Review of First Semester Calculus

Section1.1Review of First Semester Calculus

Subsection1.1.1Graphing

We'll need to know how to graph by hand some basic functions. If you have not spent much time graphing functions by hand before this class, then you should spend some time graphing the following functions by hand. When we start drawing functions in 3D, we'll have to piece together infinitely many 2D graphs. Knowing the basic shape of graphs will help us do this.

Exercise1.1.1

(a)

Provide a rough sketch of the following functions, showing their basic shapes:

\begin{equation*} \displaystyle x^2, x^3, x^4, \frac{1}{x}, \sin x, \cos x, \tan x, \sec x, \arctan x, e^x,\ln x. \end{equation*}

(b)

Use a computer algebra system, such as Wolfram Alpha, to verify your work. I suggest Wolfram Alpha, because it is now built into Mathematica 8.0+. If you can learn to use Wolfram Alpha, you will be able to use Mathematica.

You are also welcome to use Maple or Sage. On the surface both Mathematica and Maple are very similar, while Sage is meant to be more multi-purpose.

Subsection1.1.2Derivatives

In first semester calculus, one of the things you focused on was learning to compute derivatives. You'll need to know the derivatives of basic functions (found on the end cover of almost every calculus textbook). Computing derivatives accurately and rapidly will make learning calculus in high dimensions easier. The following rules are crucial.

Power rule {\((x^n)' = nx^{n-1}\)}
Sum and difference rule {\((f\pm g) = f'\pm g'\)}
Product {\((fg)' = f' g + fg'\)} and quotient rule {\(\ds\left(\frac f g\right)' = \frac{f' g - fg'}{g^2}\)}
Chain rule (arguably the most important) {\((f\circ g)' = f'(g(x))\cdot g'(x)\)}

Exercise1.1.2

Compute the derivative of \(e^{\sec x}\cos(\tan(x)+\ln(x^2+4))\text{.}\)

Show each step in your computation, making sure to show what rules you used.

Exercise1.1.3

If \(y(p) = \ds \frac{e^{p^3}\cot(4p+7)}{\tan^{-1}(p^4)}\) find \(dy/dp\text{.}\)

Again, show each step in your computation, making sure to show what rules you used.

The following problem will help you review some of your trigonometry, inverse functions, as well as implicit differentiation.

Exercise1.1.4

Use implicit differentiation to explain why the derivative of \(y=\arcsin x\) is \(\ds y'=\frac{1}{\sqrt{1-x^2}}\text{.}\)

Hint

Rewrite \(y=\arcsin x\) as \(x=\sin y\text{,}\) differentiate both sides, solve for \(y'\text{,}\) and then write the answer in terms of \(x\text{.}\)

Subsection1.1.3Integrals

Each derivative rule from the front cover of your calculus text is also an integration rule. In addition to these basic rules, we'll need to know three integration techniques. They are

{\(u\)}-substitution,
integration-by-parts, and
integration by using software.

There are many other integration techniques, but we will not focus on them. If you are trying to compute an integral to get a number while on the job, then software will almost always be the tool you use. As we develop new ideas in this and future classes (in engineering, physics, statistics, math), you'll find that \(u\)-substitution and integrations-by-parts show up so frequently that knowing when and how to apply them becomes crucial.

Exercise1.1.5

Compute \(\ds\int x\sqrt{x^2+4}dx\text{.}\)

Exercise1.1.6

Compute \(\ds\int x\sin 2x dx\text{.}\)

Exercise1.1.7

Compute \(\ds \int \arctan x dx\text{.}\)

Exercise1.1.8

Compute \(\ds \int x^2 e^{3x} dx\text{.}\)