Use the NewtonRaphson method to find a numerical approximation to the solution of x2 7 = 0 that is correct to six decimal places.
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1
Review Problems
1.1
Preliminaries
1.2
Elementary Functions
1.3
Graphing
2
Review Problems
2.1
Exponential Growth and Decay
2.2
Sequences
2.3
More Population Models
3
Review Problems
3.1
Limits
3.2
Continuity
3.3
Limits at Infinity
3.4
The Sandwich Theorem and Some Trigonometric Limits
3.5
Properties of Continuous Functions
3.6
A Formal Definition of Limits (Optional)
4
Review Problems
4.1
Formal Definition of the Derivative
4.2
The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials
4.3
The Product and Quotient Rules, and the Derivatives of Rational and Power Functions
4.4
The Chain Rule and Higher Derivatives
4.5
Derivatives of Trigonometric Functions
4.6
Derivatives of Exponential Functions
4.7
Derivatives of Inverse Functions, Logarithmic Functions, and the Inverse Tangent Function
4.8
Linear Approximation and Error Propagation
5
Review Problems
5.1
Extrema and the Mean-Value Theorem
5.2
Monotonicity and Concavity
5.3
Extrema, Inflection Points, and Graphing
5.4
Optimization
5.5
LHospitals Rule
5.6
Difference Equations: Stability (Optional)
5.7
Numerical Methods: The NewtonRaphson Method (Optional)
5.8
Antiderivatives
6
Review Problems
6.1
The Definite Integral
6.2
The Fundamental Theorem of Calculus
6.3
Applications of Integration
7
Review Problems
7.1
The Substitution Rule
7.2
Integration by Parts and Practicing Integration
7.3
Rational Functions and Partial Fractions
7.4
Improper Integrals
7.5
Numerical Integration
7.6
The Taylor Approximation
7.7
Tables of Integrals (Optional)
8
Review Problems
8.1
Solving Differential Equations
8.2
Equilibria and Their Stability
8.3
Systems of Autonomous Equations (Optional)
9
Review Problems
9.1
Linear Systems
9.2
Matrices
9.3
Linear Maps, Eigenvectors, and Eigenvalues
9.4
Analytic Geometry
10
Review Problems
10.1
Functions of Two or More Independent Variables
10.2
Limits and Continuity
10.3
Partial Derivatives
10.4
Tangent Planes, Differentiability, and Linearization
10.5
More about Derivatives (Optional)
10.6
Applications (Optional)
10.7
Systems of Difference Equations (Optional)
11
Review Problems
11.1
Linear Systems: Theory
11.2
Linear Systems: Applications
11.3
Nonlinear Autonomous Systems: Theory
11.4
Nonlinear Systems: Applications
12
Review Problems
12.1
Counting
12.2
What Is Probability?
12.3
Conditional Probability and Independence
12.4
Discrete Random Variables and Discrete Distributions
12.5
Continuous Distributions
12.6
Limit Theorems
12.7
Statistical Tools
Textbook Solutions for Calculus For Biology and Medicine (Calculus for Life Sciences Series)
Chapter 5.7 Problem 8
Question
In Example 5, we considered the equation x4 x2 = 0 (a) What happens if you choose x0 = 1 2 _ 2 in the NewtonRaphson method? Give a graphical illustration. (b) Repeat the procedure in (a) for x0 = 0.71, and compare your result with the result we obtained in Example 5 when x0 = 0.70. Give a graphical illustration and explain it in words. What happens when x0 = 0.6? (This is an example in which small changes in the initial value can drastically change the outcome.)
Solution
The first step in solving 5.7 problem number 8 trying to solve the problem we have to refer to the textbook question: In Example 5, we considered the equation x4 x2 = 0 (a) What happens if you choose x0 = 1 2 _ 2 in the NewtonRaphson method? Give a graphical illustration. (b) Repeat the procedure in (a) for x0 = 0.71, and compare your result with the result we obtained in Example 5 when x0 = 0.70. Give a graphical illustration and explain it in words. What happens when x0 = 0.6? (This is an example in which small changes in the initial value can drastically change the outcome.)
From the textbook chapter Numerical Methods: The NewtonRaphson Method (Optional) you will find a few key concepts needed to solve this.
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Title
Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
Author
Claudia Neuhauser
ISBN
9780321644688
In Example 5, we considered the equation x4 x2 = 0 (a) What happens if you choose x0 = 1
Chapter 5.7 textbook questions
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Chapter 5: Problem 1 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
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Chapter 5: Problem 2 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
Use the NewtonRaphson method to find a numerical approximation to the solution of ex = x that is correct to six decimal places.
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Chapter 5: Problem 3 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
Use the NewtonRaphson method to find a numerical approximation to the solution of x2 + ln x = 0 that is correct to six decimal places.
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Chapter 5: Problem 4 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
The equation x2 5 = 0 has two solutions. Use the NewtonRaphson method to approximate the two solutions.
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Chapter 5: Problem 5 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
Use the NewtonRaphson method to solve the equation sin x = 1 2 x in the interval (0, ).
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Chapter 5: Problem 6 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
Let f (x) = _ _ x 1 for x 1 _ 1 x for x 1 (a) Show that if you use the NewtonRaphson method to solve f (x) = 0, then the following statement holds: If x0 = 1 + h, then x1 = 1 h, and if x0 = 1 h, then x1 = 1 + h. (b) Does the NewtonRaphson method converge? Use a graph to explain what happens.
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Chapter 5: Problem 7 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
In Example 4, we discussed the case of finding the root of x1/3 = 0. (a) Given x0, find a formula for |xn|. (b) Find lim n |xn| (c) Graph f (x) = x1/3 and illustrate what happens when you apply the NewtonRaphson method.
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Chapter 5: Problem 8 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
In Example 5, we considered the equation x4 x2 = 0 (a) What happens if you choose x0 = 1 2 _ 2 in the NewtonRaphson method? Give a graphical illustration. (b) Repeat the procedure in (a) for x0 = 0.71, and compare your result with the result we obtained in Example 5 when x0 = 0.70. Give a graphical illustration and explain it in words. What happens when x0 = 0.6? (This is an example in which small changes in the initial value can drastically change the outcome.)
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Chapter 5: Problem 9 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
Use the NewtonRaphson method to find a numerical approximation to the solution of x2 16 = 0 when your initial guess is (a) x0 = 3 and (b) x0 = 4.
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Chapter 5: Problem 10 Calculus For Biology and Medicine (Calculus for Life Sciences Series) 3
Suppose that you wish to use the NewtonRaphson method to solve f (x) = 0 numerically. It just so happens that your initial guess x0 satisfies f (x0) = 0. What happens to subsequent iterations? Give a graphical illustration of your results. [Assume that f _ (x0) _= 0.
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