1. Section 2.2 Instantaneous Rates of Change
MAT 213
Brief Calculus
Section 2.2 Instantaneous Rates of Change

2. “At the instant the horse crossed the finish line, it was traveling at 42 miles per hour.”

3. There is a paradox in trying to study motion at a particular instant in time.
By focusing on a single instant,
we stop motion!!!
How can we know the speed of a horse at the instant it crosses the finish line?

4. Instantaneous Rate of Change
The instantaneous rate of change at a point on a curve is the slope of the curve at that point.
Slope is the measure of tilt of a LINE!
In the last section we discussed average rate of change as the slope of the secant line between two points
Now instantaneous rate of change gives us the slope of the tangent line at a single point

5. Local Linearity
If we look close enough near any point on a smooth curve, the curve will look like a straight line!
Example
Grapefruit

6. This is called the tangent line at that point
Local Linearity
If we look close enough near any point on a smooth curve, the curve will look like a straight line!
This is called the tangent line at that point
The slope of a graph at a point is the slope of the tangent line at that point

7. Example
On this graph, indicate where the slope of the tangent line is: positive greatest
negative zero
Find two points on the curve where the slopes are about the same.

8. Instantaneous Rate of Change
Average Rate of Change
Graphically, the average rate of change over the interval a ≤ x ≤ b is the slope of the secant line connecting (a,f(a)) with (b,f(b))
Instantaneous Rate of Change
Instantaneous rate of change at a is the slope of the line TANGENT to the curve at a single point, (a,f(a))

9. Secant line becoming tangent line

10. The Tangent Line
The tangent line at a point Q on a smooth, continuous graph is the limiting position of the secant lines between point Q and point P as P approaches Q along the graph (if the limiting position exists)

12. Except at an inflection point. Examples Concave up Concave down
The Tangent Line
General Rule
Lines tangent to a smooth, nonlinear curve do not “cut through” the graph of the curve at the point of tangency and lie completely on one side of the graph near the point of tangency
Except at an inflection point.
Examples
Concave up Concave down

13. Where instantaneous rate of change DOES NOT exist…
Point of discontinuity
Sharp point
Vertical tangent (no “run”)
Let’s look at some examples

14. Consider the graph of f(x) = |x|
Is it continuous at x = 0?
Is it differentiable at x = 0?
Let’s zoom in at 0

18. No matter how close we zoom in, the graph never looks linear at x = 0
Therefore there is no tangent line there so it is not differentiable at x = 0

19. Example- Piecewise Defined Function

20. Example
Note: This is a graph of
It has a vertical tangent at x = 0

21. In groups let’s try the following from the book
7, 25