Follow The Directions For Each Problem. You Must Show All St

Follow the directions for each problem. You must show all significant steps that you used to obtain your solution. Write clearly and legibly. Showing your work is required; however, there will be no partial credit awarded.

1) (9 pts.) The Fresnel integrals (integrals of sin(x2) and cos(x2)) occur in the problem of Fresnel diffraction in optics. Approximate to four decimal places ³ sin dxx .

2) (6 pts.) At time t, the position of a particle moving along a curve is given by ð‘¥(ð‘¡) = 𑒠− ð‘’ ð‘Žð‘›ð‘‘ ð‘¦(ð‘¡) = 3ð‘’ + ð‘’ . (a) Find all values of t at which the curve has horizontal or vertical tangent lines. (b) Find ð‘‘𑦠ð‘‘ð‘¥â„ in terms of t. (c) Find lim → , and discuss what this means about the position and path of the particle.

3) (3 pts.) Show that cosh2(x) – sinh2(x) = pts.) Let{ð‘Ž } = ( ) . Which of the following statements are true about {an} on the interval > f,0 ? (Use concepts that we have covered in class to support your answers). (a) The sequence is unbounded. (b) The sequence is bounded. (c) The sequence is monotonic and bounded. (d) The sequence is unbounded and monotonic. (e) The sequence converges. (f) The sequence conditionally converges. (g) The sequence diverges.

5) (12 pts.) Evaluate the following integral by using any two methods that were covered in class this semester (clearly state the methods used): ³ )cos( S dxxx Mehmet Akif Ozkaraaslan DUE: May 11, 2018 No Late Submissions Name:_____________ Spring pts.) In general, we do not expect power series to be useful in evaluating indeterminate forms, except when x approaches zero. Show, however, that xn x ex fo lim can be evaluated via a power series.

7) (6 pts.) The hour hand of a clock travels 12 1 as fast as the minute hand. At two o’clock the minute hand points to 12 and the hour hand points to 2. By the time the minute hand reaches 2, the hour hand points to __________.

By the time the minute hand reaches __________, the hour hand points to ________ and so on. Find a general expression for the movement of the minute hand, and a general expression for the movement of the hour hand (use sigma notation as appropriate). At what time will the two hands coincide? (Round your answer to the nearest second, and use the format hh:mm:ss).

Celestial Orbits

8) Around 1605, Johannes Kepler (1571 – 1630) proposed his first law regarding planetary motion based on celestial observations; it was later confirmed using Newton’s Laws of Motion. According to Kepler’s First Law, the orbits of planets (and other bodies such as asteroids) revolving around the Sun are ellipses with the Sun at one focus. Polar coordinates are ideally suited for describing planetary orbits, because the description of an ellipse in polar coordinates assumes that one focus of the ellipse is at the origin. Let us recall a few facts about ellipses in polar coordinates.

One equation of a conic section in polar coordinates is ð‘Ÿ = ( ) , where 0 ≤ 𜃠≤ 2ðœ‹. When the eccentricity (e) satisfies the condition 0

Halley’s Comet

One thing to keep in mind when solving the following problems is that this is a polar coordinate problem set. That means, unless otherwise indicated, there are no coordinates expressed as x and y values (however, the second problem—problem b does ask for Cartesian coordinates). Do not use results that are in a Cartesian format to solve a polar coordinate problem. Halley’s Comet has a 76-year period, and was last seen in 1985.

It has one of the most eccentric elliptical orbits (e ≅ 0.967) of all bodies orbiting the Sun. The perihelion distance is 0.57 AU and the aphelion distance is 35.08 AU. (a) Find the length of the major and minor axes of the orbit of Halley’s Comet. (b) Find an equation of the orbit in Cartesian coordinates with the origin at the center of the orbit. (c) Find an equation of the orbit in polar coordinates with the Sun (one focus) at the origin. (d) Find the slope of the path of Halley’s Comet when 𜃠= 𜋠2. (e) Plot the orbit of Halley’s Comet. (f) Did you notice anything interesting about the slope that was calculated in part d? If so, then describe what you noticed.