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

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2:18 minutes

Problem 4.8.3

Textbook Question

A graph of ƒ and the lines tangent to ƒ at x = 1, 2 and 3 are given. If x₀ = 3, find the values of x₁, x₂, and x₃, that are obtained by applying Newton’s method. <IMAGE>

Verified step by step guidance

Understand Newton's Method: Newton's method is an iterative process used to find successively better approximations to the roots (or zeroes) of a real-valued function. The formula for Newton's method is: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.

Identify the initial guess: In this problem, the initial guess x₀ is given as 3. This is the starting point for applying Newton's method.

Calculate f(x₀) and f'(x₀): To apply Newton's method, you need the value of the function f at x₀ and the value of its derivative f' at x₀. Use the graph or any given information to determine these values.

Apply Newton's formula: Use the values obtained in the previous step to calculate x₁ using the formula x₁ = x₀ - \frac{f(x₀)}{f'(x₀)}. This will give you the next approximation.

Repeat the process: Use x₁ as the new approximation and repeat the process to find x₂ and x₃. For each step, calculate f(x_n) and f'(x_n), then apply the formula x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} to find the next approximation.

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Key Concepts

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

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It starts with an initial guess and refines it using the formula x_{n+1} = x_n - f(x_n)/f'(x_n), where f is the function and f' is its derivative. This method is particularly effective for finding roots of functions and converges quickly under suitable conditions.

Tangent Lines

A tangent line to a function at a given point represents the instantaneous rate of change of the function at that point. It is defined by the slope, which is the derivative of the function at that point. In the context of Newton's Method, the tangent line at the current approximation provides a linear approximation of the function, guiding the next iteration towards the root.

Derivatives

The derivative of a function measures how the function's output changes as its input changes, essentially representing the slope of the function at any given point. It is a fundamental concept in calculus that allows us to analyze the behavior of functions, including identifying local maxima and minima, and is crucial for applying Newton's Method effectively.

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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) = sin x + x - 1; x₀ = 0.5

Textbook Question

f(x) = tan x - 2x; x₀ = 1.2

Textbook Question

f(x) = cos⁻¹ x - x; x₀ = 0.75

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 2x - x² + 2x

Textbook Question

Let ƒ(x) = 2x³ - 6x² + 4x. Use Newton’s method to find x₁ given that x₀ = 1.4. Use the graph of f (see figure) and an appropriate tangent line to illustrate how x₁ is obtained from x₀ . <IMAGE>

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) = e⁻ˣ - ((x + 4)/5)

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) = ln x - x² + 3x - 1

Textbook Question

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.

y = sin x and y = x/2

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