/*! 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 14 Find the rate of change of the d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 14

Find the rate of change of the distance between the origin and a moving point on the graph of \(y=\sin x\) if \(d x / d t=2\) centimeters per second.

Short Answer

Expert verified
The rate of change of the distance is \((2x + 2\sin x*\cos x) / d\), where d is the original distance, x is the x-coordinate of the moving point on the graph of \(y=\sin x\), and we are assuming that \(d > 0\).

Step by step solution

01

Express distance in terms of x

The distance (d) from the origin to any point on the graph of the function \(y = \sin x\) can be determined using Pythagorean theorem. As the x-coordinate of the point increases (or decreases), the y-coordinate varies as \(\sin x\), therefore the square of the distance \(d^2 = x^2 + (\sin x)^2.\)
02

Differentiate to get rate of change of distance

To find the rate of change of distance, we derive it with respect to time (t). Thus \(d d^2/ d t = d( x^2 + (\sin x)^2 )/ d t\). Applying Chain Rule for differentiation we get that \(d d^2 / dt = 2x * \frac{dx}{dt} + 2\sin x * \cos x * \frac{dx}{dt}\).
03

Substitute given rate

We substitute \(\frac{dx}{dt}\) in this equation with the given rate of 2 cm/s. So \(d d^2/ dt = 2x*2 + 2\sin x*\cos x*2\).\nHence, \(d d^2 / dt = 4x + 4\sin x*\cos x\). In order to find \(d d / dt\) we take the square root of the previous result divided by 2.

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

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{f(x)=\tan ^{2} x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4}, 1\right)}\)

Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=\cos x, \quad\left[0, \frac{\pi}{3}\right] $$

Prove that \(\arcsin x=\arctan \left(\frac{x}{\sqrt{1-x^{2}}}\right),|x|<1\)

Slope Find all points on the circle \(x^{2}+y^{2}=25\) where the slope is \(\frac{3}{4}\).

This law states that the rate of change of the temperature of an object is proportional to the difference between the object's temperature \(T\) and the temperature \(T_{a}\) of the surrounding medium. Write an equation for this law.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.