Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Differentials

Calculus

4. Applications of Derivatives

Differentials

Previous problemNext problem

3:39 minutes

Problem 2.82.2

Textbook Question

b. Estimate a solution to the equation in the given interval using a root finder.

x^5+7x+5=0; (−1,0)

Verified step by step guidance

First, understand the problem: We need to find a root of the equation x^5 + 7x + 5 = 0 within the interval (-1, 0). A root finder method, such as the bisection method, can be used to estimate the solution.

Start by evaluating the function at the endpoints of the interval. Calculate f(-1) and f(0) to determine if there is a sign change, which indicates a root exists between these points.

Apply the bisection method: Divide the interval into two halves and evaluate the function at the midpoint. If the function changes sign between the midpoint and one of the endpoints, the root lies in that subinterval.

Continue the bisection process: Repeatedly halve the interval and check for sign changes at the midpoints. This iterative process narrows down the interval where the root is located.

Stop the process when the interval is sufficiently small, or when the function value at the midpoint is close enough to zero, indicating an approximate root of the equation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Root Finding

Root finding is a numerical method used to determine the values of x for which a given function f(x) equals zero. Common techniques include the bisection method, Newton's method, and the secant method. These methods iteratively approximate the root by evaluating the function at specific points and refining the interval or estimate until a satisfactory level of accuracy is achieved.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, it must also take on any value between those two points. This theorem is crucial for establishing the existence of roots within a given interval. In the context of the equation x^5 + 7x + 5 = 0, verifying that the function changes sign over the interval (-1, 0) indicates that a root exists within that range.

Function Behavior and Graphing

Understanding the behavior of the function, including its continuity, increasing or decreasing nature, and critical points, is essential for effective root finding. Graphing the function can provide visual insights into where the function crosses the x-axis, indicating potential roots. Analyzing the function's derivative can also help identify intervals where the function is increasing or decreasing, aiding in the selection of appropriate root-finding methods.

Watch next

Master Finding Differentials with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

{Use of Tech} Investment problem A one-time investment of $2500 is deposited in a 5-year savings account paying a fixed annual interest rate r, with monthly compounding. The amount of money in the account after 5 years is a(r) = 2500(1 + r/12)⁶⁰.

a. Use Newton’s method to find the value of r if the goal is to have $3200 in the account after 5 years.

Textbook Question

How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

Textbook Question

The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of

a. the radius.

b. the surface area.

c. the volume.

Textbook Question

To find the height of a lamppost (see accompanying figure), you stand a 6-ft pole 20 ft from the lamp and measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.

<IMAGE>

Textbook Question

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.

f(x) = x ln (x + 1) -1 ; x₀ = 1.7

Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = cos x - x/7

Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x/6 - sec x on [0,8]

Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x⁵/5 - x³/4 - 1/20

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

b. Estimate a solution to the equation in the given interval using a root finder.x^5+7x+5=0; (−1,0)

Key Concepts

Root Finding

Intermediate Value Theorem

Function Behavior and Graphing

Watch next

b. Estimate a solution to the equation in the given interval using a root finder.

x^5+7x+5=0; (−1,0)