/*! 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 38 Verify that the functions \(f(x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 38

Verify that the functions \(f(x)=\sin ^{2} x\) and \(g(x)=-\cos ^{2} x\) have the same derivative. What can you say about the difference \(f-g ?\) Explain.

Short Answer

Expert verified
Answer: Yes, the functions f(x) = sin^2(x) and g(x) = -cos^2(x) have the same derivative, which is f'(x) = g'(x) = 2sin(x)cos(x). The difference between them is f(x) - g(x) = sin^2(x) + cos^2(x) = 1 for all values of x.

Step by step solution

01

Find the derivative of f(x)

To find the derivative of \(f(x) = \sin^2 x\), use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is \(u^2\) and the inner function is \(u = \sin x\). So, we first find the derivative of \(\sin x\): $$ \frac{d}{dx}(\sin x) = \cos x $$ Now we can find the derivative of \(f(x)\) using the chain rule: $$ \frac{d}{dx}(\sin^2 x) = \frac{d}{dx}(u^2) \times \frac{d}{dx}(\sin x) = 2u \cdot \cos x = 2\sin x \cos x $$
02

Find the derivative of g(x)

To find the derivative of \(g(x) = -\cos^2 x\), we again use the chain rule. In this case, the outer function is \(v^2\) and the inner function is \(v = \cos x\). So, we first find the derivative of \(\cos x\): $$ \frac{d}{dx}(\cos x) = -\sin x $$ Now we can find the derivative of \(g(x)\) using the chain rule: $$ \frac{d}{dx}(-\cos^2 x) = \frac{d}{dx}(v^2) \times \frac{d}{dx}(\cos x) = 2v \cdot -\sin x = -2\cos x \sin x $$
03

Test if the derivatives are equal

Now, compare the two derivatives, \(f'(x) = 2\sin x \cos x\) and \(g'(x) = -2\cos x \sin x\): $$ f'(x) = 2\sin x \cos x \Rightarrow\\ g'(x) = -(-2\cos x \sin x) = 2\sin x \cos x \Rightarrow\\ f'(x) = g'(x) $$ We can see that the two derivatives are indeed the same.
04

Analyze the difference between f(x) and g(x)

Now, to understand the difference \(f(x) - g(x)\), we can write it as: $$ f(x) - g(x) = \sin^2 x - (-\cos^2 x) = \sin^2 x + \cos^2 x $$ From the Pythagorean identity, we know that \(\sin^2 x + \cos^2 x = 1\). Therefore, the difference \(f(x) - g(x)\) is always equal to 1 for all values of x. In conclusion, the functions \(f(x) = \sin^2 x\) and \(g(x) = -\cos^2 x\) have the same derivative, and their difference, \(f(x) - g(x)\), is always equal to 1.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the following limits in two different ways: One of the ways should use l' H么pital's Rule. $$\lim _{x \rightarrow \infty} \frac{2 x^{3}-x^{2}+1}{5 x^{3}+2 x}$$

Concavity of parabolas Consider the general parabola described by the function \(f(x)=a x^{2}+b x+c .\) For what values of \(a, b,\) and \(c\) is \(f\) concave up? For what values of \(a, b,\) and \(c\) is \(f\) concave down?

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=4 ; v(0)=-3, s(0)=2$$

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=0.2 t ; v(0)=0, s(0)=1$$

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime}(x)=1, F^{\prime}(0)=3, F(0)=4$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.