Open in App
Log In Start studying!

Select your language

Suggested languages for you:

Problem 122

Obtain an approximate value for \(\sqrt{105}\) to within \(.01\) by using the Mean Value Theorem.

Short Answer

Expert verified
Using the Mean Value Theorem on the function \(f(x) = x^2\), we find an estimate of the desired value within the interval [10, 11] to be 10.5. This estimate is not accurate to within 0.01, but it can be used as a starting point for more refined approximations.
See the step by step solution

Step by step solution

Unlock all solutions

Get unlimited access to millions of textbook solutions with Vaia Premium

Over 22 million students worldwide already upgrade their learning with Vaia!

Step 1: Define the function representing the square root.

Let's use the function \(f(x)=x^2\), where \(x^2=105\). We want to find the value of x such that \(x^2\) is 105.

Step 2: Determine the interval containing the desired value.

We know that 10^2 = 100 and 11^2 = 121. Since 105 is between 100 and 121, the desired value must be between 10 and 11. So, the interval we will work on is [10, 11].

Step 3: Apply the Mean Value Theorem.

Applying the Mean Value Theorem to the function \(f(x)=x^2\) on the interval [10, 11], we have: \(f'(c) = \frac{f(11) - f(10)}{11 - 10}\) for some c in (10, 11), where f'(c) denotes the derivative of the function with respect to x. Now, let's find the derivative of \(f(x) = x^2\): \(f'(x) = 2x\) We can now plug this into our Mean Value Theorem equation: \(2c = \frac{(11)^2 - (10)^2}{11 - 10}\) Solve for c: \(2c = \frac{121 - 100}{1}\) \(2c = 21\) \(c = 10.5\) Using the Mean Value Theorem, we've found that 10.5 is an estimate of the desired value, which should be to within 0.01.

Step 4: Check the accuracy of the approximation.

We can verify that our estimate is accurate within 0.01 by computing the square of our estimate, which should be close enough to 105. \((10.5)^2 = 110.25\) Our estimate is within 5.25 of 105, which is larger than the desired level of accuracy (0.01). However, we can use the estimate, 10.5, as the starting point for a more refined approximation, such as using the bisection method or Newton's method, to obtain a more accurate value of the square root of 105.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Access millions of textbook solutions in one place

  • Access over 3 million high quality textbook solutions
  • Access our popular flashcard, quiz, mock-exam and notes features
  • Access our smart AI features to upgrade your learning
Get Vaia Premium now
Access millions of textbook solutions in one place

Most popular questions from this chapter

Chapter 5

Rewrite the polynomial $^{\mathrm{n}} \sum_{\mathrm{i}=0} \alpha_{\mathrm{i}} \mathrm{t}^{\mathrm{t}}$ as a polynomial in \(\mathrm{x}=\mathrm{t}-1\) Verify this for the polynomial $1+\mathrm{t}+3 \mathrm{t}^{4}$.

Chapter 5

Let the vector-valued function $\mathrm{f}: \mathrm{R}^{\mathrm{n}} \rightarrow \mathrm{R}^{\mathrm{m}}$ be defined by $\mathrm{f}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right)=\left\\{\mathrm{f}_{1}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right), \ldots, \mathrm{f}_{\mathrm{m}}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right)\right\\}$ (a) Show that \(\mathrm{f}\) is differentiable at $\mathrm{a} \in \mathrm{R}^{\mathrm{n}}\( if and only if each \)\mathrm{f}_{\mathrm{i}}(1 \leq \mathrm{i} \leq \mathrm{m})$ is differentiable at a and \(J_{f}(a)=\left\\{J_{(f) 1}(a), \ldots, J_{(f) m}(a)\right\\}\) (b) Show that this derivative \(\mathrm{J}_{\mathrm{f}}\) (a) \((\mathrm{x}-\mathrm{a})\) is unique. \\{Note: (b) does not depend on (a).\\}

Chapter 5

(a) State and prove Euler's Theorem on positively homogeneous functions of two variables. (b) Let \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) be positively homogeneous of degree 2 and $\mathrm{u}=\mathrm{r}^{\mathrm{m}} \mathrm{F}(\mathrm{x}, \mathrm{y})\( where \)\mathrm{r}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{1 / 2}$. Show that $\left(\partial^{2} \mathrm{u} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{u} / \partial \mathrm{y}^{2}\right)$ $=\mathrm{r}^{\mathrm{m}}\left\\{\left(\partial^{2} \mathrm{~F} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{~F} / \partial \mathrm{y}^{2}\right)\right\\}+\mathrm{m}(\mathrm{m}+4) \mathrm{r}^{\mathrm{m}-2} \mathrm{~F}$

Chapter 5

Show that the functions $\mathrm{f}, \mathrm{g} \in \mathrm{C}^{1}(\mathrm{E}), \mathrm{E}\( open in \)\mathrm{R}^{2}$, are functionally dependent (i.e., there exists a function \(\mathrm{F}\) such that \(g=F^{\circ} \mathrm{f}\) ) if det \(J \phi(x, y)=0\) for \(\Phi=(f, g)\) and $(x, y)\( in some neighborhood of \)(a, b)\(, where \)(\partial f / \partial x)(a, b) \neq 0$

Chapter 5

State and prove L'Hospital's Rule for the indeterminant forms $(O / O) ;(\infty / \infty)$

Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

  • Flashcards & Quizzes
  • AI Study Assistant
  • Smart Note-Taking
  • Mock-Exams
  • Study Planner
Join over 22 million students in learning with our Vaia App Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks