Vector Calculus

Susan Jane Colley

Chapter 4 Maxima and Minima in Several Variables - all with Video Answers

Educators

Section 1

Differentials and Taylor's Theorem

Problem 1

Find the Taylor polynomials $p_{k}$ of given order $k$ at the indicated point $a.$
$$
f(x)=e^{2 x}, a=0, k=4
$$

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Problem 2

Find the Taylor polynomials $p_{k}$ of given order $k$ at the indicated point $a.$
$$
f(x)=\ln (1+x), a=0, k=3
$$

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Problem 3

Find the Taylor polynomials $p_{k}$ of given order $k$ at the indicated point $a.$
$$
f(x)=1 / x^{2}, a=1, k=4
$$

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Problem 4

Find the Taylor polynomials $p_{k}$ of given order $k$ at the indicated point $a.$
$$
f(x)=\sqrt{x}, a=1, k=3
$$

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Problem 5

Find the Taylor polynomials $p_{k}$ of given order $k$ at the indicated point $a.$
$$
f(x)=\sqrt{x}, a=9, k=3
$$

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Problem 6

Find the Taylor polynomials $p_{k}$ of given order $k$ at the indicated point $a.$
$$
f(x)=\sin x, a=0, k=5
$$

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Problem 7

Find the Taylor polynomials $p_{k}$ of given order $k$ at the indicated point $a.$
$$
f(x)=\sin x, a=\pi / 2, k=5
$$

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Problem 8

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y)=1 /\left(x^{2}+y^{2}+1\right), \mathbf{a}=(0,0)$

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Problem 9

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y)=1 /\left(x^{2}+y^{2}+1\right), \mathbf{a}=(1,-1)$

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Problem 10

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y)=e^{2 x+y}, \mathbf{a}=(0,0)$

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Problem 11

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y)=e^{2 x} \cos 3 y, \mathbf{a}=(0, \pi)$

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Problem 12

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y, z)=y e^{3 x}+z e^{2 y}, \mathbf{a}=(0,0,2)$

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Problem 13

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y, z)=x y-3 y^{2}+2 x z, \mathbf{a}=(2,-1,1)$

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Problem 14

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y, z)=1 /\left(x^{2}+y^{2}+z^{2}+1\right), \mathbf{a}=(0,0,0)$

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Problem 15

Find the first- and second-order Taylor polynomials for the given function $f$ at the given point $a.$
$f(x, y, z)=\sin x y z, \mathbf{a}=(0,0,0)$

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Problem 16

Calculate the Hessian matrix $H f($ a) for the indicated function $f$ at the indicated point $a.$
$f(x, y)=1 /\left(x^{2}+y^{2}+1\right), \mathbf{a}=(0,0)$

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Problem 17

Calculate the Hessian matrix $H f($ a) for the indicated function $f$ at the indicated point $a.$
$f(x, y)=\cos x \sin y, \mathbf{a}=(\pi / 4, \pi / 3)$

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Problem 18

Calculate the Hessian matrix $H f($ a) for the indicated function $f$ at the indicated point $a.$
$f(x, y, z)=\frac{z}{\sqrt{x y}}, \mathbf{a}=(1,2,-4)$

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Problem 19

Calculate the Hessian matrix $H f($ a) for the indicated function $f$ at the indicated point $a.$
$f(x, y, z)=x^{3}+x^{2} y-y z^{2}+2 z^{3}, \mathbf{a}=(1,0,1)$

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Problem 20

Calculate the Hessian matrix $H f($ a) for the indicated function $f$ at the indicated point $a.$
$f(x, y, z)=e^{2 x-3 y} \sin 5 z, \mathbf{a}=(0,0,0)$

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Problem 21

For $f$ and $\mathbf{a}$ as given in Exercise $8,$ express the secondorder Taylor polynomial $p_{2}(x, y),$ using the derivative matrix and the Hessian matrix as in formula (10) of this section.

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Problem 22

For $f$ and $\mathbf{a}$ as given in Exercise $11,$ express the secondorder Taylor polynomial $p_{2}(x, y)$, using the derivative matrix and the Hessian matrix as in formula (10) of this section.

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Problem 23

For $f$ and $\mathbf{a}$ as given in Exercise $12,$ express the secondorder Taylor polynomial $p_{2}(x, y, z),$ using the derivative matrix and the Hessian matrix as in formula (10) of this section.

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Problem 24

For $f$ and $\mathbf{a}$ as given in Exercise $19,$ express the secondorder Taylor polynomial $p_{2}(x, y, z)$, using the derivative matrix and the Hessian matrix as in formula (10) of this section.

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Consider the function $$ f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=e^{x_{1}+2 x_{2}+\cdots+n x_{n}} $$
(a) Calculate $D f(0,0, \ldots, 0)$ and $H f(0,0, \ldots, 0)$.
(b) Determine the first- and second-order Taylor polynomials of $f$ at $\mathbf{0}$.
(c) Use formulas (3) and (10) to write the Taylor polynomials in terms of the derivative and Hessian matrices.

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Problem 26

Find the third-order Taylor polynomial $p_{3}(x, y, z)$ of $$f(x, y, z)=e^{x+2 y+3 z}$$ at (0,0,0) .

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Problem 27

Find the third-order Taylor polynomial $p_{3}(x, y, z)$ of $$f(x, y, z)=e^{x+2 y+3 z}$$ at $(0.0 .0)$

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Problem 28

Determine the total differential of the functions given in exercises.
$$
f(x, y)=x^{2} y^{3}
$$

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Problem 29

Determine the total differential of the functions given in exercises.
$$
f(x, y, z)=x^{2}+3 y^{2}-2 z^{3}
$$

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Problem 30

Determine the total differential of the functions given in exercises.
$$
f(x, y, z)=\cos (x y z)
$$

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Problem 31

Determine the total differential of the functions given in exercises.
$f(x, y, z)=e^{x} \cos y+e^{y} \sin z$

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Problem 32

Determine the total differential of the functions given in exercises.
$$
f(x, y, z)=1 / \sqrt{x y z}
$$

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Problem 33

Use the fact that the total differential $d f$ approximates the incremental change $\Delta f$ to provide estimates of the following quantities:
(a) $(7.07)^{2}(1.98)^{3}$
(b) $1 / \sqrt{(4.1)(1.96)(2.05)}$
(c) (1.1) $\cos ((\pi-0.03)(0.12))$

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Problem 34

Near the point $(1,-2,1),$ is the function $g(x, y, z)=$ $x^{3}-2 x y+x^{2} z+7 z$ most sensitive to changes in $x$ $y,$ or $z$ ?

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Problem 35

To which entry in the matrix is the value of the determinant$$\left|\begin{array}{rr}2 & 3 \\-1 & 5\end{array}\right|$$ most sensitive?

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Problem 36

If you measure the radius of a cylinder to be 2 in, with a possible error of ±0.1 in, and the height to be 3 in, with a possible error of ±0.05 in, use differentials to determine the approximate error in
(a) the calculated volume of the cylinder.
(b) the calculated surface area.

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Problem 37

A can of mushrooms is currently manufactured to have a diameter of $5 \mathrm{~cm}$ and a height of $12 \mathrm{~cm}$. The manufacturer plans to reduce the diameter by $0.5 \mathrm{~cm} .$ Use differentials to estimate how much the height of the can would need to be increased in order to keep the volume of the can the same.

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Problem 38

Consider a triangle with sides of lengths $a$ and $b$ that make an interior angle $\theta$.
(a) If $a=3, b=4,$ and $\theta=\pi / 3,$ to changes in which of these measurements is the area of the triangle most sensitive?
(b) If the length measurements in part (a) are in error by as much as $5 \%$ and the angle measurement is in error by as much as $2 \%,$ estimate the resulting maximum percentage error in calculated area.

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Problem 39

To estimate the volume of a cone of radius approximately $2 \mathrm{~m}$ and height approximately $6 \mathrm{~m},$ how accurately should the radius and height be measured so that the error in the calculated volume estimate does not exceed $0.2 \mathrm{~m}^{3}$ ? Assume that the possible errors in measuring the radius and height are the same.

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Problem 40

Suppose that you measure the dimensions of a block of tofu to be (approximately) 3 in by 4 in by 2 in. Assuming that the possible errors in each of your measurements are the same, about how accurate must your measurements be so that the error in the calculated volume of the tofu is not more than $0.2 \mathrm{in}^{3}$ ? What percentage error in volume does this represent?

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Problem 41

(a) Calculate the second-order Taylor polynomial for $f(x, y)=\cos x \sin y$ at the point $(0, \pi / 2)$
(b) If $\mathbf{h}=\left(h_{1}, h_{2}\right)=(x, y)-(0, \pi / 2)$ is such that $\left|h_{1}\right|$ and $\left|h_{2}\right|$ are no more than 0.3 , estimate how accurate your Taylor approximation is.

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Problem 42

(a) Determine the second-order Taylor polynomial of $f(x, y)=e^{x+2 y}$ at the origin.
(b) Estimate the accuracy of the approximation if $|x|$ and $|y|$ are no more than 0.1 .

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Problem 43

(a) Determine the second-order Taylor polynomial of $f(x, y)=e^{2 x} \cos y$ at the point $(0, \pi / 2)$
(b) If $\mathbf{h}=\left(h_{1}, h_{2}\right)=(x, y)-(0, \pi / 2)$ is such that $\left|h_{1}\right| \leq 0.2$ and $\left|h_{2}\right| \leq 0.1,$ estimate the accuracy of the approximation to $f$ given by your Taylor polynomial in part (a).

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