# Preview of Calculus

Site: | Saylor Academy |

Course: | MA005: Calculus I |

Book: | Preview of Calculus |

Printed by: | Guest user |

Date: | Saturday, October 12, 2024, 9:14 AM |

## Description

Read this section for an introduction to calculus.

## Two Basic Problems

Beginning calculus can be thought of as an attempt, a historically successful attempt, to solve two fundamental problems. In this section we will start to examine geometric forms of those two problems and some fairly simple ways to attempt to solve
them. At first, the problems themselves may not appear very interesting or useful, and the methods for solving them may seem crude, but these simple problems and methods have led to one of the most beautiful, powerful, and useful creations
in mathematics: Calculus.

#### First Problem: Finding the Slope of a Tangent Line

Suppose we have the graph of a function , and we want to find the equation of the line which is **tangent **to the graph at a particular point on the graph (Fig. 1). (We will give a precise
definition of tangent in Section 1.0. For now, think of the **tangent **line as the line which touches the curve at the point and stays close to the graph of near ). We know that
the point is on the tangent line, so if the x–coordinate of is , then the y–coordinate of must be and . The only other information we need to find the
equation of the tangent line is its slope, , and that is where the difficulty arises. In algebra, we needed two points in order to determine a slope, and so far we only have the point . Lets simply pick a second
point, say , on the graph of . If the x–coordinate of is (Fig. 2), then the y–coordinate is , so . The slope of the line through and
is

If we drew the graph of on a wall, put nails at the points and on the graph, and laid a straightedge on the nails, then the straightedge would have slope (Fig. 2). However, the slope can be very different from the value we want, the slope of the tangent line. The key idea is that if the point is close to the point , then the slope is close to the slope we want, . Physically, if we slide the nail at along the graph towards the fixed point , then the slope, , of the straightedge gets closer and closer to the slope, , of the tangent line. If the value of is very close to , then the point is very close to , and the value of is very close to the value of . Rather than defacing walls with graphs and nails, we can calculate

**derivative**of the function at the point , and this part of calculus is called

**differential calculus**. Chapters 2 and 3 begin differential calculus.

The slope of the tangent line to the graph of a function will tell us important information about the function and will

allow us to solve problems such as:

#### Second Problem: Finding the Area of a Shape

Suppose we need to find the area of a leaf (Fig. 4) as part of a study of how much energy a plant gets from sunlight. One method for finding the area would be to trace the shape of the leaf onto a piece of paper and then divide
the region into "easy" shapes such as rectangles and triangles whose areas we could calculate.

We could add all of the "easy" areas together to get the area of the leaf. A modification of this method would be to trace the shape onto a piece of graph paper and then count the number of squares completely inside the edge of the leaf
to get a lower estimate of the area and count the number of squares that touch the leaf to get an upper estimate of the area. If we repeat this process with smaller squares, we have to do more counting and adding, but our estimates
are closer together and closer to the actual area of the leaf. (This area can also be approximated using a sheet of paper, scissors and an accurate scale. How?)

We can calculate the area A between the graph
of a function and the x–axis (Fig. 5) by using similar methods. We can divide the area into strips of width w and determine the lower and upper values of on each strip. Then
we can approximate the area of each rectangle and add all of the little areas together to get , an approximation of the exact area. The key idea is that if w is small, then the rectangles are narrow, and the approximate
area Aw is very close to the actual area A. If we take narrower and narrower rectangles, the approximate areas get closer and closer to the actual area: .

The
process we used is the basis for a technique called **integration**, and this part of calculus is called **integral calculus**. Integral calculus and integration will begin in Chapter 4.

The
process of taking the limit of a sum of "little" quantities will give us important information about the function and will also allow us to solve problems such as:

"Find the length of the graph of over one period (from to )".

"Find the volume of a torus ("doughnut") of radius 1 inch which has a hole of radius 2 inches. (Fig. 6)""A car starts at rest and has an acceleration of feet per second per second in the
northerly direction at time t seconds. Where will the car be, relative to its starting position, after 100 seconds?"

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-1.1-Preview-of-Calculus.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.

## A Unifying Process: Limits

We used a similar processes to "solve" both the tangent line problem and the area problem. First, we found a way to get an approximate solution, and then we found a way to improve our approximation. Finally, we asked what would happen if we
continued improving our approximations "forever", that is, we "took a limit". For the tangent line problem, we let the point get closer and closer and closer to , the limit as approached .
In the area problem, we let the widths of the rectangles get smaller and smaller, the limit as approached 0. Limiting processes underlie derivatives, integrals, and several other fundamental topics in calculus, and we
will examine limits and their properties in Chapter 1.

## Two Sides Of The Same Coin: Differentiation and Integration

Just as the set–up of each of the two basic problems involved a limiting process, the solutions to the two problems are also related. The process of differentiation for solving the tangent line problem and the process of integration for solving the area problem turn out to be "opposites" of each other: each process undoes the effect of the other process.

The Fundamental Theorem of Calculus in Chapter 4 will show how this "opposite" effect works.

## Problem for Solution

The first 5 chapters present the two key ideas of calculus, show "easy" ways to calculate derivatives and integrals, and examine some of their applications. And there is more. In later chapters, new functions will be examined and ways to calculate
their derivatives and integrals will be found. The approximation ideas will be extended to use "easy" functions, such as polynomials, to approximate the values of "hard" functions such as and . And
the notions of "tangent lines" and "areas" will be extended to 3–dimensional space as "tangent planes" and "volumes".

Success in calculus will require time and effort on your part, but such a beautiful and powerful field is worth that time and effort.