Math  /  Calculus

Question(1 point)
Use Newton's method to find a solution of the equation: log(x)=42x\log (x)=4-2 x. Start with x0=3x_{0}=3. x1=x_{1}= \square x2=x_{2}= \square x3=x_{3}= \square Note: remember WeBWork can perform a lot of tedious computations for you. Copying and pasting your answer from a previous entry can help you input the next entry without having to do too much simplification on your own.

Studdy Solution

STEP 1

1. We are using Newton's method to approximate the roots of the equation log(x)=42x \log(x) = 4 - 2x .
2. The initial guess for the solution is x0=3 x_0 = 3 .
3. The function can be expressed as f(x)=log(x)4+2x f(x) = \log(x) - 4 + 2x .
4. The derivative of the function, f(x) f'(x) , is needed for Newton's method.

STEP 2

1. Define the function and its derivative.
2. Apply Newton's method to find x1 x_1 .
3. Apply Newton's method to find x2 x_2 .
4. Apply Newton's method to find x3 x_3 .

STEP 3

Define the function f(x) f(x) and its derivative f(x) f'(x) .
f(x)=log(x)4+2x f(x) = \log(x) - 4 + 2x
Calculate the derivative:
f(x)=1x+2 f'(x) = \frac{1}{x} + 2

STEP 4

Apply Newton's method to find x1 x_1 using the formula:
xn+1=xnf(xn)f(xn) x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
Substitute x0=3 x_0 = 3 into the formula:
f(3)=log(3)4+2×3 f(3) = \log(3) - 4 + 2 \times 3 f(3)=log(3)+64 f(3) = \log(3) + 6 - 4 f(3)=log(3)+2 f(3) = \log(3) + 2
f(3)=13+2 f'(3) = \frac{1}{3} + 2 f(3)=13+63 f'(3) = \frac{1}{3} + \frac{6}{3} f(3)=73 f'(3) = \frac{7}{3}
Calculate x1 x_1 :
x1=3log(3)+273 x_1 = 3 - \frac{\log(3) + 2}{\frac{7}{3}} x1=33(log(3)+2)7 x_1 = 3 - \frac{3(\log(3) + 2)}{7}

STEP 5

Apply Newton's method to find x2 x_2 using the updated x1 x_1 .
Substitute x1 x_1 into the formula:
f(x1)=log(x1)4+2x1 f(x_1) = \log(x_1) - 4 + 2x_1 f(x1)=1x1+2 f'(x_1) = \frac{1}{x_1} + 2
Calculate x2 x_2 :
x2=x1f(x1)f(x1) x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}

STEP 6

Substitute the values to find x2 x_2 :
x2=x1log(x1)4+2x11x1+2 x_2 = x_1 - \frac{\log(x_1) - 4 + 2x_1}{\frac{1}{x_1} + 2}

STEP 7

Apply Newton's method to find x3 x_3 using the updated x2 x_2 .
Substitute x2 x_2 into the formula:
f(x2)=log(x2)4+2x2 f(x_2) = \log(x_2) - 4 + 2x_2 f(x2)=1x2+2 f'(x_2) = \frac{1}{x_2} + 2
Calculate x3 x_3 :
x3=x2f(x2)f(x2) x_3 = x_2 - \frac{f(x_2)}{f'(x_2)}

STEP 8

Substitute the values to find x3 x_3 :
x3=x2log(x2)4+2x21x2+2 x_3 = x_2 - \frac{\log(x_2) - 4 + 2x_2}{\frac{1}{x_2} + 2}
The values for x1 x_1 , x2 x_2 , and x3 x_3 are obtained by substituting and calculating the expressions derived in each step. You can use a calculator or computational tool to evaluate these expressions for precise numerical answers.

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