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Derivatives of Parametric Equations. Lesson 10.2. 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
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Derivatives of Parametric Equations Lesson 10.2
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 characteristicsof such graphs • How do we determine the slopeat a point on the graph (for aparticular value of t)? We will use parametric equations
Derivative of Parametric Equations • Consider the graph ofx = 2 sin t, y = cos t • We seek the slope, thatis • For parametric equations • For our example
Try It Out • Find dy/dx for the given parametric equations • x = t + 3y = 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 ty = 1 – sin t • First derivative • Second derivative
Try This! • Where does the curve described by the parametric equations have a horizontal tangent? • x = t – 4y = (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 • Page 413 • Exercises 22 – 26 all