MT1 w soln - Midterm 1 Problem 1 Sketch the curve given in...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Midterm 1 Problem 1 Sketch the curve given in Cartesian coordinates x = t - 2 sin( t ) and y = 1 - 2 cos( t ) t [ - 3 π, 3 π ]. Find an equation of the tangent to the curve. Find the points on the curve where the tangent is vertical. How many points of self-intersection does the curve have? Solution The curve is a trochoid. Problem 34 of Section 10.1 that you had in your homework introduced it to you. Here is the sketch –4 –2 0 2 4 –10 –5 5 10 Figure 1: A trochoid The equation of the tangent to the curve is given by dy dx = 2 sin( t ) 1 - 2 cos( t ) . The tangent is vertical where dy/dx = ±∞ . It follows that it holds for the values of the parameter t such that cos( t ) = 1 / 2. Solving it we obtain t = ± π/ 3 + 2 πk , where k = - 1 , 0 , 1 since t [ - 3 π, 3 π ]. Plugging these values of the parameter into the parametric equations we get x = ± π/ 3 3 + 2 πk and y = 0, where k = - 1 , 0 , 1. Therefore, we have six points on the curve at which the tangent is vertical. From the sketch it’s clear that there are 3 points of self-intersection. Problem 2 Sketch the curve given in polar coordinates r 2 = sin(2 θ ). Find the area it encloses and the area of the surface obtained by rotating the curve about the y -axis. Solution The curve is called lemniscate of Bernoulli. We already met it in one of our discussion section. It lies in the first and the third quadrants. The sketch is given in Fig. 2. Using the symmetry of the curve we obtain the area it encloses A = 2 · 1 2 Z π/ 2 0 r 2 = Z π/ 2 0 sin(2 θ ) = 1 . 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
–0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 Figure 2: lemniscate of Bernoulli Again employing the symmetry of the curve for the area of the surface obtained by rotating the curve about the y -axis we get A y = 2 · 2 π Z π/ 2 0 r ( θ ) cos( θ ) r r 2 + dr 2 dθ.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern