Presentation on theme: "Homework, Page b. The window appears to be"— Presentation transcript:
1Homework, Page 530 1. b. The window appears to be Match the parametric equations with their graph. Identify the viewing window that seems to have been used.1.b. The window appears to be
2Homework, Page 530 5. t –2 –1 1 2 x 3 4 y –0.5 Und. 2.5 (a) Complete the table for the parametric equations and (b) plot the corresponding points5.t–2–112x34y–0.5Und.2.5
3Homework, Page 530Graph the parametric equations x = 3 – t2, y = 2t in the specified parameter interval. Use the standard viewing window.9.
4Homework, Page 530Eliminate the parameter and identify the graph of the parametric curve.13.
5Homework, Page 530Eliminate the parameter and identify the graph of the parametric curve.17.
6Homework, Page 530Eliminate the parameter and identify the graph of the parametric curve.21.
7Homework, Page 530Eliminate the parameter and identify the graph of the parametric curve.25.
8Homework, Page 530Find a parameterization for the curve.29. The line segment with endpoints (3, 4) and (6, –3).
9Homework, Page 530 33. Quadrant I Refer to the graph of the parametric equationsgiven below. Find the values of the parameter t that produce the graph in the indicated quadrant.33. Quadrant I
10Homework, Page 53037. Ben can sprint at the rate of 24 fps and Jerry sprints at the rate of 20 fps. Ben gives Jerry a 10-ft head start. The race can be modeled by the parametric equations:a. Find a viewing window to simulate a 100-yard dash. Graph simultaneously with t starting at t = 0 and t-steps of 0.05.b. Who is ahead after three seconds and by how much.Ben is ahead by 2 ft.
11Homework, Page 53041. The graph of the parametric equations x = 2 sin t and y = 2 cos t is a circle of radius 2 centered at the origin. Find an interval of values for t so that the graph is the given portion of the circle.a. The portion in the first quadrant.b. The portion above the x-axis.c. The portion to the left of the y-axis.
12Homework, Page 53045. Kirby hits a ball 4ft above the ground with an initial velocity of 120 fps at 30° above the horizontal toward a 30-ft high fence 350 ft away. Suppose that the moment Kirby hits the ball, there is a 5 fps wind gust blowing in the direction Kirby hit the ball.a. Does the ball clear the fence?
13Homework, Page 53045. Kirby hits a ball 4ft above the ground with an initial velocity of 120 fps at 30° above the horizontal toward a 30-ft high fence 350 ft away. Suppose that the moment Kirby hits the ball, there is a 5 fps wind gust blowing in the direction Kirby hit the ball.b. If so, by how much does it clear? If not, could the ball be caught?The ball clears the fence by about 1.5 ft.
14Homework, Page 53049. Orlando hits a ball 4ft above the ground with an initial velocity of 160 fps at 20° above the horizontal toward a 30-ft high fence 440 ft away. How strong in fps must a wind gust be toward or away from the fence to cause the ball to hit within a few inches of the top?
15Homework, Page 53053. The graph of the parametric equations x = t – sin t andy = 1 – cos t is a cycloid.a. What is the maximum value of y = 1 – cos t? How is that value related to the graphb. What is the distance between neighboring x-intercepts?
16Homework, Page 530A particle moves along a horizontal line so that its position at any time t is given by s (t). Write a description of the motion.57.
17Homework, Page 530Which of the following points corresponds to t = –1 in the parameterizationa.b.c.d.e.
18Homework, Page 530 65. Consider the parametric equations , and on . a. Graph the parametric equations for a = 1, 2, 3, 4 in the same window.b. Eliminate the parameters in the parametric equations to verify that they are circles. What is the radius?
19Homework, Page 530 65. Now consider the parametric equations and c. Graph the equations for a = 1 using the following pairs of values for h and k.h2–2–43k–3
20Homework, Page 53065. d. Eliminate the parameter t in the parametric equations and identify the graphs.e. Write a parameterization for the circle with center (–1 , 4) and radius 3.
25Quick Review Solutions Use the Law of Cosines to find the measure of the third side of the given triangle.4.40º8105.35º6116.47
26What you’ll learn about Polar Coordinate SystemCoordinate ConversionEquation ConversionFinding Distance Using Polar Coordinates… and whyUse of polar coordinates sometimes simplifiescomplicated rectangular equations and they are useful incalculus.
40What you’ll learn about Polar Curves and Parametric CurvesSymmetryAnalyzing Polar CurvesRose CurvesLimaçon CurvesOther Polar Curves… and whyGraphs that have circular or cylindrical symmetry often havesimple polar equations, which is very useful in calculus.
41Polar Curves and Parametric Curves Polar curves are, in reality, a special type of parametric curves, where , for all values of θ in some parameter interval that suffices to produce a complete graph (in many cases, 0 ≤ θ ≤ 2π).
42Symmetry The three types of symmetry figures to be considered are: The x-axis (polar axis) as a line of symmetry.The y-axis (the line θ = π/2) as a line of symmetry.The origin (the pole) as a point of symmetry.
43Symmetry Tests for Polar Graphs The graph of a polar equation has the indicated symmetry if either replacementproduces an equivalent polar equation.To Test for Symmetry Replace Byabout the x-axis (r,θ) (r,-θ) or (-r, π-θ)about the y-axis (r,θ) (-r,-θ) or (r, π-θ)about the origin (r,θ) (-r,θ) or (r, π+θ)