Module 25  Parametric Equations  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 25.2: Chain Rule for Parametric Equations  
This lesson investigates the procedure to find derivatives, such as , , and , for parametric equations x = f(t), y = g(t). The Chain Rule Suppose a curve is defined by the parametric equations x = f(t) y = g(t) The Chain Rule states that the derivative for the parametric curve is the ratio of to . Symbolically, Finding First Derivatives The values of the derivatives dy/dt, dx/dt, and dy/dx for a set of parametric equations can be found using the TI83. Suppose you wish to find the values of the derivatives when t = 0.5 for the parametric equations y(t) = tan t Recall that and that dy/dt represents the rate of change of y with respect to t, dx/dt represents the rate of change of x with respect to t, and dy/dx represents the rate of change of y with respect to x.
Finding Derivatives from a Graph The derivatives at a point on the graph of a parametric curve can be found by using the derivative features of the CALC menu on the Graph screen.
The graph is a hyperbola. The two diagonal lines are not part of the graph. The two lines are similar to the vertical lines that often appear in graphs of functions that have vertical asymptotes.
The calculator returns to the graph.
The value of dy/dx at t = 0.5 is approximately 2.0858. That is, the yvalues are increasing about 2.0858 times as fast as the xvalues. 25.2.1 Use the CALC menu to find the value of dy/dt and dx/dt at t = 0.5. Do these values satisfy the Chain Rule for parametric equations? Click here for the answer. Finding the Equation of the Tangent Line The slope of the tangent line is dy/dx 2.0858. To find the x and y coordinates of the point on the curve when t = 0.5, use the value feature in the CALC menu.
The coordinates of the point on the curve when t = 0.5 are approximately (1.1395, 0.5463). Using the point just found and the value of dy/dx for the slope of the line, the equation of the tangent line is Graph this function parametrically.
25.2.2 Graph the curve given by the parametric equations below and find the equation of the tangent line to the curve when t = 2. y = 1  cos(t) Click here for the answer. 

< Back  Next >  
©Copyright
2007 All rights reserved. 
Trademarks

Privacy Policy

Link Policy
