Exam 1

1. Question

   What is the derivative of $f(x)=x^7{e}^{3.9x}$, evaluated at $x=0.61$?
   
   
   1. $3.89$
   2. $5.22$
   3. $4.23$
   4. $6.86$
   5. $3.01$
   
   

   Solution

   Using the product rule for $f(x)=g(x)·h(x)$, where $g(x):=x^7$ and $h(x):=e^{3.9x}$, we obtain
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{f\text{'}(x)}& =\hfill & [g(x)·h(x)]\text{'}=g\text{'}(x)·h(x)+g(x)·h\text{'}(x)\hfill \\ \multicolumn{1}{c}{}& =\hfill & 7x^{7-1}·e^{3.9x}+x^7·e^{3.9x}·3.9\hfill \\ \multicolumn{1}{c}{}& =\hfill & e^{3.9x}·(7x^6+3.9x^7)\hfill \\ \multicolumn{1}{c}{}& =\hfill & e^{3.9x}·x^6·(7+3.9x).\hfill \end{array}}$

   
   Evaluated at $x=0.61$, the answer is
   

   ${\displaystyle e^{3.9·0.61}·0.{61}^6·(7+3.9·0.61)=5.215814.}$

   
   Thus, rounded to two digits we have $f\text{'}(0.61)=5.22$.
   
   
   1. False
   2. True
   3. False
   4. False
   5. False