phy1020.DVI

Division of infinitesimals leads to some interesting results. In general, dividing one infinitesimal number
by another often leads to afiniteresult, as we’ll see in the next section.

4.2 Differential Calculus — Finding Slopes

One important application of the calculus is that it allows us to determine the slope of a line that is not
necessarily a straight line. You’ve learned in an algebra class how to find the slope of a straight line:

slopeD

rise

run

(4.6)

In other words, pick any two points along the line, and take the change iny(y, the “rise”) divided by the
change inx(x, the “run”).
How can you calculate the slope of a line that isnotstraight — say, for example, the parabolayDx^2?
For a curved line, the slope is different at different points along the curve; it is defined to be the slope of the
straight line tangent to the curve at that point. We can calculate the slope of that tangent line by using the
calculus.
As an example, let’s take the parabolaf.x/Dx^2 and say we wish to find its slope atxD 3. We can
approximate the slope of the tangent line atxD 3 by finding the slope of the straight line connecting the point
on the parabola atxD 3 and a second point very close toxD 3. The closer the second point is toxD 3 ,
the better the approximation to the actual slopeatxD 3. For example, let the two points bexD 3 and
xD3:01. Then atxD 3 ,yDf.x/Dx^2 D 32 D 9 , and atxD3:01,yDf.x/Dx^2 D3:01^2 D9:0601.
The slope of the line connecting these points is then

slopeD

y

x

D

9:0601 9

3:01 3

D6:01 (4.7)

Now let’s try an even closer second point:xD3:001. ThenyDx^2 D3:001^2 D9:006001. Then

slopeD

y

x

D

9:006001 9

3:001 3

D6:001 (4.8)

And yet an even closer second point:xD3:0001. ThenyDx^2 D3:0001^2 D9:00060001. Then

slopeD

y

x

D

9:00060001 9

3:0001 3

D6:0001 (4.9)

The closer the second point is to 3, the closer the slope seems to be getting to 6. In other words, in thelimit
wherexgets closer and closer to 0, the slope gets closer and closer to 6 — suggesting that the slopeat
xD 3 isexactly6. We write this limit as:

slopeD lim

x! 0

y

x

D lim

x! 0

f.xCx/f.x/

.xCx/x

D lim

x! 0

f.xCx/f.x/

x

(4.10)

Sincef.x/Dx^2 in our example,

slopeD lim

x! 0

f.xCx/f.x/

x

(4.11)

D lim

x! 0

.xCx/^2 x^2

x

(4.12)

D lim

x! 0

Œx^2 C2xxC.x/^2 x^2

x

(4.13)

D lim

x! 0

2xxC.x/^2

x

(4.14)