Question

The values of a function \(f(x)\) at three different values of \(x\) are given below:
\(f(0.90) = 7.75\), \(f(1.00) = 9.00\), \(f(1.10) = 10.35\).
Using the central difference approximation method, which is the CORRECT estimate of the first differential \(f'(1.00)\) and second differential \(f''(1.00)\) respectively?

• A. 10.00 and 13.00
• B. 13.00 and 10.00
• C. 10.00 and 10.00
• D. 13.00 and 13.00

Hint:

Use f'(x0) = (f(x0+h)-f(x0-h))/(2h) and f''(x0) = (f(x0+h)-2f(x0)+f(x0-h))/h^2.

Answer 1

The Correct Option is B

Step 1: Write down the central difference formula for the first derivative:
For equally spaced points with step size \(h\), the central difference approximation for the first derivative at \(x_0\) is:
\[ f'(x_0) \approx \frac{f(x_0+h) - f(x_0-h)}{2h} \]
Step 2: Identify the known values:
Here \(x_0 = 1.00\), \(h = 0.10\), \(f(x_0-h) = f(0.90) = 7.75\), \(f(x_0) = f(1.00) = 9.00\), \(f(x_0+h) = f(1.10) = 10.35\).
Step 3: Substitute into the first derivative formula:
\[ f'(1.00) \approx \frac{10.35 - 7.75}{2(0.10)} = \frac{2.60}{0.20} = 13.00 \]
Step 4: Write down the central difference formula for the second derivative:
\[ f''(x_0) \approx \frac{f(x_0+h) - 2f(x_0) + f(x_0-h)}{h^2} \]
Step 5: Substitute the known values into the second derivative formula:
\[ f''(1.00) \approx \frac{10.35 - 2(9.00) + 7.75}{(0.10)^2} = \frac{10.35 - 18.00 + 7.75}{0.01} = \frac{0.10}{0.01} = 10.00 \]
Step 6: Compare with the given options:
The estimated values are \(f'(1.00) = 13.00\) and \(f''(1.00) = 10.00\), which matches option (B).
Final Answer:
\[ \boxed{f'(1.00) = 13.00, \ f''(1.00) = 10.00} \]