Rate of Change Over an Interval

Consider the function $Q(x) = \frac {(x+7)(x-5)}{5}$

The rate-of-change over the interval $[-3, 4]$ is $$\frac {Q(4)-Q(-3)}{4-(-3)} = \frac {4.2}{7} = 0.6$$

This is the slope of the secant line through the points $(-3, Q(-3))$ and $(4, Q(4))$.

This rate-of-change is the constant rate-of-change that would be needed in order to get from one endpoint of the interval to the other. Of course, we can see from the graph that the rate-of-change really varies over the interval.

This interval would not be a good interval to use if you were interested in the rate-of-change near $-3$.

$[-3, -2.5]$ would probably give a better interval.

The rate-of-change over the interval $[-3, -2.5]$ is $$\frac {Q(-2.5)-Q(-3)}{-2.5-(-3)} = \frac {-0.35}{0.5} = -0.7$$

This is the slope of the secant line through the points $(-3, Q(-3))$ and $(-2.5, Q(-2.5))$.

That is looking more like a tangent line, than a secant.

Rate of Change Over a Single Number?

If we make a really small interval at $-3$, then the slope of the secant line should give a good approximation of the tangent line. The slope of the secant line should approximate the slope of the tangent line.

The rate-of-change over the interval $[-3, -3+h]$ is $$\frac {Q(-3+h)-Q(-3)}{(-3+h)-(-3)} = \frac {Q(-3+h)-Q(-3)}{h} = \frac {\frac {(-3+h+7)(-3+h-5)}{5}+64}{h}$$

This is the slope of the secant line through the points $(-3, Q(-3))$ and $(-3+h, Q(-3+h))$, which is very near the tangent line at $(-3, Q(-3))$.

By moving the value of $h$ close to $0$, we can get a good estimate of the slope of the tangent line.

The slope of the line tangent to the graph at $(-3, -6.4)$ looks to be about $-0.8$.

Of course $-3$ was nothing special. We can get slopes of tangent lines at any point on the graph. Let’s move around the graph and record the tangent line slopes. We’ll record visually.

The slope of the tangent line at $-3$ is $-0.8$, so we’ll plot the point $(-3,-0.8)$ to record this information. This will be a visual encoding of the tangent slope for the domain number $-3$.

We could do this for a bunch of domain numbers.

(a)

select a domain number

(b)

select a very tiny interval beginning at that domain number

(c)

calculate the rate-of-change over the tiny interval

(d)

plot a point on the graph. First coordinate is the domain number. Second coordinate is the rate-of-change.

If we do this for every domain number, then we will have created a new function. Every domain number will be paired with the slope of the tangent line at the corresponding point.

We could graph this “slope function”.

This function, whose values are the rates-of-change of another function, is called the derivative of the other function, because it was derived from it.

$\blacktriangleright$ Let $f$ be a function. Then the derivative of $f$ is denoted as $f'$.

Through the slope of the tangent line, we have invented a way of talking about the instantaneous rate of change of a function at a number, rather than over an interval.

The function $Q(x) = \frac {(x+7)(x-5)}{5}$ is a quadratic and has a parabola for a graph. The vertex, $(-1, \tfrac {-36}{5})$, of the parabola has a horizontal tangent line, $y=\tfrac {-36}{5}$.

The slope of this tangent line is $0$. Therefore, $Q'(-1) = 0$.

The Derivative

Graph of $y = f(x) = e^{\tfrac {x}{5}}$

The value of $f'(4)$ is

positive negative