1. Given the following parametric equations. 1 t x= , y= , t = 10 t +9 t -9 a. Find an equation for the line tangent to the curve at the point defined by the given value of t. a. _____________________________________________ b. Determine the value of d2y at this point. dx 2 d2y = _________________________________ dx 2 t =10 2. Using calculus find the area of the shaded region enclosed by the following ellipse. Show work for all of the mathematics/calculus. (Give an exact answer and include the type of units on your answer.) x = 5cos t , y = 8sin t for 0 £ t £ 2p A = ______________________________________ 3. Find the length of the curve. Show all of the mathematics/calculus. (Give an exact simplified answer.) x = 6sin t - 6t cos t , y = 6 cos t + 6t sin t , 0 £ t £ p 2 L = ______________________________________ 4. Give the polar equation r 2 sin ( 2q ) = 10 . a. Find an equivalent Cartesian equation. a. _____________________________________________ b. 5. Sketch the graph of the polar equation (using the Cartesian equation.) Label the scale on our axes and show work. Using calculus find the area of the region shared by the cardioids r = 7 (1 + cos q ) and r = 7 (1 - cos q ) . You must show complete mathematical/analytic work. Give an exact simplified answer and include the type of units in your answer. Hint: Draw the graph. A = ______________________________________ 6. 3 - 3sin q 2 a. Determine the symmetry of the curve analytically. Given the polar curve r = x-axis (yes or no)? ________ y-axis (yes or no)? ________ origin (yes or no)? ________ b. Without the use of a graphing calculator or graphing software, sketch the graph of the curve. Label the scale on your axes and show complete work filling in the table below. Give exact values. q 0 p 6 p 3 p 2 2p 3 5p 6 p 7p 6 3p 2 11p 6 2p r 7. Given the equation of the conic section 7 x 2 - 28 x - 9 y 2 - 72 y = 179 . a. Identify the type of conic section and how you know. Type of Conic Section: ____________________________ How do you know? ___________________________________________________________________ ____________________________________________________________________________________ b. Rewrite the conic section in standard form. b. _____________________________________________ c. If appropriate, find the center. Support your answer. c. ___________________________ d. If appropriate, find the exact coordinates of any foci. Show work. d. ______________________ e. If appropriate, find any vertices. e. _______________________________________ f. If appropriate, find the equations of any asymptotes. f. __________________________________ g. g. _________________________________ Find the eccentricity of the conic section. 7. (Continued) h. Sketch the graph of the conic section. Make sure to show all key features and draw your axes and label the scale on your axes. 8. The given equation has one of its foci at the origin. Determine the polar coordinates of the center and major vertices with a positive r value and the equation of the directrix that corresponds to the focus at the origin. Then carefully sketch the graph of the conic section with these key points labeled. 60 Equation of directrix: _______________________ r= 12 - 6 cos q Center: __________________________ Coordinates of vertices: ______________________________
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