - 2y^2 + y^4$ 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 predictions.

Pd has an anomalous electron configuration. write the observed electron configuration of pd.

Find the first partial derivatives. S(u, v, w) = u arctan (u√w)

Verify the linear approximation at (0, 0).

$$√y+cos^2x ≈1 + 1/2y$$

The length and width of a rectangle are measured as 30 cm and 24 cm, respectively, with an error in measurement of at most 0.1cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

What is a function of three variables? How can you visualize such a function?

Find the gradient of f. f(x,y,z)=y^2*e^xyz, P(0, 1, -1), u=<3/13, 4/13, 12/13>