Question

The instantaneous rate of change of f(x) =$e^x$ at x = a is given as $e^2$ Find the value of a.

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Answer 1

We will first find the instantaneous rate of change of f(x) = $e^x$ at x = a and then compare it with $e^2$ to find the value of a. We know that the instantaneous rate of change at x = a is defined as $f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{f(a+h)-f\left(a\right)}{h}$

$\Rightarrow f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{e^{a+h}-e^a}{h}{lim}_{h\to 0}{e}^a\frac{e^h-1}{h}=e^a$

This is given as $e^2$

$\Rightarrow e^a=e^2$

$\Rightarrow a=2$