1. Section 2.2 Day 2 Basic Differentiation Rules & Rates of Change
AP Calculus AB

2. Example 1: Tangent Line
Write the equation of the tangent line for 𝑓 𝑥 = 𝑥 3 −2𝑥+10 at the point 𝑥=−2 1. Need a point: 𝑓 −2 =6 2. Need a slope: 𝑓 ′ 𝑥 =3 𝑥 2 −2 𝑓 ′ −2 =12 3. Put into point-slope form: 𝑦−6=12(𝑥+2)

3. Example 2: Tangent Line
Write the equation of the tangent line for 𝑓 𝑥 =5 cos 𝑥 + 𝑥 2 at the point 𝑥=0 1. Need a point: 𝑓 0 =5 2. Need a slope: 𝑓 ′ 𝑥 =−5 sin 𝑥 +2𝑥 𝑓 ′ 0 =0 3. Put in point-slope form: 𝑦 −5=0(𝑥−0)

4. Example 3: Horizontal Tangents
Does the curve 𝑦= 𝑥 4 −2 𝑥 2 +2 have any horizontal tangents? If so, where? 1. Horizontal tangents means the slope of the horizontal is Need slope function: 𝑦 ′ =4 𝑥 3 −4𝑥 3. Set it equal to 0 to find x values: 4 𝑥 3 −4𝑥=0 4𝑥 𝑥 2 −1 =0 4. Thus, it occurs at 𝑥=0, 𝑥=1 𝑎𝑛𝑑 𝑥=−1

5. Relationship between Position & Velocity
Let’s take a look at a function that represents the position of an object. What are the units of the slope of the function? Thus, the derivative of the position function is the velocity function.

6. Example 4: Position & Velocity pg113 Ex 9
If a billiard ball is dropped from a height of 100ft, its height, 𝑠 at time 𝑡 is given by the position function 𝑠=−16 𝑡 where 𝑠 is measured in feet and 𝑡 in seconds. A) Find the average velocity over the interval [1, 2] B) What is the velocity of the billiard ball at 4 seconds?

7. Example 4: Position & Velocity pg113 Ex 9
If a billiard ball is dropped from a height of 100ft, its height, 𝑠 at time 𝑡 is given by the position function 𝑠=−16 𝑡 where 𝑠 is measured in feet and 𝑡 in seconds.
A) Find the average velocity over the interval [1, 2]
This is AROC: 𝑓 2 −𝑓 1 2−1 = 36−84 1 =−48 𝑓𝑒𝑒𝑡/𝑠𝑒𝑐𝑜𝑛𝑑

8. Example 4: Position & Velocity pg113 Ex 9
If a billiard ball is dropped from a height of 100ft, its height, 𝑠 at time 𝑡 is given by the position function 𝑠=−16 𝑡 where 𝑠 is measured in feet and 𝑡 in seconds.
B) What is the velocity of the billiard ball at 4 seconds?
This is IROC: 𝑓 ′ 𝑥 =−32𝑡
𝑓 ′ 4 =−128 𝑓𝑒𝑒𝑡/𝑠𝑒𝑐𝑜𝑛𝑑

9. Exit Ticket for Homework
Worksheet