### Day 18 - Rates of Change, Position Function, Trig Derivatives - 09.11.14

Bell Ringer
1. What are the points in which the f(x) has a horizontal tangent line? f(x) = 4x3 - 48x

1. (2, -64)

2. (2, 64)

3. (2, -64) and (-2, 64)

4. (2, 0) and (-2, 0)

5. none of the above

2. Using the following position function, find the average velocity over the period of [1, 2]. s(t) = -16t2 + 100

1. -48

2. -40

3. -32

4. -30

5. none of the above

3. Using the function in #2, find the instantaneous velocity at t = 2.

1. -64

2. -40

3. -32

4. -30

5. none of the above

4. Using the function in #2, find the instantaneous acceleration at t = 2.

1. -64

2. -40

3. -32

4. -30

5. none of the above

Review
• Secant vs. Tangent Line
• Slope
• Definition of Derivative
• Importance of Derivative
• What does it allow us to do?
• Drawing a Tangent Line on a Graph
• Basic Differentiation Rules
• Constant Rule
• Power Rule
• Sum and Difference Rule

Lesson
• Rates of Change
• Position Function
• Sine and Cosine Derivatives
• Practice (Section 2.2)
• Checkpoint #1
• page 115 #20, #22, #24
• Checkpoint #2
• page 115 #36 and #38
• Checkpoint #3
• page 116 #54 and #64
• Checkpoint #4
• page 116 #78 and #80
• Work on Student-led Lessons!

Exit Ticket
• Exit TIcket will be posted on the board in class.
Lesson Objective(s)
• What rules can be created for finding the derivative involving:
• sine and cosine functions
• rates of change

Standard(s)
• APC.5
• Investigate derivatives presented in graphic, numerical, and analytic contexts and the relationship between continuity and differentiability.
• The derivative will be defined as the limit of the difference quotient and interpreted as an instantaneous rate of change.
• APC.6
• ​The student will investigate the derivative at a point on a curve.
• Includes:
• finding the slope of a curve at a point, including points at which the tangent is vertical and points at which there are no tangents
• using local linear approximation to find the slope of a tangent line to a curve at the point
• ​defining instantaneous rate of change as the limit of average rate of change
• approximating rate of change from graphs and tables of values.

Mathematical Practice(s)
• #1 - Make sense of problems and persevere in solving them
• #2 - Reason abstractly and quantitatively
• #3 - Construct viable arguments and critique the reasoning of others
• #4 - Model with mathematics
• #8 - Look for and express regularity in repeated reasoning