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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1:43 minutes

Problem 4.8.5

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>

Verified step by step guidance

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

First, we need to find the derivative of the function f(x) = 2x^3 - 6x^2 + 4x. The derivative, f'(x), is calculated as follows: f'(x) = \frac{d}{dx}(2x^3 - 6x^2 + 4x) = 6x^2 - 12x + 4.

Next, substitute x_0 = 1.4 into the function f(x) and its derivative f'(x) to find f(1.4) and f'(1.4).

Calculate f(1.4) = 2(1.4)^3 - 6(1.4)^2 + 4(1.4) and f'(1.4) = 6(1.4)^2 - 12(1.4) + 4.

Finally, use the Newton's method formula to find x_1: x_1 = 1.4 - \frac{f(1.4)}{f'(1.4)}. This will give you the next approximation of the root, x_1.

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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 approximate the roots of a real-valued function. It starts with an initial guess and uses the function's derivative to find successively better approximations. The formula for updating the guess is x₁ = x₀ - f(x₀)/f'(x₀), where f' is the derivative of f. This method is particularly effective when the initial guess is close to the actual root.

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In the context of Newton's Method, the derivative is crucial for determining the slope of the tangent line at the current guess, which guides the next approximation towards the root.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. In Newton's Method, the tangent line at the point (x₀, f(x₀)) is used to find the next approximation x₁. The intersection of this tangent line with the x-axis provides a new estimate for the root, illustrating how the method converges to the actual solution.

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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) = tan x - 2x; x₀ = 1.2

Textbook Question

f(x) = cos⁻¹ x - x; x₀ = 0.75

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{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

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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>

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{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)

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{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

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{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.

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Textbook Question

y = 1/x and y = 4 - x²

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