# Math Before Calculus

With a focus on looking forward to what lies ahead in calculus, this courseware is based on studying the behavior of functions with a look at how functions grow, including percentage growth. Topics include: how to read the plot of a function; the difference between formulas and equations; when estimates are preferable to exact answers; proportion and scaling; raw growth versus percentage growth; extensive experience in functions that are used in engineering and science: linear, exponential, oscillating and power functions; frequency versus period; linear and exponential data analysis; area measurement estimation; radians; unit circle; imaginary numbers; modeling linear, exponential and periodic data; linear estimation; rotations and reflections; right triangle trigonometry; inequalities.

Math Before Calculus (MBC) is designed to address several issues. For students who have been enjoying their school math, this courseware is an opportunity to learn what is important at the university level before they get to the university. For students who have been turned off by school math, this courseware is an opportunity for a new start and a fast track to the math (in context) that actually arises in science, engineering, technology, and in the workplace. In MBC, the understanding students get through visualization and experimentation minimizes the need for memorization. In MBC, only after an issue has been set up visually do the words go on. Using interactive lessons and the power of Mathematica, students using this courseware will learn hands-on through experimentation.

You may purchase a copy of our courseware without any instructional services such as mentor feedback or grading if you are simply seeking reference materials rather than enrolling for a class.

## Syllabus

**Lesson 1**: Functions and plots

*Science and math experience.*

The inside story of what functions really are. Getting control of the functions of math and science by interacting with plot commands. Analyzing the US population data. Growth and percentage growth.

**Lesson 2**: Make your own functions to get the job done

*Science and math experience.*

Calculating what you have to pay: College tuition, credit card payoff and car loans. Managing a catfish pond. Predator-prey interaction. Avoiding a DUI. Inflation. Discharge of a battery. War games including a simulation of the battle of Iwo Jima in World War II. Chaos – Just as in Jurassic Park.

**Lesson 3**: Functions folks use for calculation in science and engineering: Linear and exponential functions

*Science and math experience.*

Linear functions are those that grow by a fixed amount every time x goes up by h; exponential functions are those that grow by a fixed percentage every time x goes up by h. The base e. Consumer math: Exponential functions, bank accounts and compound interest. Use linear functions for centimeter-inch conversions, estimating the height of a tree, Chevy versus Honda and Airplane mileage. Use exponential functions for battery discharging, carbon dating, rock dating, underwater illumination, motor vehicle registrations and inflation awareness. Experiments with the natural logarithm. Linear and exponential data analysis.

**Lesson 4**: Functions folks use for calculation in science and engineering: The oscillating functions Sin[x] and Cos[x]

*Science and math experience.*

Sin[x] and Cos[x] both oscillate between -1 and 1 and repeat themselves every time x goes up by 2π. Cos[x] is a shifted version of Sin[x]. Even and odd functions. Frequency and period. The engineering oscillators: Harmonic oscillators, damped harmonic oscillators and beating oscillators. Fourier fit of periodic functions and periodic data with Sines and Cosines. Tan[x]. Morphing one function into another.

**Lesson 5**: Functions folks use for calculation in science and engineering: Power functions x^{k}.

*Science and math experience*.

Dominance in the global scale: The dominant power function in a polynomial is the highest power involved. In the global scale, exponential growth dominates power growth. Using power functions to calculate accurate values of other functions. Geometric sums and drug dosing. Experiments in factoring. Area, pictures and algebra. Using the fact a^{2}-b^{2} = (a-b)(a+b) to advantage. Young Carl Gauss, the Luke Skywalker of math. More morphing.

**Lesson 6**: Measurements

*Science and math experience.*

Scaling and proportions. Pythagorean theorem. Monte Carlo sampling estimates of area and volume measurements. The bell-shaped curve and normal probability estimates via Monte Carlo. IQ percentiles. Area and volume measurements resulting from stretching along the axes. Light bulb failures via Monte Carlo sampling.

**Lesson 7**: Trips on the unit circle, Sines and Cosines and radians

*Science and math experience.*

{Cos[t], Sin[t]} always plots out on the unit circle. To get to the position of {Cos[t], Sin[t]} on the unit circle, you start at {1, 0} and take a trip of length t on the unit circle. A trip on the unit circle of length t corresponds to t radians. Sines and Cosines in automotive mechanics: Wheel bolt circle diameter. Parametric plotting with Sines and Cosines: elliptical orbits of planets and asteroids.

**Lesson 8**: Rotation and reflections

*Science and math experience.*

Using rotations to get the formulas Cos[s + t] = Cos[s]*Cos[t] – Sin[s]*Sin[t] and Sin[s + t] = Sin[s]*Cos[t] + Sin[t]*Cos[s]. When you flip {x, y} over the line through {0, 0} and {Cos[s], Sin[s]} about {0, 0}, you get {x Cos[2 s] + y Sin[2 s], x Cos[2 s] -y Sin[2 s]}. This results from a rotation, a flip over the x-axis, followed by another rotation. Using flips to do ray tracing: Bouncing light rays off curves.

**Lesson 9**: Solving equation

*Science and math experience.*

Equations are different from formulas. When you put f[x] = x^{2} – 2x – 4 , you are giving a formula for f[x] which works no matter what x is. When you go after the particular x’s that make f[x] = 0, you are solving the equation f[x] = 0. The quadratic formula helps you solve this equation. Coming up with the deepest dip or highest crest of a quadratic and using it to get the quadratic formula without the traditional “completing the square.”

To solve for more than one variable, you generally need at least as many equations as variables.