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%  Exponential Functions                           %
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\problemgrouptitle{Exponential Functions}

\begin{problemblock}[3]{Evaluate the given limit.}

\item $\displaystyle \lim_{x \to -\infty}{e^{-x}}$
\answer{
$\displaystyle \infty$
}

\item $\displaystyle \lim_{x \to \infty}{e^{-x^6}}$
\answer{
$\displaystyle 0$
}

\item $\displaystyle \lim_{x \to \infty}{5^{e^{-x}}}$
\answer{
$\displaystyle 1$
}

\end{problemblock}

\begin{problemblock}[2]{Differentiate the given function.}

\item $f(x)=\left(x^4+1 \right) e^{-2x}$    
\answer{
$f'(x)=\left(4x^3-2x^4-2 \right) e^{-2x}$
}

\item $\displaystyle g(t)=\frac{2t^2+e^{3t}}{t^2 e^{3t}}$    
\answer{
$\displaystyle g'(t)=-6 e^{-3t}-\frac 2{t^3}$
}

\item $\displaystyle y=\frac{e^x+1}{e^x+3}$    
\answer{
$\displaystyle y'=\frac {2 e^x}{\left(e^x+3\right)^2}$
}

\item $f(x)=\cos\left(e^{2x}+x\right)$    
\answer{
$f'(x)=-\left(2e^{2x}+1\right) \sin\left(e^{2x}+x\right)$
}

\item $g(x)=\tan \left(e^x\right)+e^{\tan x}$    

\answer{
$g'(x)=e^x \sec^2\! \left(e^x\right)+\sec^2\! x \,  e^{\tan x}$
}

\item $y=\sin \sqrt{x^2+e^{2x}}$    

\answer{
$y'=\left(x+e^{2x}\right)\left( x^2+e^{2x}\right)^{-1/2} \cos \sqrt{x^2+e^{2x}}$
}

\item $\displaystyle f(x)=\frac{e^x+e^{-x}}{e^x+3e^{-x}}$    
\answer{
$\displaystyle f'(x)=\frac 4{\left( e^x+3e^{-x}\right)^2}$
}

\end{problemblock}


\begin{problemblock}{Find the equation of the tangent line to the curve at the given point.}

\item $y=x-e^{-x},\,\,\,\,\, (0,\,-1)$    
\answer{
$y=2x-1$
}

\item $x-xy+e^y+e^x=2e,\,\,\,\, (1,\, 1)$   

\answer{
$\displaystyle y=-\frac{e}{e-1}x+\frac{2e-1}{e-1}$
}

\end{problemblock}


\begin{problemblock}[2]{Evaluate the given integral.}

\item $\displaystyle\int \left( e^{2x}+e^{-x} \right)^2 \,dx$    
\answer{
$\frac{1}{4}e^{4x}+2e^x-\frac{1}{2}e^{-2x}+C$
}

\item $\displaystyle\int x e^{x^2} \,dx$    
\answer{
$\frac{1}{2} e^{x^2}+C$
}

\item $\displaystyle\int e^x \sin \left( e^x \right) \,dx$    
\answer{
$-\cos \left(e^x\right)+C$
}

\item $\displaystyle\int \frac{\left(e^{2x}+1\right)^2}{e^x} \,dx$    
\answer{
$\displaystyle \frac{1}{3} e^{3x}+2e^x-e^{-x}+C$
}

\item $\displaystyle\int_{0}^{1} \frac{e^{3x}}{\left(e^{3x}+2\right)^2} \,dx$    
\answer{
$\displaystyle\frac{1}{9}- \frac 1{3\left(e^3+2\right)}$
}

\item $\displaystyle\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \,dx$    
\answer{
$2e^2-2e$
}

\item $\displaystyle\int \left(1+e^{\tan x} \right) \sec^2\! x  \,dx$    
\answer{
$\tan x+e^{\tan x}+C$
}

\item $\displaystyle\int \frac 1{\left( e^x+e^{-x}\right)^2}  \,dx$    
\answer{
$\displaystyle -\frac 1{2\left(e^{2x}+1 \right)}+C$
}

\end{problemblock}

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