We compute the instantaneous growth rate by computing the limit of average growth rates.
- If represents the position of an object on a line (with respect to time), the rate of change gives the velocity of the object.
- If represents the velocity of an object with respect to time, the rate of change gives the acceleration of the object.
- If represents the revenue generated by selling objects, the rate of change gives us the marginal revenue, meaning the additional revenue generated by selling one additional unit. Note, there is an implicit assumption that is quite large compared to .
- If represents the cost to produce objects, the rate of change gives us the marginal cost, meaning the additional cost generated by selling one additional unit. Again, there is an implicit assumption that is quite large compared to .
- The rate of change of a function can help us approximate a complicated function with a simple function.
- The rate of change of a function can be used to help us solve equations that we would not be able to solve via other methods.
We’ve been computing average rates of change for a while now,
More precisely, the average rate of change of a function is given by as the input changes from to .
What happens if we compute the average rate of change of for each value of as gets closer and closer to ? In other words, what is the meaning of the limit provided that the limit exists?
Naturally, we call this limit the instantaneous rate of change of the function at .
- Find the expression for the average rate of change of between the points and . Now evaluate the function, Simplify,
- Find the average rate of change of between the points and , . Now substitute in for the function, Simplify the top, Factor, Factor and cancel,
- Find the instantaneous rate of change of at the point . So, the
instantaneous rate of change is the limit, as , of average rates of change of
between points and , for .
We have already computed an expression for the average rate of change for all . Therefore,
within a completely different context.
Have a look at the figure below. The figure depicts a graph of the function , two points on the graph, and , and a secant line that passes through these two points.
This is exactly the expression for the average rate of change of as the input changes from to !
As before, we can ask ourselves: What happens as gets closer and closer to ? In other words, what is the meaning of the limit of slopes of secant lines through the points and as gets closer and closer to ?
How can we interpret the limit provided that the limit exists?
This scenario is illustrated in the figure below. What happens as ?
Therefore, this limit deserves a special name that could be used regardless of the context.
The definition of the derivative allows us to define a tangent line precisely.
Naturally, by the point-slope equation of the line, it follows that the tangent line is given by the equation
But, most functions are not linear, and their graphs are not straight lines. Therefore, the computation of the derivative is not as simple as in the previous example.
Find the slope of the tangent line to the curve at the point .
Start by writing out the definition of the derivative, Multiply by to clear the fraction in the numerator, Combine like-terms in the numerator,
Take the limit as goes to , We are looking for an equation of the line through the point with slope . The point-slope formula tells us that the line has equation given by or . We can confirm our results by looking at the graph of and the line .
The object has velocity at time .
Below we can see the graph of and the tangent line at , with a slope of . Notice, again, how the line fits the graph of the function near the point .