diff --git a/meanValueTheorem/digInMeanValueTheorem.tex b/meanValueTheorem/digInMeanValueTheorem.tex index 9ab6f39..8d3a2f7 100644 --- a/meanValueTheorem/digInMeanValueTheorem.tex +++ b/meanValueTheorem/digInMeanValueTheorem.tex @@ -298,7 +298,7 @@ We calculate $f(0) = \answer[given]{1}$ and $f(3)=\answer[given]{e^3}$. That means the average rate of change of $f$ on the interval $[0, 3]$ is $\dfrac{f(3)-f(0)}{3-0}=\answer[given]{\dfrac{e^3-1}{3}}$. - The value of $c$ will be in the interval $(0, 3)$ will satisfy $f'(c) = \dfrac{e^3-1}{3}$. + The value of $c$ will be in the interval $(0, 3)$ and will satisfy $f'(c) = \dfrac{e^3-1}{3}$. Since $f'(x) = e^x$, this means $c$ will be the solution to the equation: $\displaystyle e^c = \dfrac{e^3-1}{3}$. Taking the natural logarithm of both sides of this equation gives: \[c = \ln\left( \dfrac{e^3-1}{3} \right). \] @@ -312,7 +312,7 @@ In conclusion, the Mean Value Theorem relates the function $f$ and its derivative, $f'$. Since the derivative has many interpretations, e.g. instantaneous rate of change, slope of the tangent line, velocity, it is no surprise that we can use the MVT in different contexts. -Therefore, if the the function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then the MVT guarantees that +Therefore, if the function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then the MVT guarantees that \begin{enumerate} \item there is a point $c$ in $(a,b)$ where the instantaneous rate of change of $f$ is equal to the average rate of change of $f$ over the interval $[a,b]$;