1. Section 12-3 Tangent Lines and velocity (day2)

2. Essential Question
How can I find the slope of a line tangent at a point?
How can I find average and instantaneous velocity?

3. Warm-up
Find the slope of the line tangent to
y = x2 + 7 at x = 6. 12

4. Now if we wanted to find the slope of a line tangent at x = 9, we’d have to repeat our process.
OR DO WE???

5. To do this, simply leave the slope in terms of x.
Let’s find the equation of for the slope of the tangent line at ANY point x.
To do this, simply leave the slope in terms of x.
Example 1 – Find an equation for the slope of the graph y = x3 at any point x.
y = 3x2

6. NOW…what would the slope of the line be at:
YOU TRY!
Example 2 – Find an equation for the slope of the graph y = x2 – 4x +2 at any point x.
y = 2x – 4
NOW…what would the slope of the line be at:
x = 4
x = -5
x = ½
m = 4
m = -14
m = -3

7. AVERAGE VS INSTANTANEOUS VELOCITY

8. AVERAGE VELOCITY
Example 3 – The distance in miles that a runner in a marathon has traveled after a certain amount of time t in hours is f(t) = 1.3t2 + 12t. What is the runner’s average velocity between the second and third hour?

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

10. INSTANTANEOUS VELOCITY

11. Example 4 –Tourists standing on a 300-foot-tall viewing tower often drop coins into the fountain below. The height of a coin falling from the tower after t seconds is given by h(t) = 300 – 16t2. Find the velocity v(t) of the coin at 2 seconds.
-64ft/sec
The negative sign indicates that the height of the coin is decreasing

12. Instantaneous Velocity at Any Point
Works the same way slope at a graph at any point works.
EXAMPLE 5 – The distance in feet of a water rocket from the ground after t seconds is given by s(t) = 90t – 16t2. Find the expression fro the instantaneous velocity v(t) of the rocket at any time t.
V(t) = 90 – 32t

13. HW: Worksheet 12-3(day2)