1. Drill: Find the limit of each of the following.
No Limit
You will want to get this as one fraction first! Start by multiplying numerator and denominator by 2(2+x)

2. Lesson 4: Rates of Change and Tangent Lines

3. Average Rates of Change
Average rate of change of a quantity over a period of time is the amount of change divided by the time it takes.
In general, the average rate of change of a function over an interval is the amount of change divided by the length of that interval.

4. Example
Find the average rate of change of f(x) = x3 – x over the interval [1, 3].
f(1) = 0 and f(3) = 24
Using the formula:
24-0 = 24 =

5. Tangent Line
The tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point.
As it passes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point.

6. Finding Slope and Tangent Line
Find the slope of the parabola y = x2 at the point P(2,4). Write an equation for the tangent to the parabola at this point.

7. Finding Slope and Tangent Line
Formula:
𝑥=2
f(2) = 4
=4+h

8. Finding Slope and Tangent Line
To determine the slope, always substitute 0 for h:
Slope = = 4.
Find the equation using the point P(2, 4) and the slope from above of 4.
y – 4 = 4(x – 2)
y – 4= 4x – 8
y = 4x - 4

9. Example: Finding Slope and Tangent Line
Find the slope of the parabola y = x2 + 1 at the point P(2, 5). Write an equation for the tangent to the parabola at this point.

10. You try: Find the slope of the parabola at the point P
You try: Find the slope of the parabola at the point P. Then write an equation for the tangent to the parabola at this point.
y = 2x2 – 1, P (2, 7)
2 2+ℎ 2 −1−7 ℎ
When h = 0, slope = 8
2 4+4ℎ+ ℎ 2 −8 ℎ
8+8ℎ+ 2ℎ 2 −8 ℎ
8ℎ+ 2ℎ ℎ = 8 + 2h
𝑦=8𝑥−9

11. Warm-up: Find the slope of the parabola y = 2x2 at the point P(2,8)
Warm-up: Find the slope of the parabola y = 2x2 at the point P(2,8). Write an equation for the tangent to the parabola at this point.
Slope
Formula:
𝑥=2
f(2) = 8
=8+2h

12. Warm-Up: Finding Slope and Tangent Line
Equation:
y – 8 = 8(x – 2)
y – 8= 8x – 16
y = 8x - 8

13. Definition: Slope of a Curve at a Point
The slope of line tangent to a point on a curve y = f (x) at the point (a, f (a)) is provided the limit exists.

14. Example 1: f(x) = 7 + x2 Find the slope at x = a = 7+ 𝑎+ℎ 2 − 7+ 𝑎 2 ℎ
= 7+ 𝑎+ℎ 2 − 7+ 𝑎 2 ℎ
= 7+ 𝑎 2 +2𝑎ℎ+ ℎ 2 − 7+ 𝑎 2 ℎ
= 2𝑎ℎ+ ℎ 2 ℎ =2𝑎+ℎ=2a

15. Example 1
b) Where is the slope equal to 4? 2a = 4 a = 2 c) What happens to the graph at the point (a, f(a))? If m = 2a; when a < 0, slope is negative; when a = 0, slope is 0; and when a > 0, slope is positive.

16. Example 2: Let a) Find the slope of the curve at x = a.

19. Example Exploring Slope and Tangent
Let b) Where does the slope equal ? (look part a for the slope.)

20. Example :Exploring Slope and Tangent
Let c) What happens to the tangent to the curve at the point for different values of a? The slope of f(x) is , which will always be positive! Therefore, the slope is always positive for any value of a.

21. You try.
Find the slope of f at x = a. Where is the slope equal to p?

23. Note that the slope will not always be positive or
negative. When a<0, the slope is positive. When
a = 0, the slope is undefined, and when a>0, the
slope is negative. (Look at your calc table!)

24. Find the slope of f at x = a. Where is the slope equal to p?

25. Warm-Up Graded CW
You may use your notes, but you may not use consult with your classmates.
Please remember that you may not have your phone must remain in your pocket/backpack/purse

26. Definitions The difference quotient:
The normal line to a curve at a point is the line perpendicular to the tangent at that point.
Instantaneous rate of change of position with respect to time t:

27. Example Finding a Normal Line
Write an equation for the normal to the curve f (x) = x2 – 3 at x = 2.

28. Example Investigating Free Fall
A rock breaks loose from the top of a tall cliff. Find the instant speed of this falling rock at t = 3 sec.

29. Closure
Write an equation for the normal to the curve f(x) = -x2 + 5 at x = 1

30. Closure: Write an equation for the normal to the curve f(x) = -x2 + 5 at x = 1