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