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Chapter 2: Problem 1

Suppose \(s(t)\) is the position of an object moving along a line at time \(t \geq 0 .\) What is the average velocity between the times \(t=a\) and \(t=b ?\)

Short Answer

Expert verified
Answer: The formula to calculate the average velocity is: Average velocity = \(\frac{s(b) - s(a)}{b-a}\), where \(s(t)\) represents the position of the object at any time \(t \geq 0\).

Step by step solution

01

Identify the given information

We are given that \(s(t)\) represents the position of the object at any time \(t \geq 0\). We need to find the average velocity between the times \(t=a\) and \(t=b\).
02

Calculate the change in position

The change in position is simply the difference between the positions at times \(t=a\) and \(t=b\), or \(s(b) - s(a)\).
03

Calculate the change in time

The change in time between \(t=a\) and \(t=b\) is the difference between the times: \(b - a\).
04

Find the average velocity

We can now find the average velocity by dividing the change in position by the change in time, as follows: Average velocity = \(\frac{s(b) - s(a)}{b-a}\)

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Most popular questions from this chapter

Suppose you park your car at a trailhead in a national park and begin a 2 -hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2 -hr hike back to your car. Assume the lake is 3 mi from your car. Let \(f(t)\) be your distance from the car \(t\) hours after 7 A.M. on Friday morning and let \(g(t)\) be your distance from the car \(t\) hours after 7 A.M. on Sunday morning. a. Evaluate \(f(0), f(2), g(0),\) and \(g(2)\). b. Let \(h(t)=f(t)-g(t) .\) Find \(h(0)\) and \(h(2)\). c. Use the Intermediate Value Theorem to show that there is some point along the trail that you will pass at exactly the same time of morning on both days.

$$\begin{aligned} &\text {a. Use a graphing utility to estimate } \lim _{x \rightarrow 0} \frac{\tan 2 x}{\sin x}, \lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin x}, \text { and }\\\ &\lim _{x \rightarrow 0} \frac{\tan 4 x}{\sin x} \end{aligned}$$ b. Make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{\tan p x}{\sin x},\) for any real constant \(p\)

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The magnitude of the electric field at a point \(x\) meters from the midpoint of a \(0.1-\mathrm{m}\) line of charge is given by \(E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}(\text { in units of newtons per coulomb }, \mathrm{N} / \mathrm{C}).\) Evaluate \(\lim _{x \rightarrow 10} E(x)\).

Suppose \(f\) is continuous at \(a\) and assume \(f(a)>0 .\) Show that there is a positive number \(\delta>0\) for which \(f(x)>0\) for all values of \(x\) in \((a-\delta, a+\delta) .\) (In other words, \(f\) is positive for all values of \(x\) in the domain of \(f\) and in some interval containing \(a .)\)
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