# Calculus Introduction

For many students, just the thought of taking calculus is pretty scary. Calculus may be difficult, but much of is really just advanced algebra, geometry, and trigonometry – stuff that you have already learned.

You need to be willing to do some WORK! You can’t learn calculus just by listening to me or taking a pill. ## What is Calculus

So what is calculus? It takes the ordinary rules of algebra and geometry and tweaks them so they can be used on more complicated problems.

Example – pushing a box up a flat ramp is simple math, but pushing it up a real life hill (a curve) requires calculus. The energy to push the box is constant for the ramp, but on a curve it is constantly changing. You have to break up the curving incline into small chunks and calculate each chunk separately. If you break each piece up small enough, it becomes practically straight.

So, in a nutshell, calculus takes a problem that can’t be done with regular math because things are constantly changing (curves on a graph) and zooms in on the curve until it becomes straight and then uses regular math to finish the problem.

What makes calculus unusual and fantastic is that it actually zooms in infinitely. In fact, everything you do in calculus involves infinity in one way or another, because if something is constantly changing, it is changing infinitely often from each infinitesimal moment to the next.

Here are some real life examples using regular math and calculus:

• Electric companies need to know how much cable is needed to 1) go diagonally across a park (regular math) or 2) hanging in a catenary pattern from two towers (calculus).
• Architects need to calculate the area of a roof to know the cost and figure the weight on it with and without snow when it is 1) a peaked roof, but essentially flat (regular math) and 2) a complicated, non-spherical shape like the Houston Astrodome (calculus).
• Calculating the lead needed to hit a moving object for 1) a quarterback throwing to his receiver (regular math) and 2) NASA finding the lead for the Viking 1 aimed at Mars where both Earth and Mars have different elliptical orbits and the relative speed of both planets is constantly changing, not to mention the constantly changing gravitational pull of the Earth, Moon, Mars, and the sun during it’s trip (calculus).

In the real world, relationships are rarely as simple as a straight line graph. This is what makes calculus so useful.

## Big Ideas of Calculus

Calculus is based on two big ideas – differentiation and integration.

Differentiation is the process of finding a derivative, the fancy calculus name for the slope (or steepness) of a curve. The slope is a simple rate, like miles per hour or profit per item. Slope, as we learned in Algebra is the change in y divided by the change in x, or rise over run. As we study more and more complicated problems we find that the slope for any shape (except a straight line) is constantly changing but can be described as an equation. In calculus we look for the instantaneous rate of change - the slope of the line tangent to a curve at a single point.

For example, when you drive your car your average speed stays fairly constant, but your instantaneous speed could be changing. The instantaneous speed is what a police officer uses to decide if you need a speeding ticket.

Integration is used to find the area between the graph of a curve and the x-axis. It is basically just fancy addition. At its simplest, integration is the process of cutting up an area into small sections and then adding up the areas of all the little bits to get the total area.

Example – you can easily calculate the area of a simple rectangle, but to calculate the area under a curve it is hard because you don’t have an area formula for this funny shape. What you do is zoom in until each little piece looks pretty much like an ordinary trapezoid with flat sides (a triangle sitting on top of a rectangle), find the area of each tiny trapezoid, and then add them all up.

# Quick Review of your book

Appendices – starting on p 577, this provides lots of review material and formulas

Glossary – p 635 has definitions of frequently used words and terms

Selected Answers – p 647 provides answers to Exercise odd problems and all Quick Review problems. Use these to make sure you understand how to do the assignments.

## Selected Calculus Topics Comments, questions, or you just need some help? Send an e-mail to Mrs. Shearer Last Updated: 09/07/2011