Error

Calculus Level 4

Given that $f(x)$ satisfy the equation $\displaystyle \int f(x) \, dx = x f(x) + \dfrac{e^{-x^2}}{\sqrt \pi} + C,$ where $C$ is the constant of integration.

If we know that $f(0) = 0$ and $\displaystyle \lim_{n\to\infty} f(n) = 1$ , compute $f'(0)$ to 2 decimal places.

The answer is 1.13.

1 solution

Rishabh Jain
Feb 9, 2016

Differentiating both sides, we get: $\large \color{#EC7300}{f(x)}=xf'(x)+ \color{#EC7300}{f(x)}-2x\dfrac{e^{-x^2}}{\sqrt{\pi}}$ $\large \Rightarrow f'(x)=\dfrac{2e^{-x^2}}{\sqrt{\pi}}$ $\Large f'(0)=\dfrac{2}{\sqrt{\pi}}=\huge\boxed{\color{#007fff}{1.13}}\\ ~~~~~~~~~~~~~(\small\color{#D61F06}{{\text{to 2 decimal places}}})$

Two questions: (1) How do you know that $f$ is differentiable?, (2) Why can you divide by $x$ and then proceed to plug in $x=0$ ?

Otto Bretscher - 5 years, 4 months ago

Extra credit: find the function. Hint:the title

Aareyan Manzoor - 5 years, 4 months ago

Is it even possible to integrate f'(x) obtained?? (to find f(x)..

Rishabh Jain - 5 years, 4 months ago

error function

Aareyan Manzoor - 5 years, 4 months ago

Error

The answer is 1.13.

1 solution

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