### 1.3 If you thought practice makes perfect, you could be right.

Calculus is an integral part of a meteorologist’s training. The ability to solve problems with calculus differentiates meteorologists from weather readers. You should know how to perform both indefinite and definite integrals. Brush up on the derivatives for variables raised to powers, logarithms, and exponentials. We will take many derivatives with respect to time and to distance.

#### Need Extra Practice?

Visit the Khan Academy website that explains calculus with lots of examples, practice problems, and videos. You can start with single variable calculus, but may find it useful for more complicated calculus problems.

### Simple Integrals and Derivatives That are Frequently Used to Describe the Behavior of Atmospheric Phenomena

1.

$$\begin{array}{l}\frac{da}{dt}=-ka\\ \frac{da}{a}=-kdt\\ {\displaystyle \underset{{a}_{o}}{\overset{{a}_{1}}{\int}}\frac{da}{a}}=-{\displaystyle \underset{{t}_{o}}{\overset{{t}_{1}}{\int}}kdt}\\ \mathrm{ln}({a}_{1})-\mathrm{ln}({a}_{0})=-k\left({t}_{1}-{t}_{0}\right)\\ \mathrm{ln}({a}_{1}/{a}_{0})=-k\left({t}_{1}-{t}_{0}\right)\\ {a}_{1}/{a}_{0}={e}^{\left(-k\left({t}_{1}-{t}_{0}\right)\right)}=\mathrm{exp}\left(-k\left({t}_{1}-{t}_{0}\right)\right)\\ {a}_{1}={a}_{0}{e}^{\left(-k\left({t}_{1}-{t}_{0}\right)\right)}={a}_{0}\mathrm{exp}\left(-k\left({t}_{1}-{t}_{0}\right)\right)\end{array}$$

2. $p={p}_{o}{e}^{\left(-z/H\right)}\text{\hspace{1em}};\text{\hspace{1em}}{\displaystyle \underset{0}{\overset{\infty}{\int}}pdz=\text{\hspace{0.17em}}\text{?(Dothedefiniteintegral.)}}$

$${{\displaystyle \underset{0}{\overset{\infty}{\int}}p\text{\hspace{0.17em}}dz=-H{p}_{o}{e}^{-z/H}|}}_{0}^{\infty}=-H{p}_{o}\left(0-1\right)={p}_{o}H$$

3. $p={p}_{0}{e}^{\left(-\frac{z}{H}\right)};\frac{1}{p}\frac{dp}{dz}=?$

$$\frac{dp}{dz}=-\frac{1}{H}{p}_{0}{e}^{\frac{-z}{H}}=-\frac{1}{H}p;\frac{1}{p}\frac{dp}{dz}=-\frac{1}{H}$$

4. $\begin{array}{l}\frac{d\mathrm{ln}(ax)}{dt}=?\text{}\frac{d\mathrm{ln}(ax)}{dt}=\text{}\frac{1}{ax}\frac{d(ax)}{dt}=\text{}\frac{1}{ax}\frac{adx}{dt}=\frac{1}{x}u\text{,whereu=velocity}\\ \end{array}$

5. $d(\mathrm{cos}(x))=?\text{}d(\mathrm{cos}(x))=-\mathrm{sin}(x)dx\text{}$

### You have the power.

Often in meteorology and atmospheric science you will need to manipulate equations that have variables raised to powers. Sometimes, you will need to multiply variables at different powers together and then rearrange your answer to simplify it and make it more useful. In addition, it is very likely that you will need to invert an expression to solve for a variable. The following rules should remind you about powers of variables.

#### Laws of Exponents

$${a}^{x}{a}^{y}={a}^{x+y}$$ $${\left(ab\right)}^{x}={a}^{x}{b}^{x}$$ $${\left({a}^{x}\right)}^{y}={a}^{xy}$$ $${a}^{-x}=\frac{1}{{a}^{x}}$$ $$\frac{{a}^{x}}{{a}^{y}}={a}^{x-y}$$ $${a}^{0}=1$$ $${\left(\frac{a}{b}\right)}^{x}={a}^{x}{\left(\frac{1}{b}\right)}^{x}={\left(\frac{1}{a}\right)}^{-x}{b}^{-x}={\left(\frac{b}{a}\right)}^{-x}$$If $a={b}^{x}$ , then raise both sides to the exponent $\frac{1}{x}$ to move the exponent to the other side: ${a}^{\frac{1}{x}}={\left({b}^{x}\right)}^{\frac{1}{x}}={b}^{\frac{x}{x}}=b$

If ${a}^{x}{b}^{y}$ , and you want to get an equation with *a* raised to no power, then raise both sides to the exponent $\frac{1}{x}$ : ${\left({a}^{x}{b}^{y}\right)}^{\frac{1}{x}}={\left({a}^{x}\right)}^{\frac{1}{x}}{\left({b}^{y}\right)}^{\frac{1}{x}}=a{b}^{\frac{y}{x}}=\text{newconstant}$

This brief video (7:42) sums up these important rules:

Are you ready to give it a try? Solve the following problem on your own. After arriving at your own answer, click on the link to check your work. Here we go:

#### Check Your Understanding

$$x=a{y}^{b}$$What does y equal?

**Click for answer.**

$$\begin{array}{l}{x}^{1/b}={\left(a{y}^{b}\right)}^{1/b}={a}^{1/b}{\left({y}^{b}\right)}^{1/b}={a}^{1/b}y\\ y={x}^{1/b}/{a}^{1/b}={\left(\frac{x}{a}\right)}^{1/b}\end{array}$$

#### Quiz 1-2: Solving integrals and differentials.

Now it's time to to take another quiz. Again, I *highly* recommend that you begin by taking the Practice Quiz before completing the graded Quiz, since it will make you more competent and confident to take the graded Quiz : ).

- Go to the Canvas and find
**Practice Quiz 1-2**. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz. - When you feel you are ready, take
**Quiz 1-2**. You will be allowed to take this quiz only**once**. This quiz is timed, so after you start, you will have a limited amount of time to complete it and submit it. Good luck!