MA005: Calculus I

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 $y = f(x)$, and we want to find the equation of the line which is tangent to the graph at a particular point $P$ 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 $P$ and stays close to the graph of $y = f(x)$ near $P$). We know that the point $P$ is on the tangent line, so if the x–coordinate of $P$ is $x = a$, then the y–coordinate of $P$ must be $y = f(a)$ and $P = (a, f(a))$. The only other information we need to find the equation of the tangent line is its slope, $m_{tan}$, 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 $P$. Lets simply pick a second point, say $Q$, on the graph of $y = f(x)$. If the x–coordinate of $Q$ is $b$ (Fig. 2), then the y–coordinate is $f(b)$, so $Q = (b, f(b))$. The slope of the line through $P$ and $Q$ is

$m_{PQ} = \dfrac{rise}{run} = \dfrac{f(b) – f(a)}{b – a}$

If we drew the graph of $y = f(x)$ on a wall, put nails at the points $P$ and $Q$ on the graph, and laid a straightedge on the nails, then the straightedge would have slope $m_{PQ}$ (Fig. 2). However, the slope $m_{PQ}$ can be very different from the value we want, the slope $m_{tan}$ of the tangent line. The key idea is that if the point $Q$ is close to the point $P$, then the slope $m_{PQ}$ is close to the slope we want, $m_{tan}$. Physically, if we slide the nail at $Q$ along the graph towards the fixed point $P$, then the slope, $m_{PQ} = \dfrac{f(b) – f(a)}{b – a}$, of the straightedge gets closer and closer to the slope, $m_{tan}$ , of the tangent line. If the value of $b$ is very close to $a$, then the point $Q$ is very close to $P$, and the value of $m_{PQ}$ is very close to the value of $m_{tan}$. Rather than defacing walls with graphs and nails, we can calculate

$m_{PQ} = \dfrac{f(b) – f(a)}{b – a}$

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 $y = f(x)$ 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 $y = f(x)$ on each strip. Then we can approximate the area of each rectangle and add all of the little areas together to get $A_w$, 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: $\mathrm{A} =limit _{w \rightarrow 0} A_w$.

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 $y = sin(x)$ over one period (from $x = 0$ to $x = 2π$)".

"A car starts at rest and has an acceleration of $5 + 3sin(t)$ 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
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