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GBK Precalculus Jordan Johnson

GBK Precalculus Jordan Johnson. Today’s agenda. Greetings Review HW due yesterday (~20 min): Section 4-4, #29-43 odd. Intro (~5 min) Finish Pendulum Lab (~20 min) Lesson: Parametric Equations (~25 min) Homework Clean-up. Objectives. Understand how parametric equations work:

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GBK Precalculus Jordan Johnson

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  1. GBK PrecalculusJordan Johnson

  2. Today’s agenda • Greetings • Review HW due yesterday (~20 min): • Section 4-4, #29-43 odd. • Intro (~5 min) • Finish Pendulum Lab (~20 min) • Lesson: Parametric Equations (~25 min) • Homework • Clean-up

  3. Objectives • Understand how parametric equations work: • Graph them. • Know what they mean. • Use them to model motion. • Where possible, eliminate their parameter to produce one equation you can analyze. • Note: Today’s examples will use trig functions, but parametric equations won’t always.

  4. Parametric Equations • Not all relations we might want to graph are functions: • We could solve for y… • …but that gives us arather messy solution.

  5. Parametric Equations, cont. • Instead of having y depend on x, we could make x and y both be dependent variables (functions). • This requires a third variable, conventionally t. • Sometimes this is also preferable because y doesn’t conceptually depend on x anyway. • As in the pendulum problem.

  6. Pendulum Motion Worksheet • Steps 1-3: Obtain: • the period of the swing that results from releasing the pendulum from 30cm off in the x-dimension • the period that results from pushing the pendulum from vertical to 20cm off in the y-dimension. • Step 4: Derive equations. • Step 5: Compute values at various times. • Steps 6-10: Graph and analyze. • Step 11: Algebraically, remove t if possible.

  7. Ellipses and Parametric Equations • The parametric equations for an ellipse are: • x = h + a cos t • y = k + b sin t • (h, k) is the ellipse’s center. • a and b are the ellipse’s x- and y-radii, respectively.

  8. Examples • Put your grapher in parametric mode, and plot: • x = 5 cos t • y = 7 sin t • Use degree mode. • x- and y-ranges should be about [-8,8]. • t-range should be [0°,360°].

  9. Examples • Put your grapher in parametric mode, and plot: • x = -1 + 9 cos t • y = 3 + 2 sin t • Use radian mode. • x-range should be about [-11,10]. • y-range should be about [-6,6]. • t-range should be [0,2].

  10. “Eliminating the Parameter” • Idea: • Substitute to eliminate t. • Use trig identities and/or algebra. • Example: • x = 3t + 1, y = t2  t= x-1/3 (by division) y = (x-1/3)2(bysubstitution) = (x2)/9– 2x/9+ 1/9 • So these equations are for a parabola.

  11. “Eliminating the Parameter” • Idea: • Substitute to eliminate t. • Use trig identities and/or algebra. • Example: • x = 2 cos t, y = 3 sin t  x2 = 4 cos2t, y2 = 9 sin2t(by squaring)  (x2)/4 = cos2t, (y2)/9= sin2t(by division)  (x2)/4+ (y2)/9= cos2t + sin2t (by addition)  (x2)/4 + (y2)/9 = 1 (by the Pyth. identity)

  12. Homework • From Section 4-5 (pp. 180-182), do: • Reading Analysis questions. • Exercises Q1-Q10. • Exercises 1-9 odd, 15-19 odd. • Due Monday, 1/10.

  13. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and side tables). • See you tomorrow!

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