Homework 5¶

Due: February 20, 2018, 8 a.m.¶

Please give a complete, justified solution to each question below. A single-term answer without explanation will receive no credit.

Please complete each question on its own sheet of paper (or more if necessary), and upload to Gradsescope.

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\dydx}{\frac{dy}{dx}} \newcommand{\proj}{\textrm{proj}} % For boldface vectors: \renewcommand{\vec}[1]{\mathbf{#1}} $$

1. A friend was asked to find the equation of the tangent plane to the surface $z = x^3 - \arctan (y-1) $ at the point $(x, y) = (2, 1)$. The friend's answer was $$z = 3x^2(x-2) - \frac{1}{2-2y+y^2}(y-1)+8.$$

At a glance, without doing any computation, how do you know that this is incorrect? What mistake did the friend make?
Answer the question correctly. \end{enumerate}

2. Estimate the quantity $$ \frac{8.01}{\sqrt{1.99\times2.01}} $$ "by hand" using a technique similar to that done in class.

3. Let $$ f(x,y) = \ln (y-x^2) $$

Find $L(x,y)$, the linearization of $f$ around the point $(0,1)$ and show that $L$ has the same first-order partial derivatives as $f$ at that point.
Find constants $a_1,\ldots,a_6$ such that the function $$M(x,y) = a_1 + a_2x +a_3y +a_4x^2 +a_5 xy +a_6 y^2$$ has the same
- value,
- first-order partial derivatives, and
- second-order partial derivatives as $f$ at $(0,1)$.

4. Find the value of $\partial x/\partial z$ at the point $(1, -1, -3)$ if the equation $$ xz + y \ln(x) - x^2 + 4 = 0$$ defines $x$ as a function of the two independent variables $y$ and $z$.

5. If $f(u, v, w)$ is differentiable and $u = x-y, v= y-z$, and $w = z-x$, show that $$ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z} = 0. $$

6. A function $f(x, y)$ is called homogeneous of degree $p$ if $$ f(tx, ty) = t^p f(x, y)$$ for all $t$. Show that any differentiable, homogeneous function of degree $p$ satisfies Euler's Theorem: $$ x f_x(x, y) + y f_y(x, y) = p f(x, y). $$ Hint: Define $g(t) = f(tx, ty)$ and compute $g'(1)$.