Gilbert Strang; Edwin “Jed” Herman

Review Exercises

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

394.

The domain of $f (x, y) = x^{3} {sin}^{−1} (y)$ is $x =$ all real numbers, and $− π \leq y \leq π .$

395.

If the function $f (x, y)$ and all its second derivatives are continuous everywhere, then $f_{x y} = f_{y x} .$

396.

The linear approximation to the function of $f (x, y) = 5 x^{2} + x tan (y)$ at $(2, π)$ is given by $L (x, y) = 22 + 21 (x - 2) + (y - π) .$

397.

$(\frac{3}{4}, \frac{9}{16})$ is a critical point of $g (x, y) = 4 x^{3} - 2 x^{2} y + y^{2} - 2 .$

For the following exercises, sketch the function in one graph and, in a second, sketch several level curves.

398.

$f (x, y) = e^{− (x^{2} + 2 y^{2})} .$

399.

$f (x, y) = x + 4 y^{2} .$

For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it.

400.

$lim_{(x, y) \to (1, 1)} \frac{4 x y}{x - 2 y^{2}}$

401.

$lim_{(x, y) \to (0, 0)} \frac{4 x y}{x - 2 y^{2}}$

For the following exercises, find the largest region of continuity for the function.

402.

$f (x, y) = x^{3} {sin}^{−1} (y)$

403.

$g (x, y) = ln (4 - x^{2} - y^{2})$

For the following exercises, find all first derivatives, full or partial, as appropriate.

404.

$f (x, y) = \sqrt{x^{2} - y^{2}}$

405.

$u (x, y) = x^{4} - 3 x y + 1, x = 2 t, y = t^{3}$

For the following exercises, find all second partial derivatives.

406.

$g (t, x) = 3 t^{2} - sin (x + t)$

407.

$h (x, y, z) = \frac{x^{3} e^{2 y}}{z}$

For the following exercises, find the equation of the tangent plane to the specified surface at the given point.

408.

$z = x^{3} - 2 y^{2} + y - 1$ at point $(1, 1, −1)$

409.

$z = e^{x} + \frac{2}{y}$ at point $(0, 1, 3)$

410.

Approximate $f (x, y) = e^{x^{2}} + \sqrt{y}$ at $(0.1, 9.1) .$ Write down your linear approximation function $L (x, y) .$ How accurate is the approximation to the exact answer, rounded to four digits?

411.

Find the differential $d z$ of $h (x, y) = 4 x^{2} + 2 x y - 3 y$ and approximate $Δ z$ at the point $(1, −2) .$ Let $Δ x = 0.1$ and $Δ y = 0.01 .$

412.

Find the directional derivative of $f (x, y) = x^{2} + 6 x y - y^{2}$ in the direction $v = i + 4 j .$

413.

Find the maximal directional derivative magnitude and direction for the function $f (x, y) = x^{3} + 2 x y - cos (π y)$ at point $(3, 0) .$

For the following exercises, find the gradient.

414.

$c (x, t) = e {(t - x)}^{2} + 3 cos (t)$

415.

$f (x, y) = \frac{\sqrt{x} + y^{2}}{x y}$

For the following exercises, find and classify the critical points.

416.

$z = x^{3} - x y + y^{2} - 1$

For the following exercises, use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints.

417.

$f (x, y) = x^{2} y, x^{2} + y^{2} = 4$

418.

$f (x, y) = x^{2} - y^{2}, x + 6 y = 4$

419.

A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of $5 %$ in height and $2 %$ in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height $6$ cm and radius $2$ cm.

420.

A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of $2$ ft/sec and $3$ ft/sec, respectively. Find the rate at which the liquid level is rising when the length is $14$ ft, the width is $10$ ft, and the height is $4$ ft.