/*! 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 17 A bug is moving along the parabo... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 17

A bug is moving along the parabola \(y=x^{2} .\) At what point on the parabola are the \(x\) - and \(y\) -coordinates changing at the same rate? (Source: Calculus, Tom M. Apostol, Vol. 1, John Wiley \& Sons, New York, 1967.)

Short Answer

Expert verified
Answer: The point is (\(\frac{1}{2}\), \(\frac{1}{4}\)).

Step by step solution

01

Differentiate the Equation

Since the bug is moving along the parabola y = x^2, we will differentiate the equation with respect to time. To do this, we first differentiate y = x^2 with respect to x, then use the chain rule to differentiate with respect to time: \(\frac{dy}{dx} = 2x\) Now, using the chain rule, we differentiate each side with respect to time 't': \(\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = 2x \cdot \frac{dx}{dt}\)
02

Setting the Derivatives Equal

We are told that the x-coordinate and y-coordinate are changing at the same rate, which means: \(\frac{dx}{dt} = \frac{dy}{dt}\) Now, substitute the expression for \(\frac{dy}{dt}\) from Step 1: \(\frac{dx}{dt} = 2x \cdot \frac{dx}{dt}\)
03

Solving for x

Now we need to solve the equation for x. First, we divide both sides of the equation by \(\frac{dx}{dt}\), as it cannot be zero (the bug is moving): \(1 = 2x\) Next, divide both sides of the equation by 2: \(x = \frac{1}{2}\)
04

Finding y

Now that we have the value of x, we will use the equation of the parabola, \(y = x^2\), to find the value of y: \(y = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
05

Writing the Final Answer

We found that the x-coordinate is \(\frac{1}{2}\) and the y-coordinate is \(\frac{1}{4}\). Therefore, the point on the parabola where the x- and y-coordinates are changing at the same rate is: \((\frac{1}{2}, \frac{1}{4})\)

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