Problem 4. For each of the following functions, prove whether or not it defines an inner product on the givenvector space.(a) On R²,(8.2)= ac-bd.(b) On P,(c) On Mnxn,(f(x),9(x)) = f* f'(x)9(x)dx.0(A, B) = tr(A+B).

Step 1: Part (a): Part (b): Part (c):

Answered: Problem 4. For each of the following…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

See similar textbooks

Similar questions

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
6. Sketch: f(x) = [x+ 2] -1, on [-2,2]
Plz
Let φ1(x) = 1, φ2(x) = x/|x|, φ3(x) = x^2 − 1/3 for −1 ≤x ≤1.Show that (φ1,φ2,φ3) form an orthogonal set.
Assume that f: R→ R and g: R → R are infinitely differentiable functions. Prove the following generalization of the product rule: dn dxn n (f(z)g(2)) - (1) (M) (2) g-) (x)}) = k=0
Suppose you want to find an approximate representation for a function z = f(x, y) that is valid near x = 1, y = 2. Namely, you want to have f (x, y) ~ bo + b1,0(x − 1) + bo,1(y−2)+ b2,0(x − 1)² + b₁,1(x − 1)(y − 2) + bo,2(y − 2)² +.... a. Use the ideas from the beginning of this chapter to find the coefficients. b. What are the four coefficients for the third order terms that have the form (x - 1)³, (x - 1)²(y − 2), (x − 1)(y-2)², (y-2)³? c. How many coefficients are needed for the fourth order terms? d. Find the Taylor series approximation up to order two near x = 0, y = л of f(x, y) = ex cos(y).
7. Evaluate F dr from (0,0) to (3,9) along the curve y=x if F(x, y,z) = 4 xyi + 3xy}
4. Let [a, b] be an interval in R and let be a contimuous function on [a, b). (i) Check that the function r+ (b- a)x + a is a bijection from [0, 1] to [a, b] and its inverse is I+E. (ii) For r € [0, 1], let g(x) = »((b-a)x+a) (which is equivalent to say that (r) = g(). Check that g is continuous on [0, 1]. (iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges uniformly to g on [0, 1]. Construct a sequence of polynomials (pn)n such that || – Pn|.ja.bj = ||g – (iv) Deduce the Weiestrass approximation theorem for the continuous function on [a, b].

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

Problem 4. For each of the following functions, prove whether or not it defines an inner product on the given vector space. (a) On R², (8.2) = ac- bd. (b) On P, (c) On Mnxn, (f(x),9(x)) = f* f'(x)9(x)dx. 0 (A, B) = tr(A+B).