The graph below shows an exponential function \(f(x)=a^x\) and its derivative \(f'(x)\text{.}\) Choose all the options that describe the constant \(a\text{.}\)
\begin{align*}
\amp (a)\;\; a \lt 0\amp
\amp (b)\;\;a \gt 0\amp
\amp (c)\;\;a \lt 1\\
\amp (d)\;\;a \gt 1\amp
\amp (e)\;\; a \lt e\amp
\amp(f)\;\; a \gt e
\end{align*}
3
True or false: \(\ds\diff{}{x}\{e^x\}=xe^{x-1}\)
4
A population of bacteria is described by \(P(t)=100e^{0.2t}\text{,}\) for \(0 \leq t \leq 10\text{.}\) Over this time period, is the population increasing or decreasing?
We will learn more about the uses of exponential functions to describe real-world phenomena in Section 3.3.
Exercises — Stage 2
5
Find the derivative of \(f(x)=\dfrac{e^{x}}{2x}\) .
6
Differentiate \(f(x)=e^{2x}\text{.}\)
7
Differentiate \(f(x)=e^{a+x}\text{,}\) where \(a\) is a constant.
8
For which values of \(x\) is the function \(f(x)=xe^x\) increasing?
9
Differentiate \(e^{-x}\text{.}\)
10
Differentiate \(f(x)=(e^x+1)(e^x-1)\text{.}\)
11
A particle's position is given by
\begin{equation*}
s(t)=t^2e^t.
\end{equation*}
When is the particle moving in the negative direction?
Exercises — Stage 3
12
Let \(g(x)=f(x)e^x\text{,}\) for a differentiable function \(f(x)\text{.}\) Give a simplified formula for \(g'(x)\text{.}\)
Functions of the form \(g(x)\) are relatively common. If you remember this formula, you can save yourself some time when you need to differentiate them. We will explore this more in Question 2.14.2.19, Section 2.14.
13
Which of the following functions describe a straight line?
