Calculus 2 Unit 10 (Test 2)

• Parametric Equations

  x = f(t)

  y = f(t)

  (t) = parameter
• How to sketch the parametric equations

  1. Use the bounds that the problem gives you: ex: -2 < t < 4
  2. Create a table to organize and graph the points
  3. When you finish finding your points, graph them and put arrows to indicate what direction they are going in
• How to eliminate the parameter

  1. Use one of the equations to find t =. Example: y = t + 1, find t = y -1.
  2. Substitute t = y -1 into x = t^2 - 2t. You get x = (y-1)^2 - 2(y-1) and that simplifies to x = (y-1)(y-1) - 2y + 2.
  3. Final answer is x = y^2 - 4y + 3
• How to eliminate the parameter when you have y=6sin(x) and x=6cos(x)

  1. Isolate sin(x) and cos(x). Ex: x= 6cos(t), cos(t) = x/6
  2. Fill it into the x^2 + y^2 = 1 equation, sin^2(x) + cos^2(x) = 1
  3. Final answer (y/6)^2 + (x/6)^2 = 1, y^2 + x^2 = 36.
• How to eliminate the parameter when you have x=5cos(theta) and y=sec^2(theta)

  1. Since sec(x) = 1/cos(x), we can use this and find cos(x). So cos(x) = x/5.
  2. Fill it into the equation: y = sec^2(theta), y= 1/cos^2(theta) = 1/(x/5)^2
  3. Final answer: y= 1/(x/5)^2
• x = f(t)

  y = g(t)

  dy/dx = dy/dt * dx/dt

  with respect to dx/dt cannot equal 0
• d^2y/dx^2 = d/dx * dy/dx

  = (d/dx(dy/dt))/ (dx/dt)
• How to find if at a point there are tangents, find their equations

  1. Find the t values. When y=0, find the values of t.
  2. Find dx/dt and dy/dt, then find dy/dx
  3. Evaluate dy/dx at the t values, usually 2
  4. Fill in the t values, with the x and y values ( original equations ), into the y - y1 = m(x-x1), t being the slope value
• Area under a parametric curve formula

  $A\ =$$\ \int_{a}^{b}g\left(t\right)f'\left(t\right)dt$

  f'(t) = dx/dt

  dx = f'(t)dt
• Parametric curves arc length formula

  L = int from a to b sqrt ( (dx/dt)^2 + (dy/dt)^2 ) dt

  $L\ =$\ \int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}dt
• Polar Coordinates

  ( r, theta)

  r = distance from the origin

  theta = the angle (ex: 0, pi, 2pi )
• If theta is positive: counter-clockwise from the polar axis

  If theta is negative: clockwise from the polar axis
• r can be positive or negative

  if r > 0, it is positive, graph it regularly

  if r < 0, it is negative, reflect it, go in the opposite direction
• cos (theta) = x/r

  x = r cos(theta)
• sin(theta) = y/r

  y = r sin(theta)
• tan(theta) = y/x ( where x does not equal 0)

  r^2 = x^2 + y^2
• To change from polar to rectangular coordinates, use the formulas

  x = r cos(theta) and y = r sin(theta)
• To change from rectangular to polar coordinates, use the formulas,

  r^2 = x^2 + y^2 and tan(theta) = y/x
• Hyperbola graph

  
• Limacon Graph

  
• Cardioid Graph

  
• Ellipse Graph

  
• Rose Curve

  
• soh = sin

  cah = cos

  toa = tan

  cho = csc

  sha = sec

  cao = cot
• How to sketch a polar curve

  1. First find the points to the polar curve by graphing cartesian points. Use theta = 0, pi/2 , pi, 3pi/2, 2pi
  2. Use a table to organize the points and graph it using a polar curve. Follow the direction and draw arrows to indicate its direction.
• Cos(2theta)

  Use a smaller interval

  theta= 0, pi/4, pi/2, 3pi/4, pi, 5pi/4, 3pi/2, 7pi/4, 2pi
• x = f(theta) cos(theta)

  y = f(theta) sin(theta)

  so,

  dx/dtheta = dr/dtheta cos(theta) - r sin(theta)

  dy/dtheta = dr/dtheta sin(theta) + r cos(theta)
• dy/dx =

  (dr/dtheta sin(theta) + r cos(theta))/

  (dr/dtheta cos(theta) - r sin(theta))
• a^2 + 2ab + b^2

  Remember to try and see if you can factor the terms under the square root! ( arc length )