A differentiable function $f(x)$ is such that $f(-0.1) = 2, f(0) = 1, \text{ and } f(0.1) = 2$. Then, $\frac{d^2f}{dx^2}$ at $x = 0$ is ______ (answer in integer).
Approximating Second Derivative Using Central Difference
The problem asks for the value of the second derivative, $\frac{d^2f}{dx^2}$, at $x = 0$. We are given values of the differentiable function $f(x)$ at three points: $f(-0.1) = 2$, $f(0) = 1$, and $f(0.1) = 2$.
We can approximate the second derivative using the central difference formula. This formula is suitable when we have function values at equally spaced points around the point of interest.
Applying the Central Difference Formula
The central difference formula for the second derivative is:
$ \frac{d^2f}{dx^2}(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} $In this problem:
- The point of interest is $x = 0$.
- The step size is $h = 0.1$, as the points are $-0.1, 0, 0.1$.
- $f(x+h) = f(0.1) = 2$.
- $f(x) = f(0) = 1$.
- $f(x-h) = f(-0.1) = 2$.
Calculating the Second Derivative
Substitute the values into the formula:
$ \frac{d^2f}{dx^2}(0) \approx \frac{f(0.1) - 2f(0) + f(-0.1)}{(0.1)^2} $ $ \frac{d^2f}{dx^2}(0) \approx \frac{2 - 2(1) + 2}{(0.1)^2} $ $ \frac{d^2f}{dx^2}(0) \approx \frac{2 - 2 + 2}{0.01} $ $ \frac{d^2f}{dx^2}(0) \approx \frac{2}{0.01} $ $ \frac{d^2f}{dx^2}(0) \approx 200 $The approximated value of the second derivative at $x = 0$ is 200.
Important Questions from Differentiability
What is the value of f'(x) at x = 4 from the following table of values?
x 1 2 3 4 f(x) 20 22 27 35 The set of all points, where the function \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {1 - {{\rm{e}}^{ - {{\rm{x}}^2}}}} \) is differentiable, is
Let f be a differentiable function defined for all x ∈ R such that f(x3) = x5 for all x ∈ R, x ≠ 0. Then the value of \(\dfrac{df}{dx} (8)\) is:
If \(f(x)=\displaystyle\sum_{n-0}^{2k}\left(a_n|x|^n+b_n\ \sin^2x\right)\), where \(a_i^{'}\)s and \(b_i^{'}\)s (0 ≤ i ≤ k) are real constants, then f(x) is:
The set of all point where the function f(x) = 2x|x| is differentiable, is:
