Derivatives of Parametric Equations Lesson 10.2

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

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

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?

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

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

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?

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