CK-12-Calculus

1.4. The Calculus http://www.ck12.org

We note that the curve now looks very much like a straight line. If we were to overlay this view with a straight line
that intersects the curve at( 1 , 1 ),our picture would look like this:

We can make the following observations. First, this line would appear to provide a good estimate of the value of
f(x)forx−values very close tox= 1 .Second, the approximations appear to be getting closer and closer to the actual
value of the function as we take points on the line closer and closer to the point( 1 , 1 ).This line is calledthe tangent
line tof(x)at( 1 , 1 ).This is one of the basic situations that we will explore in calculus.
Tangent Line to a Graph
Continuing our discussion of the tangent line tof(x)at( 1 , 1 ),we next wish to find the equation of the tangent line.
We know that it passes though( 1 , 1 ),but we do not yet have enough information to generate its equation. What
other information do we need?(Answer: The slope of the line.)
Yes, we need to find the slope of the line. We would be able to find the slope if we knew a second point on the line.
So let’s choose a pointPon the line, very close to( 1 , 1 ).We can approximate the coordinates ofPusing the function
f(x) =x^2 ; henceP(x,x^2 ).Recall that for points very close to( 1 , 1 ),the points on the line are close approximate
points of the function. Using this approximation, we can compute the slope of the tangent as follows:
m= (x^2 − 1 )/(x− 1 ) =x+1 (Note: We choose points very close to( 1 , 1 )but not the point itself, sox 6 =1).
In particular, forx= 1 .25 we haveP( 1. 25 , 1. 5625 )andm=x+ 1 = 2. 25 .Hence the equation of the tangent line, in
point slope form isy− 1 = 2. 25 (x− 1 ).We can keep getting closer to the actual value of the slope by takingPcloser
to( 1 , 1 ),orxcloser and closer tox= 1 ,as in the following table:

P(x,y) m

( 1. 2 , 1. 44 ) 2. 2

( 1. 15 , 1. 3225 ) 2. 15

( 1. 1 , 1. 21 ) 2. 1

( 1. 05 , 1. 1025 ) 2. 05

( 1. 005 , 1. 010025 ) 2. 005

( 1. 0001 , 1. 00020001 ) 2. 0001

As we get closer to( 1 , 1 ),we get closer to the actual slope of the tangent line, the value 2.We call the slope of the
tangent line at the point( 1 , 1 )the derivative of the functionf(x)at the point( 1 , 1 ).
Let’s make a couple of observations about this process. First, we can interpret the process graphically as finding
secant lines from( 1 , 1 )to other points on the graph. From the diagram we see a sequence of these secant lines and