Question

A. Determine whether the Mean Value Theorem applies to the following function on the given interval left bracket a comma b right bracket[a,b].b. If? so, find or approximate the? point(s) that are guaranteed to exist by the Mean Value Theorem.c. Make a sketch of the function and the line that passes through left parenthesis a comma f left parenthesis a right parenthesis right parenthesis(a,f(a)) and left parenthesis b comma f left parenthesis b right parenthesis right parenthesis(b,f(b)). Mark the points P? (if they? exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P.f(x) = e^x ; [0,ln15]
Show step by step.

Answer 1

$f(x)=e^x$ is continuous and differentiable on its entire domain, so the mean value theorem holds for any interval you pick. We have

$f'(x)=e^x$

and the MVT says there is some $c\in(0,\ln15)$ such that

$f'(c)=\dfrac{f(\ln15)-f(0)}{\ln15-0}$

So we have

$e^c=\dfrac{15-1}{\ln15}=\dfrac{14}{\ln15}$

$\implies c=\ln\dfrac{14}{\ln15}\approx1.642$