Math Problem Statement
V 2 . 0 3 4
ا ل ش ك ل
ا ل ع ش ر ي
ا ل ط ب ي ع ي
ل ل ر ق م
4 5 . 2
ه و O a .
0 . 4 5 x 1 0 1
ي ا
ب .
0 . 4 5 x 1 0
2
ي ا
ج . 0 . 4 5 X 1 0
1
ي ا
د .
0 . 0 4 5
ب ا س ت خ د ا م
ت ق ر ي ب
ث ل ا ث ي
ا ل أ ر ق ا م
ل ل ت ق ر ي ب
ا ل أ و ل
ل ج ذ ر
ا ل م ع ا د ل ة f ( x )
0 ،
ب ط ر ي ق ة
ن ي و ت ن
ي ك و ن p 1
= 3 . 5 2
م ع pe
3 .
إ ذ ا
ك ا ن ت f ( 3 )
= 0 . 4 8 ،
ف أ ي
م م ا
ي ل ي
ي م ك ن
أ ن
ي ك و ن
ق ي م ة f ' ( 3 ) a .
UD 0 2
0 . 9 2 3
ي ا
ب .
0 . 8 2 2
ي ا
ج .
0 . 9 2
ي ا
د .
0 . 9 2 3
Solution
The problem appears to be related to Newton's method, which is used to approximate the roots of a function .
Given:
- with
The question asks for the value of , the derivative of at .
Newton's method formula for finding the next approximation is:
Using the information provided:
\quad \text{and} \quad p_e = 3$$ Given that $$p_2 = p_e = 3$$, this implies: $$3 = 3.52 - \frac{0.48}{f'(3)}$$ To solve for $$f'(3)$$: $$\frac{0.48}{f'(3)} = 3.52 - 3 = 0.52$$ $$f'(3) = \frac{0.48}{0.52}$$ Calculating this: $$f'(3) = \frac{0.48}{0.52} \approx -0.923$$ So the correct choice for $$f'(3)$$ is **Option A: -0.923**. Would you like more details on Newton's method or any other part of this problem? Here are some follow-up questions that might interest you: 1. How does Newton's method work for finding roots? 2. What are the limitations of Newton's method? 3. How do you interpret the derivative $$f'(x)$$ in this context? 4. What happens if $$f'(x)$$ is close to zero in Newton's method? 5. How would a different initial guess $$p_1$$ affect the convergence? **Tip:** Newton's method is highly sensitive to the choice of the initial guess. A poor guess can lead to divergence or slow convergence.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Newton's Method
Derivatives
Formulas
Newton's method formula: p_{n+1} = p_n - \frac{f(p_n)}{f'(p_n)}
Theorems
-
Suitable Grade Level
Advanced High School