Use Newton's method to approximate a root of the equation cos(đĽ2+5)=đĽ3 as follows:
Let đĽ1=1 be the initial approximation.
The second approximation đĽ2 is____
f(x) = cos(x^2+5)-x^3
f'(x) = -2x sin(x^2+5)-3x^2
x_n+1 = x_n - f(x_n)/f'(x_n)
x0 = 1
n...........x_n
1 0.98368
2 0.98315
3 0.98315
To use Newton's method to approximate the root of the equation cos(x^2+5) = x^3, we follow these steps:
1. Start with an initial approximation: đĽ1 = 1 (given)
2. Compute the function value and the derivative at đĽ1:
- Evaluate f(đĽ1) = cos(đĽ1^2 + 5) - đĽ1^3
- Evaluate f'(đĽ1), the derivative of f, which is needed for the Newton's method. Here, f'(đĽ) = -sin(đĽ^2 + 5) * 2đĽ - 3đĽ^2.
3. Use the formula for Newton's method to find the second approximation, đĽ2:
đĽ2 = đĽ1 - f(đĽ1) / f'(đĽ1)
Evaluate the expression and find đĽ2.
Note: Repeat steps 2 and 3 to find subsequent approximations until you reach the desired level of accuracy.
Let's calculate đĽ2 using the given information:
1. đĽ1 = 1 (given)
2. Evaluate f(đĽ1):
f(đĽ1) = cos(đĽ1^2 + 5) - đĽ1^3
= cos(1^2 + 5) - 1^3
= cos(6) - 1
â -0.982
Evaluate f'(đĽ1):
f'(đĽ1) = -sin(đĽ1^2 + 5) * 2đĽ1 - 3đĽ1^2
= -sin(1^2 + 5) * 2(1) - 3(1)^2
= -sin(6) * 2 - 3
â -4.029
3. Use the formula for Newton's method:
đĽ2 = đĽ1 - f(đĽ1) / f'(đĽ1)
= 1 - (-0.982) / (-4.029)
â 1.243
Therefore, the second approximation đĽ2 is approximately 1.243.
To use Newton's method to approximate the root of the equation, we need to follow these steps:
1. Start with an initial approximation, đĽ1 = 1.
2. Calculate the function value and its derivative at đĽ1:
đ(đĽ1) = cos(đĽ1^2 + 5) - đĽ1^3
đ'(đĽ1) = -2đĽ1sin(đĽ1^2 + 5) - 3đĽ1^2
3. Use the Newton's method formula to find the second approximation đĽ2:
đĽ2 = đĽ1 - (đ(đĽ1) / đ'(đĽ1))
Substitute the values:
đĽ2 = 1 - (cos(1^2 + 5) - 1^3) / (-2(1)sin(1^2 + 5) - 3(1)^2)
Calculate đĽ2 using the given formula.
Calculate đ(đĽ2) and đ'(đĽ2) to further refine the approximation if necessary.
Therefore, đĽ2 is the second approximation value obtained from the given formula.