2.1: Derivatives and Rates of Change

We will study a special type of limit called a derivative. These arise when we are looking at an instantaneous rate of change, the slope of a tangent line, or a velocity.

Tangents

In Section 1.4 we described the intuition behind creating a tangent line.

We saw it was a limit. Here's the definition:

The tangent line to the curve $y = f(x)$ at the point $P(a, f(a))$ is the line through $P$ with slope \[m = \lim_{x\rightarrow a}\dfrac{f(x) - f(a)}{x - a}\] if this limit exists.
Find an equation of the tangent line to the parabola $y = x^2$ at the point $P(1,1)$.

The tangent line limit is an indeterminate form. Here is an animation of the previous example:

Notice how the expression $f(x) - f(a)$ approaches $0$ and $x - a$ also approaches 0.

Because this limit is an indeterminate form, we need to simplify the expression before applying the limit.

In practice, it's easier to simplify the following equivalent expression of the slope of the tangent line: \[m = \lim_{h \rightarrow 0} \dfrac{f(a + h) - f(a)}{h}\]

Find an equation of the tangent line to the hyperbola $y = 3/x$ at the point $(3, 1)$.

The tangent line is sometimes called the slope of the curve because if you zoom in far enough, the curve almost looks like a straight line.

Velocities

In Section 1.4 we described dropping a ball from the CN tower and defined its velocity to be the "limit" of the average velocities over shorter and shorter time periods.

Recall that $s = f(t)$ gives the position of an object moving in a straight line $t$ seconds after rest, called the position function.

Over the time interval $[a, a+h]$, the change in position is $f(a + h) - f(a)$. Thus the average velocity is \[\text{average velocity} = \dfrac{\text{displacement}}{\text{time}} = \dfrac{f(a + h) - f(a)}{h}\]

But looking at shorter and shorter time periods of the average velocity requires $h \rightarrow 0$. Thus we have the following definition:

The instantaneous velocity $v(a)$ at time $t = a$ is the limit of the average velocities \[v(a) = \lim_{h \rightarrow 0} \dfrac{f(a + h) - f(a)}{h}\]

We now redo the problem in 1.4 but with this limit.

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.

Derivatives

Notice how the same limit \[\lim_{h\rightarrow 0} \dfrac{f(a + h) - f(a)}{h}\] occurred when we wanted to calculate a "rate of change." This limit appears often enough so we give it a name:

The derivative of a function $f$ at a number $a$, denoted by $f'(a)$ is \[f'(a) = \lim_{h\rightarrow 0} \dfrac{f(a+h) - f(a)}{h}\] if this limit exists.

The derivative can take on three interpretations:

  1. It is the slope of the tangent line at $x = a$.
  2. It is the "instantaneous rate of change" at $x = a$. For example:
    • If $s(t)$ is a position function, then $s'(a)$ is the instantaneous velocity at the moment in time $t = a$.
    • Think of this interpretation as how fast the heights $f(x)$ are changing at the $x$-value $a$.
  3. It is the slope of the curve at $x = a$. In particular, you should think of the tangent line as an "approximation" to the function $f(x)$.

Find the derivative of the function $f(x) = x^2 - 8x + 9$ at the number $a$.

The slope of the tangent line is the derivative.

Thus the tangent line to $y = f(x)$ at $(a, f(a))$ is the line through $(a, f(a))$ whose slope is $f'(a)$.

The general equation can be described with point-slope form.

Suppose $f(x)$ is a function. The equation of the tangent line at the point $(a, f(a))$ is \[y - f(a) = f'(a) (x - a)\]
Find an equation of the tangent line to the parabola $y = x^2 - 8x + 9$ at the point $(3, -6)$.