The Geometry of Derivatives
Revisiting the "Tangent Problem"
In a previous lecture, we have investigated the idea of the so-called "Tangent Problem" where we considered how to find the equation for the tangent line to a given curve at a specified point. This process required us to look at the limit of the slopes of the secant lines approaching the tangent line.
Suppose our curve has equation and we are tasked with finding the equation of the tangent line to the curve at the point . We then pick a point nearby , where and we can compute the slope of the secant line by
We can let the point approach the point by letting approach . If, during this process, the slope of the secant line approaches a number , then we have the following definition.
Definition
The tangent line to the curve at the point is the line through with slope
provided that this limit exists.
Example 1
Find an equation of the tangent line to the curve given by at the point .
Note that even though this graphic represents the point approaching the point from the right-hand side, we know from the previous chapter that for the limit to exist, the left- and right-hand limits must agree, and so we could have considered the limit from either direction.
Solution
Using the slider in the graph below to take the limit graphically, we guess that the slopes of the secant lines are approaching 2.
We can, of course, verify this is true by evaluating the limit given by the definition for the tangent line.
For the curve defined by we have that the slope of the tangent line at the point is given by
since and so .
Using the methods of the previous chapter (and a little bit of algebra), we can evaluate this limit as follows.
Thus, the slope of the tangent line is exactly what we suspected!
We are not done yet, though, because we were asked to find the EQUATION of the tangent line, not just its slope. To find an equation of the tangent line, we will use the point-slope formula for a line with slope and point on the line .
For our line this is given by:
Thus, the equation for the tangent line to the curve defined by at the point is given by
Connecting the "Tangent Problem" and The Derivative
It is not immediately obvious how the derivative of a function and the "tangent problem" are connected, but to see the connection, we first need to look at the definition for the derivative of a function at a specified point.
Definition
The derivative of a function at a number , is denoted by , is
if this limit exists.
We see that the derivative of the function at the point is given in terms of the limit of the difference quotient
that we have seen in the past.
An attentive reader would notice that this formulation bears more than a little similarity to the limit which defines the slope of the tangent line.
Indeed, if we let then we can rewrite this to and will approach 0 if and only if approaches . Translating these ideas into the definition of the derivative, we get
which is exactly the definition of the slope of the tangent line to the curve at the point which we discussed in great detail earlier.
Therefore, the tangent line to at the point is the line through whose slope is equal to , the derivative of at .
Example 2
Find for
Solution
From the definition of we have
Replacing our values we have
Finding a common denominator for the terms in the numerator and simplifying yields
Multiplying by numerator and denominator of this new expression by the reciprocal of the numerator gives us
which simplifies to
Crossing out the common factor of in the numerator and denominator results
and by taking the limit we have
Therefore,
Question 1
Using the given graphic as an aid, verify that the solution we found for Example 2 is correct for the values of Explain your findings below.