/*! 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 12 True-False Determine whether the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 12

True-False Determine whether the statement is true or false. Explain your answer. If \(f\) is continuous on a closed interval \([a, b]\) and differentiable on \((a, b),\) then there is a point between \(a\) and \(b\) at which the instantaneous rate of change of \(f\) matches the average rate of change of \(f\) over \([a, b] .\)

Short Answer

Expert verified
True. The statement aligns with the Mean Value Theorem, given the conditions are satisfied.

Step by step solution

01

Understand the Statement

The statement is invoking the Mean Value Theorem (MVT) for derivatives. According to the MVT, if a function is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), there exists at least one point \(c\) in \((a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\). This means the instantaneous rate of change (the derivative) at some point \(c\) is equal to the average rate of change over the entire interval \([a, b]\).
02

Check the Conditions of the Mean Value Theorem

Verify that the conditions for applying the Mean Value Theorem are met: The function \(f\) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). The problem statement explicitly states that these conditions are met.
03

Apply the Mean Value Theorem

Since both conditions (continuity on \([a, b]\) and differentiability on \((a, b)\)) for the Mean Value Theorem are satisfied, we can apply the theorem. According to the MVT, there indeed exists at least one point \(c\) in \((a, b)\) for which \(f'(c) = \frac{f(b) - f(a)}{b - a}\). This satisfies the requirement that the instantaneous rate of change equals the average rate of change.
04

Conclusion

The statement is true because it correctly restates the conclusion of the Mean Value Theorem under the given conditions, which have been verified. Therefore, there is indeed a point \(c\) in \((a, b)\) where the derivative equals the average rate of change over \([a, b]\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuity
Continuity is a fundamental idea in calculus and mathematics as a whole. It means that a function doesn't have any sharp jumps, gaps, or breaks on the interval we are considering. More precisely, a function is continuous at a point if the limit of the function as it approaches that point from both sides is equal to the function's value at that point.
- If a function is continuous on a closed interval \([a, b]\), it means you can trace the function from start to finish without lifting your pencil.
For the Mean Value Theorem (MVT), continuity is crucial because without it, we could have sudden discontinuities that would make it impossible to find a point that equates the average and instantaneous rates of change.
- In simple words, continuity ensures that there is no sudden 'jump' in the value of the function, which is why it's a key requirement in many theorems like MVT.
Differentiability
Differentiability is another important concept, closely related to continuity. A function is differentiable at a point if it has a derivative there, meaning you can find a precise value for the slope of the tangent to the function at that specific point.
- If a function is differentiable on an interval \((a, b)\), it means we can calculate the derivative, or instantaneous rate of change, at every point within this interval.
It's important to note that if a function is differentiable at a point, it must also be continuous there. However, a function can be continuous at a point without being differentiable there 鈥� think of a sharp corner or cusp.
Differentiability is essential for the Mean Value Theorem because it allows us to confidently say that there exists a point within the interval where the instantaneous rate of change matches the average rate of change.
- Without differentiability, we wouldn't be able to find the 'c' where this equality holds.
Instantaneous Rate of Change
The instantaneous rate of change of a function at a given point is the value of the derivative at that point. It represents the slope of the tangent line to the curve at that particular point.
When we talk about the instantaneous rate of change, we mean the rate at which the function's value is changing right at that moment.
- Mathematically, it's expressed as \(f'(c)\), where \(c\) is the point of interest.
The Mean Value Theorem relates this instantaneous rate of change to the average rate of change over the interval \([a, b]\).
The average rate of change is found by the formula \(\frac{f(b) - f(a)}{b - a}\), representing how much the function changes, on average, per unit interval.
- According to the MVT, there's at least one point within the interval where these two rates are equal, linking the concept of instantaneous rate of change to the bigger picture of the function's overall change.

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

Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=\sqrt{x+1} ;[0,3] $$

Let \(f(x)=x^{3}-4 x\) (a) Find the equation of the secant line through the points \((-2, f(-2))\) and \((1, f(1))\). (b) Show that there is only one point \(c\) in the interval \((-2,1)\) that satisfies the conclusion of the Mean-Value Theorem for the secant line in part (a). (c) Find the equation of the tangent line to the graph of \(f\) at the point \((c, f(c)) .\) (d) Use a graphing utility to generate the secant line in part (a) and the tangent line in part (c) in the same coordinate system, and confirm visually that the two lines seem parallel.

Sketch a reasonable graph of \(s\) versus \(t\) for a mouse that is trapped in a narrow corridor (an \(s\) -axis with the positive direction to the right) and scurries back and forth as follows. It runs right with a constant speek and forth as for a while, then gradually slows down \(t 0.6 \mathrm{m} / \mathrm{s}\), then quickly speeds up to \(2.0 \mathrm{m} / \mathrm{s}\), then gradually slows to a stop but immediately reverses direction and quickly speeds up to \(1.2 \mathrm{m} / \mathrm{s}\).

An annuity is a sequence of equal payments that are paid or received at regular time intervals. For example, you may want to deposit equal amounts at the end of each year into an interest-bearing account for the purpose of accumulating a lump sum at some future time. If, at the end of each year, interest of \(i \times 100 \%\) on the account balance for that year is added to the account, then the account is said to pay \(i \times 100 \%\) interest, compounded annually. It can be shown that if payments of \(Q\) dollars are deposited at the end ofeach year into an account that pays \(i \times 100 \%\) compounded annually, then at the time when the \(n\) th payment and the accrued interest for the past year are deposited, the amount \(S(n)\) in the account is given by the formula $$S(n)=\frac{Q}{i}\left[(1+i)^{n}-1\right]$$ Suppose that you can invest \(\$ 5000\) in an interest-bearing account at the end of each year, and your objective is to have \(\$ 250,000\) on the 25th payment. Approximately what annual compound interest rate must the account pay for you to achieve your goal? [Hint: Show that the interest rate \(i\) satisfies the equation \(50 i=(1+i)^{25}-1,\) and solve it using Newton's Method.]

If \(p(x)\) is a polynomial, discuss the usefulness of knowing zeros for \(p, p^{\prime},\) and \(p^{\prime \prime}\) when determining information about the graph of \(p .\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.