Find the best approximation of...
Let \(P_{2}(\mathbb{R})\) have the inner product,
$$ \langle\mathbf{p}, \mathbf{q}\rangle=\int_{-1}^{1} p(x) q(x) d x, \quad \forall \mathbf{p}, \mathbf{q} \in P_{2}(\mathbb{R}) . $$
Find the best approximation of \(f(x)=x^{3}+x^{4}\) by polynomials in \(P_{2}(\mathbb{R})\).
Homework Answers
Suppose x = (1, 0, -1) and y = (-2, 4, 8) are vectors in R3. Find a vector z
Suppose \(\mathbf{x}=\langle 1,0,-1\rangle\) and \(\mathbf{y}=\langle-2,4,8\rangle\) are vectors in \(\mathbb{R}^{3} .\) Find a vector \(\mathbf{z} \in \mathbb{R}^{3}\) such that \(2 \mathbf{x}+\mathbf{y}+2 \mathbf{z}=\mathbf{0} .\)\(\mathbf{z}=\)
Notice that these polynomials form an orthogonal set with this inner product. Find the best 1²-13...
Notice that these polynomials form an orthogonal set with this inner product. Find the best 1²-13 Let P2 have the inner product given by evaluation at -5, -1, 1, and 5. Let po(t) = 2, P1(t)=t, and q(t) = 12 approximation to p(t) = t by polynomials in Span{Po.P1,9}. The best approximation to p(t) = t by polynomials in Span{Po.P2,q} is
== Let P3 have the inner product given by evaluation at -3, -1, 1, and 3....
== Let P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 4, p1(t)=t, and t² – 5 q(t) = Notice that these polynomials form an orthogonal set with this inner product. Find the best 4 approximation to p(t) = tº by polynomials in Span{P0,21,9}. The best approximation to p(t) = tº by polynomials in Span{Po.21,93 is
Consider the region
4. [5 pts.] Consider the region \(D\), outside the circles \(C_{2}\) and \(C_{3}\) and inside the circle \(C_{1}\) in the figure below and a vector field \(\vec{F}(x, y)=\langle P(x, y), Q(x, y)\rangle\). Assume we know that \(\oint_{C_{2}} \vec{F} \cdot d \vec{r}=\oint_{C_{3}} \vec{F} \cdot d \vec{r}=-2 \pi\), and \(Q_{x}-P_{y}=2\) on an open region containing \(D .\) UseGreen's Theorem to find \(\oint_{C_{1}} \vec{F} \cdot d \vec{r}\).
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be def...
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be defined by ull , ).Consider the target function f (x) with the approximating space P e', and work 2. Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials p (x) through degree four. Standardize your polynomials such that p, (1) 1 (b) Find the best degree 4 approximation to f(x) using the specified norm, and working with this...
4 2-5 Notice that these polynomials form an Let P3 have the inner product given by...
4 2-5 Notice that these polynomials form an Let P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 2, P (t) = 4t, and act) = orthogonal set with this inner product. Find the best approximation to p(t) = tº by polynomials in Span{Po-P1:9). The best approximation to p(t) = tº by polynomials in Span{Po.P7.93 is
Suppose that the functions
Suppose that the functions \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}, g: \mathbb{R}^{3} \rightarrow \mathbb{R}\), and \(h: \mathbb{R}^{3} \rightarrow \mathbb{R}\) are continuously differentiable and let \(\left(x_{0}, y_{0}, z_{0}\right)\) be a point in \(\mathbb{R}^{3}\) at which$$ f\left(x_{0}, y_{0}, z_{0}\right)=g\left(x_{0}, y_{0}, z_{0}\right)=h\left(x_{0}, y_{0}, z_{0}\right)=0 $$and$$ \left\langle\nabla f\left(x_{0}, y_{0}, z_{0}\right), \nabla g\left(x_{0}, y_{0}, z_{0}\right) \times \nabla h\left(x_{0}, y_{0}, z_{0}\right)\right\rangle \neq 0 $$By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a...
Question 2: For this question, consider the non-standard pairing on the space of real polynomials P...
Question 2: For this question, consider the non-standard pairing on the space of real polynomials P given by g) = Lif(t)g(x).rº dr. (a) Prove that (,) defines an inner product on P. (b) Let O be the set of odd polynomials, i.e. f(r) € P such that f(x)= -f(-r). Show that is a subspace of P. (c) Explain why g() = 5x2 - 3 is in 0+ (the orthogonal complement of O with respect to (>). (d) Let P<2 denote...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe de...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
Given Show that
Given$$ \vec{F}(x, y)=\left\langle\frac{2 x^{3}+2 x y^{2}-2 y}{x^{2}+y^{2}}, \frac{2 y^{3}+2 x^{2} y+2 x}{x^{2}+y^{2}}\right\rangle $$Show that \(\int_{C} \vec{F} \cdot d \vec{r}=4 \pi\) for any positively oriented simple closed curve that encloses the origin.
Suppose \(\mathbf{x}=\langle 1,0,-1\rangle\) and \(\mathbf{y}=\langle-2,4,8\rangle\) are vectors in \(\mathbb{R}^{3} .\) Find a vector \(\mathbf{z} \in \mathbb{R}^{3}\) such that \(2 \mathbf{x}+\mathbf{y}+2 \mathbf{z}=\mathbf{0} .\)\(\mathbf{z}=\)
Notice that these polynomials form an orthogonal set with this inner product. Find the best 1²-13 Let P2 have the inner product given by evaluation at -5, -1, 1, and 5. Let po(t) = 2, P1(t)=t, and q(t) = 12 approximation to p(t) = t by polynomials in Span{Po.P1,9}. The best approximation to p(t) = t by polynomials in Span{Po.P2,q} is
== Let P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 4, p1(t)=t, and t² – 5 q(t) = Notice that these polynomials form an orthogonal set with this inner product. Find the best 4 approximation to p(t) = tº by polynomials in Span{P0,21,9}. The best approximation to p(t) = tº by polynomials in Span{Po.21,93 is
4. [5 pts.] Consider the region \(D\), outside the circles \(C_{2}\) and \(C_{3}\) and inside the circle \(C_{1}\) in the figure below and a vector field \(\vec{F}(x, y)=\langle P(x, y), Q(x, y)\rangle\). Assume we know that \(\oint_{C_{2}} \vec{F} \cdot d \vec{r}=\oint_{C_{3}} \vec{F} \cdot d \vec{r}=-2 \pi\), and \(Q_{x}-P_{y}=2\) on an open region containing \(D .\) UseGreen's Theorem to find \(\oint_{C_{1}} \vec{F} \cdot d \vec{r}\).
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be defined by ull , ).Consider the target function f (x) with the approximating space P e', and work 2. Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials p (x) through degree four. Standardize your polynomials such that p, (1) 1 (b) Find the best degree 4 approximation to f(x) using the specified norm, and working with this...
4 2-5 Notice that these polynomials form an Let P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 2, P (t) = 4t, and act) = orthogonal set with this inner product. Find the best approximation to p(t) = tº by polynomials in Span{Po-P1:9). The best approximation to p(t) = tº by polynomials in Span{Po.P7.93 is
Question 2: For this question, consider the non-standard pairing on the space of real polynomials P given by g) = Lif(t)g(x).rº dr. (a) Prove that (,) defines an inner product on P. (b) Let O be the set of odd polynomials, i.e. f(r) € P such that f(x)= -f(-r). Show that is a subspace of P. (c) Explain why g() = 5x2 - 3 is in 0+ (the orthogonal complement of O with respect to (>). (d) Let P<2 denote...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
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