1.4: The Tangent and Velocity Problems

Goals:

The Tangent Problem

Consider the following graph of a function. *drawn in class*

The tangent line of a graph of a function $f(x)$ is a line that intersects the graph at one point $(a, f(a))$ near that point.

We will describe what "near the point" means in Section 1.5.

How do we actually create this line? Here's how:
  1. Start with the point you want the tangent line to be at (called point of tangency).
  2. Pick another point near the point of tangency.
  3. You now have two points. Draw a line through it (called the secant line).
  4. Move the other point closer and closer to the point of tangency.
  5. If you move this point closer forever, the secant line will be as close as you want to the tangent line.

Here is an example of the above process with an animation.

This process of "moving closer and seeing what's at the end" is called a limit, which will be described in the next section.

It turns out the slope of the tangent line (dotted line above) is called the derivative.

The Velocity Problem

When driving a car, your speedometer tells you the exact speed you are going.

That number at a specific time point is called the instantaneous velocity; the word instantaneous is meant to describe that velocity you see happens at one single point in time only.

You can find the instantaneous velocity if you have a function $s(t)$ which describes the total distance traveled after $t$ seconds.

Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the instantaneous velocity of the ball after 5 seconds.

Again, finding instantaneous velocity involves moving a quantity closer to another one. This is a limit!

Therefore, to understand the derivative and instantaneous velocity, we need to understand limits.

Let's do this now.