Page 19 - Calc for the Clueless

B. Find the average velocity 3 < t < 5.

C. Find the instantaneous velocity at t = 5.

A.

B.

C.

At t=5, V inst=2t +5 =2(5) + 5 =15 feet per second.

Note 1

∆t is usually very small when compared to t.

Note 2

The derivative does not always exist. If y = |x|, the derivative does not exist at x = O, since all secant lines on
the left have slope -1 and all on the right have slope 1. These lines never approach one line.

In almost all courses, you are asked to do some problems using the definition of derivative. This is really a
thorough exercise in algebra with just a touch of limits. Let us do three examples.

Example 3—

2
Using the definition of derivative, find f'(x) if f(x) = 3x + 4x - 5.

If you have done your algebra correctly, all remaining terms at this point will have at least ∆x multiplying them.
If there is a fraction, all terms in the top will have at least ∆x.