1. 2.4 Rates of Change & Tangent Lines

2. Average Rate of Change
The average rate of change of a quantity over a period of time is the slope on that interval of time.
Ex.:
Find the average rate of change of f(x) = x3 – x over the interval [1, 3].

3. Secant & Tangent Lines Secant lines touch a graph at two points.
The slope of a secant line represents the AVERAGE RATE OF CHANGE of a function over a given interval.
(1, 1)
(5, 7)

4. Secant & Tangent Lines
A tangent line touches a graph at one point only.
Tangent lines determine the direction of a body’s (graph’s) motion at every point along its path.
Tangent lines represent the INSTANTANEOUS RATE OF CHANGE. (the slope at an actual point, not over an interval)
Here’s a question we can answer soon:
A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds?

5. Tangent Lines
The more secant lines you draw, the closer you are getting to a tangent line.
SOUND FAMILIAR TO SOME CONCEPT WE’VE DONE???

6. Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
The slope at (1, 1) can be better approximated by a secant line through (3, 9).

7. Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
The slope at (1, 1) can be even better approximated by a secant line through (2, 4).

8. Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
An even better approximation for the slope at (1, 1) would be to use a secant line through (1.1, 1.21).
How long could we continue to do this?

9. Tangent Lines
Example: Consider the function f(x) = x2 for x ≥ 0. What is the slope of the curve at (1, 1)?
What about using the point (1+h, (1+h)2) to find the slope at (1, 1)? (where h is a small change)
If h is a small change, I can say h 0. Therefore the slope of a tangent line at (1, 1) is 2.

10. Tangent Lines The slope of the curve at the point is:
The slope of a curve at a point is the same as the slope of the tangent line to the curve at that point.

11. Tangent Lines
Example: Find the slope of the parabola y = x2 at the point where x = 2. Then, write an equation of the tangent line at this point.

12. Tangent Lines
Example: Find the slope of the parabola at the point (2, 4). Then, write an equation of the tangent line at this point.
The slope of a line tangent to the parabola at (2, 4) is m = 4.
To find the equation of the tangent line, use y = mx + b
Since m = 4 and b = -4, the equation of the tangent line is
y = 4x – 4

13. Tangent Lines Example: Let . Find the slope of the curve at x = a.
(get common denominator)

14. Tangent Lines Example: Let . Where does the slope equal ?
We just found that the slope at any point a of f(x) is
Therefore, when does ?
Substituting in these a values into x in the original function, we see the graph has a slope of -1/4 at
(2, 1/2) and (2, -1/2)

15. Tangent Lines The following statements mean the same thing:
The slope of y = f(x) at x = a
The slope of the tangent line to y = f(x) at x = a
The Instantaneous rate of change of f(x) with respect to x at x = a

16. Normal Lines
The normal line to a curve at a point is the line that is perpendicular to the tangent line at that point.
Example: Write an equation for the normal to the curve f(x) = 4 – x2 at x = 1.
Slope of tangent line:
(Slope of Tangent line)

17. Normal Lines
The normal line to a curve at a point is the line that is perpendicular to the tangent line at that point.
Example: Write an equation for the normal to the curve f(x) = 4 – x2 at x = 1.
Slope of normal line
Slope of tangent line
Normal Line:
Normal Line:

18. Wrapping it Back Together
Remember the problem from an earlier slide?:
A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds?
Speed/Velocity is an INSTANTANEOUS RATE OF CHANGE.
Free fall equation: y =-16t2

19. Wrapping it Back Together