/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 114 Conjecture Let \(f\) be a differ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 114

Conjecture Let \(f\) be a differentiable function of period \(p .\) (a) Is the function \(f^{\prime}\) periodic? Verify your answer. (b) Consider the function \(g(x)=f(2 x) .\) Is the function \(g^{\prime}(x)\) periodic? Verify your answer.

Short Answer

Expert verified
Yes, the derivative of a periodic function is also periodic. The derivative of a function \(g(x) = f(2x)\) is only periodic if the period of the derivative of \(f\) is twice the period of \(g\).

Step by step solution

01

Determining if the derivative of a periodic function is periodic

A function \(f(x)\) is said to be period if there exists a positive real number \(p\) such that for all \(x\) and all integers \(n\), \(f(x+n p)=f(x)\). If \(f\) is differentiable, then the derivative \(f'(x)\) would have the property \(f'(x+np)=f'(x)\), for any integer \(n\). Thus the derivative of a periodic function is also periodic.
02

Setting up the function \(g(x)=f(2x)\) and its derivative

The function \(g(x)\) is a transformation of \(f(x)\), where we replace \(x\) with \(2x\). We can differentiate \(g\) with respect to \(x\) to get: \[g'(x) = (f(2x))' = 2f'(2x) \] because of the chain rule of differentiation.
03

Determining if \(g'(x)\) is periodic

The derivative function \(g'(x)\) is periodic if there exists a \(p > 0\) such that \(g'(x + p) = g'(x)\), for all \(x\) and all integers \(n\). Given our definition of \(g'(x)\), we find \(g'(x + p) = 2f'(2x + 2p)\), which by the periodicity of \(f'(x)\), is equal to \(2f'(2x) = g'(x)\) if \(2p\) is a period of \(f'(x)\). Thus \(g'(x)\), in this case, would only be periodic if \(p\) is equal to half the period of \(f'\).
04

Conclusion

The derivative of a periodic function \(f'(x)\) is also periodic with the same period as \(f\). However, for a function \(g(x)\) defined as a transformation of \(f(x)\), its derivative \(g'(x)\) is periodic only if the period of \(f'(x)\) fits the transformation rule.

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Most popular questions from this chapter

Determining Differentiability In Exercises 89 and \(90,\) determine whether the function is differentiable at \(x=2\) . $$ f(x)=\left\\{\begin{array}{ll}{\frac{1}{2} x+1,} & {x<2} \\ {\sqrt{2 x},} & {x \geq 2}\end{array}\right. $$

True or False? In Exercises \(93-96\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is continuous at a point, then it is differentiable at that point.

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are $$\begin{array}{l}{P_{1}(x)=f^{\prime}(a)(x-a)+f(a) \text { and }} \\\ {P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)}\end{array}$$ In Exercises 123 and \(124,\) (a) find the specified linear and quadratic approximations of \(f,(b)\) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\) . $$ f(x)=\tan x ; \quad a=\frac{\pi}{4} $$

True or False? In Exercises \(125-128\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=(1-x)^{1 / 2},\) then \(y^{\prime}=\frac{1}{2}(1-x)^{-1 / 2}\).

Area The included angle of the two sides of constant equal length \(s\) of an isosceles triangle is \(\theta\) . (a) Show that the area of the triangle is given by \(A=\frac{1}{2} s^{2} \sin \theta .\) (b) the angle \(\theta\) is increasing at the rate of \(\frac{1}{2}\) radian per minute. Find the rates of change of the area when \(\theta=\pi / 6\) and \(\theta=\pi / 3 .\) (c) Explain why the rate of change of the area of the triangle is not constant even though \(d \theta / d t\) is constant.
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