Merced College; Don Power

 

CALCULUS, CH 11 – STUDY GUIDE

 

1.  Eliminate the parameter:  x = 3tan(t)-3,  y = 2sec(t)+2.  What type of conic section do you get?  Sketch its graph.

 

2.  Find the slope of the tangent line to the graph of x = 2t2+1,  y = t3-1 at t = 2.

Use the second derivative to determine whether the graph is concave upward or concave downward at that point.

 

3.  Set up an integral for the length of the curve for a function given parametrically
(e.g. x = 2t2+1,  y = t3-1, from t=1 to t=3)

 

4.  Sketch and find the area enclosed by r = 2+2sin(q)

 

5.  Sketch and find the area enclosed by one loop of  r = 2cos(5q)

 

6.  Find the derivative of a polar equation and use it to determine the point(s) at which the function has vertical tangents and horizontal tangents.  Ex:  r =4+4cos(q)

 

7.  Convert a polar equation to rectangular form.  Example:  r = 4cos(q) - 3sin(q)

 

8.  Convert a rectangular equation to polar form.  Example:  y = 3x2

 

9.  Find the vertex, focus, and p for the parabola  .  Sketch.

 

10.  Find the vertices, foci and eccentricity, and sketch:

 

11.  Find the vertices, foci and asymptotes, and sketch:  9x2 - y2 -36x - 6y + 18 = 0

 

12.  Find an equation of the parabola with vertex (3,-2) and focus (3,1).

 

13.  Find an equation of the ellipse with eccentricity  and vertices .

            (Notice from the vertices that a = 5, not 3).

 

14. Find an equation of the hyperbola with asymptotes y = ±4x+5 and foci (0,6) and (0,4)

 

15.  Find the equations of the tangent line and the normal line to the hyperbola 5x2-4y2=16 at the point in the first quadrant where x=4.

 

16. Find the equations of the tangent line and the normal line to the ellipse 3x2+2y2=30 at the point in the first quadrant where x=3.

 

17.  Find the volume of the solid formed by revolving the graph of 4x2+y2=1 about the   x-axis.

 

18. Find the volume of the solid formed by revolving the region bounded by the graphs of 4x2-8y2=1, y=0 and x=3 about the x-axis.

 

19.       For the conic section  

a.   Use the discriminant to determine whether this object is an ellipse, a hyperbola or                       a parabola

  1. Find the exact values of sinq and cosq and find the specific conversion formulas for x and y for the rotation to eliminate the xy term.

c.   Apply the conversion formulas to find the new equation in terms of

The relevant formulas are.

 

20.       Sketch the graph of   in polar coordinates;

            Based on the graph, identify the

                        Type of conic section

Center  (in rectangular coordinates)

                        Foci     (in rectangular coordinates)

                        Vertices            (in rectangular coordinates)

                        Eccentricity

                        Equation of the conic (in rectangular coordinates)

 

Return to:  Merced College; Don Power               Updated 01/16/07 by Don Power