1. Derivatives of Parametric Equations
Lesson 10.2

2. We will use parametric equations
Studying Graphs
Recall that a function, y = f(x) is intersected no more than once by a vertical line
Other graphs exist that are not functions
We seek to study characteristics of such graphs
How do we determine the slope at a point on the graph (for a particular value of t)?
We will use parametric equations

3. Derivative of Parametric Equations
Consider the graph of x = 2 sin t, y = cos t
We seek the slope, that is
For parametric equations
For our example

4. Try It Out Find dy/dx for the given parametric equations
x = t + 3 y = t2 + 1
What is the slope of the line tangent to the graph when t = 2?
What is the slope of the line tangent to the graph when x = 2?

5. Second Derivatives
The second derivative is the derivative of the first derivative
But the first derivative is a function of t
We seek the derivative with respect to x
We must use the chain rule

6. Second Derivatives
Find the second derivative of the parametric equations
x = 3 + 4cos t y = 1 – sin t
First derivative
Second derivative

7. Try This!
Where does the curve described by the parametric equations have a horizontal tangent?
x = t – 4 y = (t 2 + t)2
Find the derivative
For what value of t does dy/dx = 0?

8. Human Cannonball Let's model this motion with parametrics

9. Human Cannonball
Using some software we can get ordered pairs of various locations on the image

10. Human Cannonball
We know

11. Human Cannonball Say his height is 6 ft
We have (412, 121) and (421, 132) as ordered pairs for his height

12. Human Cannonball
Now how far did he travel
We have 314 units

13. Human Cannonball
Let's use the parametric equations to determine the velocity

14. Assignment Lesson 10.2A Page 412 Exercises 1 – 21 Lesson 10.2B
Exercises 22 – 26 all