# 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. $$\frac{d a}{d t}=-k a$$

$$\frac{d a}{a}=-k d t$$

$$\int_{a_{o}}^{a_{1}} \frac{d a}{a}=-\int_{t_{o}}^{t_{1}} k d t$$

$$\ln \left(a_{1}\right)-\ln \left(a_{0}\right)=-k\left(t_{1}-t_{0}\right)$$

$$\ln \left(a_{1} / a_{0}\right)=-k\left(t_{1}-t_{0}\right)$$

$$a_{1} / a_{0}=e^{\left(-k\left(t_{1}-t_{0}\right)\right)}=\exp \left(-k\left(t_{1}-t_{0}\right)\right)$$

$$a_{1}=a_{0} e^{\left(-k\left(t_{1}-t_{0}\right)\right)}=a_{0} \exp \left(-k\left(t_{1}-t_{0}\right)\right)$$

2. $$p=p_{o} e^{(-z / H)} \quad ; \quad \int_{0}^{\infty} p d z=? \quad$$ (Do the definite integral.)

$$\int_{0}^{\infty} p d z=-\left.H p_{o} e^{-2 I H}\right|_{0} ^{\infty}=-H p_{o}(0-1)=p_{o} H$$

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

$$\frac{d p}{d z}=-\frac{1}{H} p_{0} e^{\frac{-z}{H}}=-\frac{1}{H} p ; \frac{1}{p} \frac{d p}{d z}=-\frac{1}{H}$$

4. $$\frac{d \ln (a x)}{d t}=? \quad \frac{d \ln (a x)}{d t}=\frac{1}{a x} \frac{d(a x)}{d t}=\frac{1}{a x} \frac{a d x}{d t}=\frac{1}{x} u,$$ where $$u=$$ velocity

5. $$d(\cos (x))=? \quad d(\cos (x))=-\sin (x) d x$$

### 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

\begin{aligned} a^{x} a^{y} &=a^{x+y} \\(a b)^{x} &=a^{x} b^{y} \\\left(a^{x}\right)^{y} &=a^{x y} \\ a^{-x} &=\frac{1}{a^{x}} \\ \frac{a^{x}}{a^{y}} &=a^{x-y} \\ a^{0} &=1 \end{aligned}

$$\left(\frac{a}{b}\right)^{x}=a^{x}\left(\frac{1}{b}\right)^{x}=\left(\frac{1}{a}\right)^{-x} b^{-x}=\left(\frac{b}{a}\right)^{-x}$$

\begin{aligned} \text { If } a=& b^{x}, \text { then raise both sides to the exponent } \frac{1}{x} \text { to move the } \\ & \text { exponent to the other side: } a^{\frac{1}{x}}=\left(b^{x}\right)^{\frac{1}{x}}=b^{\frac{x}{x}}=b \end{aligned}

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 { new constant }

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

Rules of Exponents

Click Answer for transcript of the Rules of Exponents.

Exercise

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

$$x^{1 / b}=\left(a y^{b}\right)^{1 / b}=a^{1 / b}\left(y^{b}\right)^{1 / b}=a^{1 / b} y$$

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

#### 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 : ).

1. 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.
2. 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!

This page titled 1.3: If you thought practice makes perfect, you could be right is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by William Brune (John A. Dutton: e-Education Institute) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.