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Exercises 2.8 Exercises

1.

Compute
\begin{equation*} \lim_{\langle x,y \rangle \to \langle 1,-1 \rangle} \langle x^2+y, e^{xy} \rangle. \end{equation*}

2.

Define \(f \colon \R^2 \to \R\) by
\begin{equation*} f(x,y) = \begin{cases} \dfrac{x^2y^2}{x^2+y^2}, & \langle x,y \rangle \ne \langle 0,0 \rangle, \\ 0, & \langle x,y \rangle = \langle 0,0 \rangle. \end{cases} \end{equation*}
Show that \(f\) is continuous at \(\langle 0,0 \rangle\text{.}\)

3.

Show that
\begin{equation*} \lim_{\langle x,y \rangle \to \langle 0,0 \rangle} \frac{x^2-y^2}{x^2+y^2} \end{equation*}
does not exist.

4.

Let \(F \colon \R^2 \to \R^2\) be defined by
\begin{equation*} F(x,y)=\langle x^2+3y, xy-y^2 \rangle. \end{equation*}
  1. Compute the Jacobian matrix of \(F\) at a general point \(\langle x,y \rangle\text{.}\)
  2. Compute \(DF(\langle 1,2 \rangle)\text{.}\)

5.

Let \(f(x,y)=x^2y+y^3\text{.}\)
  1. Find the tangent plane to the graph of \(f\) at the point above \(\langle 1,-1 \rangle\text{.}\)
  2. Use the linear approximation at \(\langle 1,-1 \rangle\) to estimate \(f(1.02,-0.97)\text{.}\)

6.

Let \(f \colon \R^2 \to \R^2\) and \(g \colon \R^2 \to \R\) be given by
\begin{equation*} f(x,y)=\langle x^2-y, x+y^2 \rangle, \qquad g(u,v)=u^2+uv. \end{equation*}
Compute \(D(g \circ f)(\langle 1,1 \rangle)\text{.}\)

7.

Let \(f(x,y,z)=xe^{yz}\text{.}\)
  1. Compute \(\nabla f(1,0,2)\text{.}\)
  2. Compute the directional derivative of \(f\) at \(\langle 1,0,2 \rangle\) in the direction \(\vu=\dfrac{1}{3}\langle 2,-1,2 \rangle\text{.}\)

8.

Let \(f(x,y)=x^2+2xy+3y^2-4x\text{.}\)
  1. Compute \(\nabla f(1,-1)\text{.}\)
  2. Find the unit vector at \(\langle 1,-1 \rangle\) in which \(f\) increases most rapidly.
  3. Find the maximum directional derivative of \(f\) at \(\langle 1,-1 \rangle\text{.}\)

9.

Define \(h \colon \R^2 \to \R\) by
\begin{equation*} h(x,y) = \begin{cases} \dfrac{x^2y}{x^2+y^2}, & \langle x,y \rangle \ne \langle 0,0 \rangle, \\ 0, & \langle x,y \rangle = \langle 0,0 \rangle. \end{cases} \end{equation*}
Show that \(h\) is differentiable at \(\langle 0,0 \rangle\text{,}\) and compute \(Dh(\langle 0,0 \rangle)\text{.}\)

10.

Find the second-order Taylor polynomial of \(f(x,y)=e^x \sin y\) at \(\langle 0,0 \rangle\text{.}\)

11.

Find and classify all stationary points of \(f(x,y)=x^3-3x+y^2\text{.}\)

12.

Consider \(f(x,y)=x^4-y^4\text{.}\)
  1. Show that \(\langle 0,0 \rangle\) is a stationary point.
  2. Show that the second derivative test is inconclusive at \(\langle 0,0 \rangle\text{.}\)
  3. Determine whether \(\langle 0,0 \rangle\) is a local maximum, a local minimum, or a saddle point.

13.

Use Lagrange multipliers to find the maximum and minimum values of \(f(x,y)=xy\) subject to the constraint \(x^2+4y^2=4\text{.}\)

14.

Find the point on the line \(2x-y=3\) that is closest to the origin.

15.

Let \(F \colon \R^2 \to \R^2\) be defined by
\begin{equation*} F(x,y)=\langle e^x\cos y, e^x\sin y \rangle. \end{equation*}
  1. Show that \(F\) has a \(C^1\) local inverse near \(\langle 0,0 \rangle\text{.}\)
  2. Compute \(D(F^{-1})(\langle 1,0 \rangle)\text{.}\)

16.

Let
\begin{equation*} F(x,y)=x^2+xy+y^3-3. \end{equation*}
Show that near \(x=1\text{,}\) the equation \(F(x,y)=0\) determines \(y\) as a \(C^1\) function \(y=g(x)\) with \(g(1)=1\text{.}\) Compute \(g'(1)\text{.}\)
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