The inn er produ ct of two functions f and g on [a,b] is given by (f, 9) = | 5 (#) g (x) dr. Two distinct functions f and g are said to be ort hogonal if (f,g) = 0. Show that the foll owing set of functions is orthogonal on [–n, 7] . {sin x, sin 2x, sin 3x,...cos x, cos 2x, cos 3x, ...}

The inn er produ ct of two functions f and g on [a,b] is given by (f, 9) = | 5 (#) g (x) dr. Two distinct functions f and g are said to be ort hogonal if (f,g) = 0. Show that the foll owing set of functions is orthogonal on [–n, 7] . {sin x, sin 2x, sin 3x,...cos x, cos 2x, cos 3x, ...}

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.3: The Addition And Subtraction Formulas

Problem 80E

See similar textbooks

Related questions

Q: The set of functions {sin nx}, n = 1, 2, 3, ... , is %3D orthogonal on the interval [ -n, n]. Show…

A: Given- The set of functions sin nx, n = 1, 2, 3, ⋯, is orthogonal on the interval -π, π To Show-…

Q: Verify that the three conditions of the hypothesis of Rolle's Theorem hold for the function f (x) =…

A: 1. f(x) is continuous on the closed interval [a,b] 2. f(x) is differentiable on the open interval…

Q: Explain why the function has at least one zero in the given interval.

A: Given function is fx=x2-2-cosx over interval 0, π.

Q: 3. Show that the functionf on [-1,1] defined as if f (r) = 1+e if x = 0 is discontinuous at x = = 0.…

Q: Differentiate f(x) = 4sin (6z – 7) – 4 cos (). Enter your answer as a function of æ in the box…

Q: Does there exist a continuous function f(x) such that f(0) all x in (0, 2)? 10, f(2) = 2 and f'(x)…

A: Let's find.

Q: Find the values of a and b that make f(x) continuous for all real x. be* + a + 1, ax² + b(x +3), a…

A: The function f(x) is said to be continuous if the left-hand limit is equal to the right-hand limit…

Q: Show that the function is continuous. (x) = 3x cos 3x

A: Show that the function is continuous. (x) = 3x cos 3x

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

Q: Let f be the function defined by: sin x, x <0 x2, 0 <x < 1 2 - x, 1<x < 2 x-3, x22 f(x) = %3D For…

Q: 2. Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous

A: Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous.

Q: Explain why the function f(x)=cos.x-xmust have at least one zero on the interval [0,5]

Q: Prove or disprove: The function f : [0, 1] → R defined by x sin (¹) if 0 < x≤1, {₁ 0 if x = 0, f(x)…

A: Theorem: If f : K→R is continuous and K⊂R is compact, then f is uniformly continuous on K. Given:…

Q: To plot the function f(x) == %3D 2x over [0,5], we use fplot('exp^(-x) /2*x'[0,5)…

A: Since we are given with multiple number of questions and our guideline is solve the first question…

Q: Suppose f(x) = – 2x – cos r. (a) Show that f is strictly decreasing on R. (b) Hence, justify that 2x…

Q: Determine whether the functions are linearly dependent or linearly independent on the specified…

A: To check the linear dependence/independence of such functions, the concept of Wronskian can be used:…

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

Q: Show that the function f defined on [0, 1] by the conditions f=2e when<a<를, -1,2, 3, p=1, 2, 3,...…

Q: Prove that f(x) =- 2x(sinx+ tanx), is an odd function, where [] denotes %3D X+ 2n -3 the greatest…

Q: Explain why the function has at least one zero in the given interval. Function Interval f(x) = x2 -…

Q: 11. Determine the absolute extreme values of the function fx) = 2 cosx- x on the interval 0<x<2x.

A: We are authorized to solve only 1 question, please repost remaining Extrema

Q: Show that the function f defined on [0, 1]'by the conditions 1 f(x)=2rx when 7=1, 2, 3,...... És…

Q: Show that given functions are orthogonal on the indicated interval. f1(x) = ex, f2(x)= xe-x - e-x…

A: given functions are orthogonal on the i interval [0,2] are f1(x) = ex, f2(x)= xe-x - e-x .

Q: Show that the function is continuous on its domain. State the domain. Vx + x cosx

A: fx=x4+x3 cosx The domain is the set of all positive real numbers including zero. That is, the domain…

Q: Let f(x) be twice differentiable and invertible function such that f'(x) >0 and f"(x) > 0, prove х,…

Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx<0x20≤x<12−x1≤x<2x−3x≥2

Q: Verify that Rolle's Theorem can be applied to the function f(z)= 1³ – 9z2 + 26z – 24 on the interval…

A: Given f(x)=x3-9x2+26x-24, x∈[2,4]f(2)=23-9×22+26×2-24=0f(4)=43-9×42+26×4-24=0 hence, the Rolle's…

Q: 1 Show that f (x) = is Libschitz function on %3D x +1

Q: Is the set {sin nx , cos mx} an orthogonal set with respect to the inner product on the set of all…

Q: Find the intervals on which f is increasing and the intervals on which it is decreasing. f(x) = - 2…

A: Given, f(x) = 2cox - x x ∈ [ 0 , 2π] f'(x) = 2 (-sinx) - 1 = -2sinx -1…

Q: Verify that Rolle's Theorem can be applied to the function f (z) =2³ – 10z2 +27z – 18 on the…

Q: Determine if a function f : R2 → R defined by f (x; y) = { x sin(1/x2+y2) , if (x, y) ≠ (0,0) and 0…

Q: Show that the function f defined on [0, 1] by the conditions 1 f(x)=2rx when 7=1, 2, 3,...... És…

Q: Verify that the function f (x) = x² – 8x + 15 defined over the interval [3, 5], satisfies Rolle's…

A: using role's theorem: if a function is continuous and differentiable on the closed interval [a b]…

Q: Find any points of inflection (as ordered pairs) on the graph of f (x) = x+ 2 cosx in the interval…

Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

A: What is Continuous Function: A function is referred to be continuous at a certain point in its…

Q: Suppose that f(1) = 5, ƒ(4) = -8, f'(1) = 4, f'(4) = -2, and f" is continuous. Find the value of |…

Q: Verify that Rolle's Theorem can be applied to the function f (z) = 23 – 10z2 +27x – 18 on the…

Q: Show that Rolle’s theorem is false for the function f(x) = |x| on the interval [−2, 2]. How can a…

A: Given function f(x)=|x|interval [-2,2] We apply Rolle's theorem only when the function is continuous…

Q: Explain why the function has at least two zeros in the interval [1, 5]. f(x) = 2 cos x

A: To show the function has at least two zeros .

Q: If a function is real and continuous in the region from a to b and f(a) and f(b) and if both have…

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: Intermediate Value Theorem: It states that if f is a continuous function whose domain contains the…

Q: Verify that the function f(x) = x 2 − 4x − 11 defined over the interval [0, 4] satisfies Rolle’s…

A: Rolle's Theorem: Rolle's theorem states that if a function f(x) is continuous on [a, b] and…

Q: Suppose that the function f: R → R has the property that -x² < f(x)<x² for all x. Prove that fis…

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

Q: Find all points cc satisfying the conclusion of the MVT for the function…

Q: (i) Consider the function such that f(x) = 2x if 0SxS1 and f(x) = x-1 if x> 1. Is f continuous over…

Question

The inn er produ ct of two functions f and g on [a,b] is given by
(f, g) = |
f (x) g (x) dx.
Two distinct functions f and g are said to be ort hogonal if (f, g) = 0. Show that the following
set of functions is orthogonal on [–7, "] .
%3D
{sin x, sin 2x, sin 3x, ...cos x, cos 2x, cos 3x,...}

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps with 5 images

SEE SOLUTION Check out a sample Q&A here