4.2.E: Best Affine Approximations (Exercises)

Exercise \(\PageIndex{1}\)

Find the best affine approximation for each of the following functions at the specified point \(\mathbf{c}\).

(a) \(f(x, y)=\left(x^{2}+y^{2}, 3 x y\right), \mathbf{c}=(1,2) \)

(b) \(g(x, y, z)=(\sin (x+y+z), x y \cos (z)), \mathbf{c}=\left(0, \frac{\pi}{4}, \frac{\pi}{4}\right)\)

(c) \(h(s, t)=\left(3 s^{2}+t, s-t, 4 s t^{2}, 4 t-s\right), \mathbf{c}=(-1,3)\)

Answer

(a) \(A(x, y)=\left[\begin{array}{ll}
2 & 4 \\
6 & 3
\end{array}\right]\left[\begin{array}{l}
x-1 \\
y-2
\end{array}\right]+\left[\begin{array}{l}
5 \\
6
\end{array}\right]=\left[\begin{array}{l}
2 x+4 y-5 \\
6 x+3 y-6
\end{array}\right]\)

(c) \(A(s, t)=\left[\begin{array}{rr}
-6 & 1 \\
1 & -1 \\
36 & -24 \\
-1 & 4
\end{array}\right]\left[\begin{array}{l}
s+1 \\
t-3
\end{array}\right]+\left[\begin{array}{c}
6 \\
-4 \\
-36 \\
13
\end{array}\right]=\left[\begin{array}{c}
-6 s+t-3 \\
s-t \\
36 s-24 t+72 \\
-2+4 t
\end{array}\right]\)

Exercise \(\PageIndex{2}\)

Each of the following functions parametrizes a surface \(S\) in \(\mathbb{R}^3\). In each case, find parametric equations for the tangent plane \(P\) passing through the point \(f\left(s_{0}, t_{0}\right)\). Plot \(S\) and \(P\) together.

(a) \(f(s, t)=(t \cos (s), t \sin (s), t),\left(s_{0}, t_{0}\right)=\left(\frac{\pi}{2}, 2\right)\)

(b) \(f(s, t)=\left(t^{2} \cos (s), t^{2}, t^{2} \sin (s)\right),\left(s_{0}, t_{0}\right)=(0,1)\)

(c) \(f(s, t)=(\cos (s) \sin (t), \sin (s) \sin (t), \cos (t)),\left(s_{0}, t_{0}\right)=\left(\frac{\pi}{2}, \frac{\pi}{4}\right)\)

(d) \(f(s, t)=(3 \cos (s) \sin (t), \sin (s) \sin (t), 2 \cos (t)),\left(s_{0}, t_{0}\right)=\left(\frac{\pi}{4}, \frac{\pi}{4}\right)\)

(e) \(f(s, t)=((4+2 \cos (t)) \cos (s),(4+2 \cos (t)) \sin (s), 2 \sin (t)),\left(s_{0}, t_{0}\right)=\left(\frac{3 \pi}{4}, \frac{\pi}{4}\right)\)

Answer

(a) \(x=-2 s+\pi, y=t, z=t\)

(c) \(x=-\frac{1}{\sqrt{2}}\left(s-\frac{\pi}{2}\right)\)

(e) \(\begin{aligned}
& x=-(2 \sqrt{2}+1)\left(s-\frac{3 \pi}{4}\right)+\left(t-\frac{\pi}{4}\right)-2 \sqrt{2}-1\\
&y=-(2 \sqrt{2}+1)\left(s-\frac{3 \pi}{4}\right)-\left(t-\frac{\pi}{4}\right)+2 \sqrt{2}+1\\
&y=\sqrt{2}\left(t-\frac{\pi}{4}\right)+\sqrt{2}
\end{aligned}\)

Exercise \(\PageIndex{3}\)

Let \(S\) be the graph of a function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\). Define the function \(F: \mathbb{R}^{2} \rightarrow \mathbb{R}^3 \) by \(F(s, t)=(s, t, f(s, t))\). We may find an equation for the plane tangent to \(S\) at \(\left(s_{0}, t_{0}, f\left(s_{0}, t_{0}\right)\right)\) using either the techniques of Section 3.3 (looking at \(S\) as the graph of \(f\)) or the techniques of this section (looking at \(S\) as a surface parametrized by \(F\)). Verify that these two approaches yield equations for the same plane, both in the special case when \(f(s,t) = s^2 + t^2 \) and \(\left(s_{0}, t_{0}\right)=(1,2)\), and in the general case.

Exercise \(\PageIndex{4}\)

Use the chain rule to find the derivative of \(f \circ g\) at the point \(\mathbf{c}\) for each of the following.

(a) \(f(x, y)=\left(x^{2} y, x-y\right), g(s, t)=\left(3 s t, s^{2}-4 t\right), \mathbf{c}=(1,-2)\)

(b) \(f(x, y, z)=(4 x y, 3 x z), g(s, t)=\left(s t^{2}-4 t, s^{2}, \frac{4}{s t}\right), \mathbf{c}=(-2,3)\)

(c) \(f(x, y)=\left(3 x+4 y, 2 x^{2} y, x-y\right), g(s, t, u)=\left(4 s-3 t+u, 5 s t^{2}\right), \mathbf{c}=(1,-2,3)\)

Answer

(a) \(D(f \circ g)(1,-2)=\left[\begin{array}{cc}
720 & -468 \\
-8 & 15
\end{array}\right]\)

(c) \(D(f \circ g)(1,-2,3)=\left[\begin{array}{ccc}
92 & -89 & 3 \\
10920 & -9880 & 1040 \\
-16 & 17 & 1
\end{array}\right]\)

Exercise \(\PageIndex{5}\)

Suppose

\[ \begin{aligned}
&x=f(u, v), \\
&y=g(u, v),
\end{aligned} \]

and

\[ \begin{aligned}
&u=h(s, t), \\
&v=k(s, t).
\end{aligned} \]

(a) Show that

\[ \frac{\partial x}{\partial s}=\frac{\partial x}{\partial u} \frac{\partial u}{\partial s}+\frac{\partial x}{\partial v} \frac{\partial v}{\partial s} \nonumber \]

and

\[ \frac{\partial x}{\partial t}=\frac{\partial x}{\partial u} \frac{\partial u}{\partial t}+\frac{\partial x}{\partial v} \frac{\partial v}{\partial t} . \nonumber \]

(b) Find similar expressions for \(\frac{\partial y}{\partial s}\) and \(\frac{\partial y}{\partial t}\).

Exercise \(\PageIndex{6}\)

Use your results in Exercise 5 to find \(\frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}\), and \(\frac{\partial y}{\partial t}\) when

\[ \begin{aligned}
&x=u^{2} v , \\
&y=3 u-v,
\end{aligned} \]

and

\[ \begin{aligned}
&u=4 t^{2}-s^{2} , \\
&v=\frac{4 t}{s} .
\end{aligned} \]

Answer

\(\begin{aligned}
& \frac{\partial x}{\partial s}=(2 u v)(-2 s)+\left(u^{2}\right)\left(-\frac{4 t}{s^{2}}\right)\\
&\frac{\partial x}{\partial t}=(2 u v)(8 t)+\left(u^{2}\right)\left(\frac{4}{s}\right)\\
&\frac{\partial y}{\partial s}=(3)(-2 s)+(-1)\left(-\frac{4 t}{s^{2}}\right)\\
&\frac{\partial y}{\partial t}=(3)(8 t)+(-1)\left(\frac{4}{s}\right)
\end{aligned}\)

Exercise \(\PageIndex{7}\)

Suppose \(T\) is a function of \(x\) and \(y\) where

\[ \begin{aligned}
&x=r \cos (\theta) , \\
&y=r \sin (\theta) .
\end{aligned} \]

Show that

\[ \frac{\partial T}{\partial r}=\frac{\partial T}{\partial x} \cos (\theta)+\frac{\partial T}{\partial y} \sin (\theta) \nonumber \]

and

\[ \frac{\partial T}{\partial \theta}=-\frac{\partial T}{\partial x} r \sin (\theta)+\frac{\partial T}{\partial y} r \cos (\theta) . \nonumber \]

Exercise \(\PageIndex{8}\)

Suppose the temperature at a point \((x,y)\) in the plane is given by

\[ T=100-\frac{20}{\sqrt{1+x^{2}+y^{2}}} . \nonumber \]

(a) If \((r , \theta )\) represents the polar coordinates of \((x,y)\), use Exercise 7 to find \(\frac{\partial T}{\partial r}\) and \(\frac{\partial T}{\partial \theta}\) when \(r=4\) and \(\theta = \frac{\pi}{6}\).

(b) Show that \(\frac{\partial T}{\partial \theta}=0\) for all values of \(r\) and \(\theta\). Can you explain this result geometrically?

Answer

(a) \(\left.\frac{\partial T}{\partial r}\right|_{r=4, \theta=\frac{\pi}{6}}=\frac{80}{17 \sqrt{17}},\left.\frac{\partial T}{\partial \theta}\right|_{r=4, \theta=\frac{\pi}{6}}=0\)

(b) The level curves of \(T\) are circles.

Exercise \(\PageIndex{9}\)

Let \(T\) be the torus parametrized by

\[ \begin{aligned}
&x=(4+2 \cos (t)) \cos (s) , \\
&y=(4+2 \cos (t)) \sin (s) , \\
&z=2 \sin (t) ,
\end{aligned} \]

for \(0 \leq s \leq 2 \pi\) and \(0 \leq t \leq 2 \pi\).

(a) If \(U\) is a function of \(x\), \(y\), and \(z\), find general expressions for \(\frac{\partial U}{\partial s}\) and \(\frac{\partial U}{\partial t}\).

(b) Suppose

\[ U=80-40 e^{-\frac{1}{20}\left(x^{2}+y^{2}+z^{2}\right)} \nonumber \]

gives the temperature at a point \((x,y,z)\) on \(T\). Find expressions for \(\frac{\partial U}{\partial s}\) and \(\frac{\partial U}{\partial t}\) in this case. What is the geometrical interpretation of these quantities?

(c) Evaluate \(\frac{\partial U}{\partial s}\) and \(\frac{\partial U}{\partial t}\) in the particular case \(s=\frac{\pi}{4}\) and \(t=\frac{\pi}{4}\).