Ellipse Problem: The ellipse in Figure 4-7g has the parametric equations x = 3 cos t y =

Chapter 4, Problem 5

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Ellipse Problem: The ellipse in Figure 4-7g has the parametric equations x = 3 cos t y = 5 sin t Figure 4-7g a. Confirm by grapher that these equations give the graph in Figure 4-7g. b. Find an equation for dy/dx. c. Evaluate the point (x, y) when t = /4, and find dy/dx when t = /4. On a copy of Figure 4-7g, draw a line at this point (x, y) that has slope dy/dx. Is the line tangent to the graph? d. Determine whether this statement is true or false: When t = /4, the point (x, y) is on a line through the origin that makes a 45-degree angle with the x- and y-axes. e. Use your equation for dy/dx from part b to find all the points where the tangent line is vertical or horizontal. Show these points on your graph. f. Eliminate the parameter t and thus confirm that your graph actually is an ellipse. This elimination can be done by cleverly applying the Pythagorean property for sine and cosine.