Use the binomial series to expand the function as a power series. State the radius of convergence.

$$(1-x)^{2 / 3}$$

Use a binomial series to represent each function as a Maclaurin series, and find the interval of convergence. $f(x)=\frac{1}{(1+x)^{3 / 4}}$

Use the binomial series to expand the function as a power series. State the radius of convergence. \

$$\dfrac{x}{\sqrt{4+x^2}}$$

Use the binomial series to expand the function as a power series if

$$f(x) = \frac{5}{(4 + x)^3}.$$

Find the shortest distance from the point (2, 0, -3) to the plane x + y + z = 1.

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 of those points. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions.
$f(x, y)=4+x^3+y^3-3 x y$

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 of those points. Explain your reasoning. Then use the Second Derivatives Test to confirm your predictions.

$$f(x, y)=4+x^3+y^3-3 x y$$

Find the directional derivative of f at the given point in the indicated direction.

$$f(x, y,z)=x^2y+x√1+z,$$

(1, 2, 3). in the direction of v = 2i + j - 2k

Find the directional derivative of $f$ at the given point in the direction indicated by the angle $\theta$.

$$f(x, y)=y^x, \quad(1,2), \quad \theta=\pi / 2$$

Use implicit differentiation to find the derivative of $y$ with respect to $x$ at the given point.

$$\ln (x^2-3 y)=x-y-1 ;(2,1)$$

A function f and a point P are given. Let $\theta$ correspond to the direction of the directional derivative. Write the directional derivative at $P$ as a function of $\theta$; call this function $g$. $f(x, y)=10-2 x^2-3 y^2 ; P(3,2)$

Find the gradient of the function and the maximum value of the directional derivative at the given point. Function: f(x, y) = x² + 2xy Point: (1, 0)

Find the gradient of the function and the maximum value of the directional derivative at the given point.

Function: $w=x y^2 z^2$ Point: $(2,1,1)$

Given $\phi=x y+y z+x z$, find gradient $\phi$ at point (1,2,3) and the directional derivative of $\phi$ at the same point in the direction toward the point (3,4,4).