2.7 Tangents, Velocities, and Other Rates of Change

We now rejoin the exciting world of tangent lines and derivatives. We have already gone over most of this in section 2.1. To find instantaneous rate of change, the derivative, or the slope of the tangent line (which all are the same thing), we are basically finding the average velocity (v=d/t), for example, over a time period of zero. This seemingly undefined equation can be solved with some tricky algebra. We use the equation lim x->a [f(x)-f(a)]/(x-a). To find the limit, we must factor x-a out of the top. We are basically removing the discontinuity to make our limit computable.

This equation can also be written as lim h->0 [(f(a+h)-f(a)]/h, where h is the difference of the x-values between the two points that you are finding the slope from. The last step using this equation then, would be to replace h with a zero. Therefore, you must get h out of the denominator of the fraction by factoring. By using these methods, you can solve for a derivative algebraically.

You may have noticed that this ability is very handy in certain types of real-world applications. You can use these methods to find the velocity of an object at an instant in time. More generally, the derivative is how fast the function is changing in an instant, or the instantaneous rate of change. All of these are also the same as the slope of the tangent line at a point.

Now for an example: find the slope of the tangent line of the parabola x^2 at the point (2,4).

You can use whichever method you want.

First Method:
lim x->2 [(f(x)-f(2)]/(x-2) First substitute the values into the equation
=lim x->2 (x^2-4)/(x-2) Factor the top.
=lim x->2 [(x-2)(x+2)]/(x-2) Cancel out the (x-2) from the top and bottom
=lim x->2 x+2 substitute 2 in for x
= 4

This method happens to be easier for this problem, but we can use the other method too.

Second Method:
=lim h->0 [f(2+h)-f(2)]/h Substitute into the equation
=lim h->0 [(2+h)^2-4]/h Apply the function
=lim h->0 (h^2+4h+4-4)\h Simplify
=lim h->0 (h^2+4h)\h Factor out an h on top
=lim h->0 [h(h+4)]\h Cancel out the h's.
=lim h->0 h+4 Substitute 0 in for h
= 4

Here is a helpful link that covers the topics presented in this lesson:
http://tutorial.math.lamar.edu/AllBrowsers/2413/Tangents_Rates.asp

Isabella, you are the next in line.

In case you were wondering (and I know you are), here are the first 20 digits of pi. I expect everybody to have them memorized by tomorrow.

pi=3.1415926535897932384...

Don't worry, you will get more digits throughout the year.