Homework 11: Taylor’s Theorem

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Graded Problems

Give a $2\times 2$ matrix $A=\left [ \begin {array}{cc} a & b \\ c & d \end {array} \right ]$ with $a > 0$ and $\det (A) >0$ for which the quadratic form $p(\mathbf {x})=\mathbf {x}^TA\mathbf {x}$ is NOT positive definite. Does this mean the theorem in problem 6 is incorrect?

Prove the following theorem: Let $A=\left [ \begin {array}{cc} a & b \\ b & c \end {array} \right ]$ be a symmetric $2\times 2$ matrix. If det $(A)<0$ , then the quadratic form $p(\mathbf {x})=\mathbf {x}^TA\mathbf {x}$ is indefinite, regardless of the value of $a$ .

(Hint: think about the cases $a>0$ , $a=0$ and $a<0$ . In two cases you can apply Sylvester’s Theorem. In the third case, you’ll have to do some work by hand to show $p(x,y)$ can have both positive and negative values.)

Professional Problem

Complete the online peer review form posted on moodle.

Completion Packet

Find the symmetric matrix that represents each quadratic form.

(a): $r(x_1,x_2,x_3,x_4)=x_3^2-x_2x_3+x_1x_4$
(b): $t(\mathbf {x})=\mathbf {x}^T\left [ \begin {array}{cccc} 6 & 1 & 8 & -2 \\ 0 & 5 & 1 & 9 \\ 0 & 0 & -2 & 0 \\ 0 & 0 & 0 & 0 \end {array} \right ]\mathbf {x}$

Prove the following theorem without using Sylvester’s theorem: Let $A=\left [ \begin {array}{cc} a & b \\ b & c \end {array} \right ]$ be a symmetric $2\times 2$ matrix. If $a>0$ and $\det (A)>0$ , then the quadratic form $p(\mathbf {x})=\mathbf {x}^TA\mathbf {x}$ is positive definite.

(Hint: Write out $p$ in terms of the variables $x$ and $y$ , then complete the square with respect to $x$ and collect the remaining terms.)

(a): Write the Taylor series for $f(x) = \sin (x)$ centered at $x = 0$ .
(b): Find the second-order Taylor approximation for $f(x,y) = \sin (xy)$ centered at $(0,0)$ , using your answer to part (a).
(c): Verify your answer to part (b), by computing the second-order Taylor approximation for $f(x,y) = \sin (xy)$ directly.

(a): Write the Taylor series for $f(x) = e^{x}$ centered at $x = 0$ .
(b): Find the second-order Taylor approximation for $f(x,y) = e^{x^2+y^2}$ centered at $(0,0)$ , using your answer to part (a).
(c): Verify your answer to part (b), by computing the second-order Taylor approximation for $f(x,y) = e^{x^2+y^2}$ directly.

Compute the Hessian matrix for each function at the given point.

(a): $f(x,y) = \sqrt {xy}$ at $(1,1)$
(b): $f(x,y) = \cos (x) + x^2\sin (y)$ at $(0, \pi )$
(c): $f(x,y,z) = \frac {1}{x^2+y^2+z^2+1}$ at $(0,0,0)$

Consider the function $f(x,y) = \frac {x}{x+y}$ and the point $\mathbf {a}=(1,2)$ .

(a): Find the first-order Taylor polynomial of $f$ at $\mathbf {a}$ .
(b): Find the second-order Taylor polynomial of $f$ at $\mathbf {a}$ .
(c): Express the second-order Taylor polynomial using the derivative matrix and the Hessian matrix, as in formula (10) of section 4.1 of the textbook.

Consider the function $f(x,y) = x^2e^{y}$ and the point $\mathbf {a}=(1,0)$ .

(a): Find the first-order Taylor polynomial of $f$ at $\mathbf {a}$ .
(b): Find the second-order Taylor polynomial of $f$ at $\mathbf {a}$ .
(c): Express the second-order Taylor polynomial using the derivative matrix and the Hessian matrix, as in formula (10) of section 4.1 of the textbook.