3.3.E: Best Affine Approximations (Exercises)

Exercise \(\PageIndex{1}\)

For each of the following, find the best affine approximation to the given function at the specified point \(\mathbf{c}\).

(a) \(f(x, y)=3 x^{2}+4 y^{2}-2, \mathbf{c}=(2,1)\)

(b) \(g(x, y)=y^{2}-x^{2}, \mathbf{c}=(1,-2)\)

(c) \(g(x, y)=y^{2}-x^{2}, \mathbf{c}=(0,0)\)

(d) \(f(x, y, z)=-\log \left(x^{2}+y^{2}+z^{2}\right), \mathbf{c}=(1,0,0)\)

(e) \(h(w, x, y, z)=w^{2}+x^{2}+3 y^{2}=2 z^{2}, \mathbf{c}=(1,2,-2,1)\)

Answer

(a) \(A(x, y)=12 x+4 y-12\)

(c) \(A(x, y)=0\)

(e) \(A(w, x, y, z)=2 w+4 x-12 y-4 z-19\)

Exercise \(\PageIndex{2}\)

For each of the following, find the equation of the plane tangent to the graph of \(f\) for the given point \(\mathbf{c}\). Plot the graph and the tangent plane together.

(a) \(f(x, y)=4 x^{2}+y^{2}, \mathbf{c}=(1,-1)\)

(b) \(f(x, y)=\sqrt{9-x^{2}-y^{2}}, \mathbf{c}=(2,1)\)

(c) \(f(x, y)=9-x^{2}-y^{2}, \mathbf{c}=(2,-2)\)

(d) \(f(x, y)=3 y^{2}-x^{2}, \mathbf{c}=(1,-1)\)

Answer

(a) \(8 x-2 y-z=5\)

(c) \(4 x-4 y+z=17\)

Exercise \(\PageIndex{3}\)

Suppose \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}\) is the best affine approximation to \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}\) at \(\mathbf{c}\). Explain why \(|\nabla f(\mathbf{c}) \cdot \mathbf{h}|\) is a good approximation for \(|f(\mathbf{c}+\mathbf{h})-f(\mathbf{c})|\) when \(\|\mathbf{h}\| \) is small. That is, explain why \(|\nabla f(\mathbf{c}) \cdot \mathbf{h}|\) is a good approximation for the error in approximating \(f(\mathbf{c}+\mathbf{h})\) by \(f(\mathbf{c})\).

Exercise \(\PageIndex{4}\)

Suppose \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}\) is defined by \(f(x, y, z)=x y z\).

(a) Find the best affine approximation to \(f\) at (3,2,4).

(b) Suppose \(x\), \(y\), and \(z\) represent the length, width, and height of a box. Suppose you measure the length to be \(3 \pm h\) centimeters, the width to be \(2 \pm h\) centimeters, and the height to be \(4 \pm h\) centimeters. Use the best affine approximation from (a) to approximate the maximum error you would make in computing the volume of the box from these measurements.

Answer

(a) \(A(x, y, z)=8 x+12 y+6 z-48\)

(b) \(26 h\)

Exercise \(\PageIndex{5}\)

A metal plate is heated so that its temperature at a point (x,y) is

\[ T(x, y)=50 y^{2} e^{-\frac{1}{5}\left(x^{2}+y^{2}\right)} . \nonumber \]

A bug moves along the ellipse parametrized by

\[ \alpha(t)=(\cos (t), 2 \sin (t)) . \nonumber \]

Find the rate of change of temperature for the bug at times \(t=0, t=\frac{\pi}{4},\) and \(t=\frac{\pi}{2}\).

Answer

\(\left.\frac{d T}{d t}\right|_{t=0}=0 ;\left.\frac{d T}{d t}\right|_{t=\frac{\pi}{4}}=140 e^{-\frac{1}{2}} ;\left.\frac{d T}{d t}\right|_{t=\frac{\pi}{2}}=0\)

Exercise \(\PageIndex{6}\)

Let \(x\), \(y\), and \(z\) be the length, width, and height, respectively, of a box. Suppose the box is increasing in size so that when \(x = 3\) centimeters, \(y = 2\) centimeters, and \(z = 5\) centimeters, the length is increasing at rate of 2 centimeters per second, the width at a rate of 4 centimeters per second, and the height at a rate of 3 centimeters per second.

(a) Find the rate of change of the volume of the box at this time.

(b) Find the rate of change of the length of the diagonal of the box at this time.

Exercise \(\PageIndex{7}\)

Suppose \(w=-\log \left(x^{2}+y^{2}+z^{2}\right)\) and \((x, y, z)=(4 t, \sin (t), \cos (t)\). Find

\[ \left.\frac{d w}{d t}\right|_{t=\frac{\pi}{3}} . \nonumber \]

Answer

\(\left.\frac{d w}{d t}\right|_{t=\frac{\pi}{3}}=-\frac{96 \pi}{16 \pi^{2}+9} \approx-1.807\)

Exercise \(\PageIndex{8}\)

The kinetic energy \(K\) of an object of mass \(m\) moving in a straight line with velocity \(v\) is

\[ K=\frac{1}{2} m v^{2} . \nonumber \]

If, at time \(t=t_0\), \(m = 2000\) kilograms, \(v = 50\) meters per second, \(m\) is decreasing at a rate of 2 kilograms per second, and \(v\) is increasing at a rate of 1.5 meters per second per second, find

\[ \left.\frac{d K}{d t}\right|_{t=t_{0}} . \nonumber \]

Exercise \(\PageIndex{9}\)

Each of the following equations specifies some curve in \(\mathbb{R}^2\). In each case, find an equation for the line tangent to the curve at the given point \(\mathbf{a}\).

(a) \(x^{2}+y^{2}=5, \mathbf{a}=(2,1)\)

(b) \(2 x^{2}+4 y^{2}=18, \mathbf{a}=(1,-2)\)

(c) \(y^{2}-x=0, \mathbf{a}=(4,-2)\)

(d) \(y^{2}-x^{2}=5, \mathbf{a}=(-2,3)\)

Answer

(a) \(2 x+y=5\)

(c) \(x+4 y=-4\)

Exercise \(\PageIndex{10}\)

Each of the following equations specifies some surface in \(\mathbb{R}^3\). In each case, find an equation for the plane tangent to the surface at the given point \(\mathbf{a}\).

(a) \(x^{2}+y^{2}+z^{2}=14, \mathbf{a}=(2,1,-3)\)

(b) \(x^{2}+3 y^{2}+2 z^{2}=9, \mathbf{a}=(2,-1,1)\)

(c) \(x^{2}+y^{2}-z^{2}=1, \mathbf{a}=(1,2,2)\)

(d) \(x y z=6, \mathbf{a}=(1,2,3)\)

Answer

(a) \(2 x+y-3 z=14\)

(c) \(x+2 y-2 z=1\)

Exercise \(\PageIndex{11}\)

Suppose \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}\) is differentiable at \((a,b)\), \(f(a,b) = c\), and \(\frac{\partial}{\partial y} f(a, b) \neq 0\). Let \(C\) be the level curve of \(f\) with equation \(f(x,y) = c\). Show that

\[ y=-\frac{\frac{\partial}{\partial x} f(a, b)}{\frac{\partial}{\partial y} f(a, b)}(x-a)+b \nonumber \]

is an equation for the line tangent to \(C\) at \((a,b)\).