1.
The Power Rule for derivatives states:
\begin{gather*}
\frac{d}{dx}\left( x^n \right) =
\end{gather*}
Worksheet Calculus Review
In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.
2.
The Power Rule for antiderivatives states:
\begin{gather*}
\int\left( x^n \right)\ dx =
\end{gather*}
3.
Practice with the following derivatives:
(a)
\(\displaystyle \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) = \)
(b)
\(\displaystyle \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) = \)
4.
Practice with the following antiderivatives:
(a)
\(\displaystyle \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx = \)
(b)
\(\displaystyle \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx = \)
Theorem 90. Fundamental Theorem of Calculus.
If \(F(x)\) is an antiderivative of \(f(x)\)—i.e., if \(F'(x) = f(x)\)—then:
\begin{gather*}
\int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a).
\end{gather*}
If \(f(x)\) is a probability density function (don’t worry about what that means for now), then integrals such as \(\int_a^b f(x)\ dx\) are used to calculate certain probabilities, as are values of the form \(F(x)\text{.}\)
5.
Practice with the following integrals:
(a)
\(\displaystyle \int_0^1 \frac{3}{2} x^{3/2} \ dx = \)
(b)
\(\displaystyle \int_0^x 3e^{-3t} \ dt = \)
