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Chapter 2: Problem 13

Find the rate of change of the distance between the origin and a moving point on the graph of \(y=x^{2}+1\) if \(d x / d t=2\) centimeters per second.

Short Answer

Expert verified
In order to find the rate at which the distance changes, substitute the given \(d x / d t=2\) cm/s into the derived expression for \(d D/ d t\) from the equation \(D^{2}=x^{2}+(x^{2}+1)^{2}\), and solve.

Step by step solution

01

Determine the distance

We first need to find the distance between the origin and the point on the graph of the function. Using the Euclidean distance formula in two dimensions, the distance (D) from the origin (0,0) to a point (x, y) on the graph is given by \(D=\sqrt{x^{2}+y^{2}}\). Given that \(y=x^{2}+1\), we substitute into the formula for D to give \(D=\sqrt{x^{2}+(x^{2}+1)^{2}}\).
02

Simplify the distance expression

We can simplify this equation by squaring both sides to eliminate the square root, giving \(D^{2}=x^{2}+(x^{2}+1)^{2}\) and then simplifying by collecting like terms.
03

Derive the distance expression with respect to time

As we want to find \(d D / d t\) and given \(d x / d t=2\) centimeters (cm) per second, we take the derivative of both sides of the equation derived from step 2 with respect to time (t). Here, we employ the chain rule of differentiation where necessary, noting that \(d x^{2} / d t=2 x d x / d t\). It is also important to remember to include all Constant C in the equation.
04

Substitute \(d x / d t\) into the derived equation

Substitute the given \(d x / d t=2\) into the derived expression for \(d D/ d t\) obtained in step 3 to solve for \(d D/ d t\) .
05

Simplify to find \(d D / d t\)

Simplify the equation to find the value of \(d D / d t\), which is the rate of change of the distance between the moving point and the origin. This is the answer to the problem.

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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. \(\frac{\text { Function }}{y=\cos 3 x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4},-\frac{\sqrt{2}}{2}\right)}\)

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