I know this is a lot of stuff, but I want to be certain that I have all of this information correct. If there are any errors, I feel like they would come from the first problem or from on the last one. Many thanks to anyone willing to help me out with any of this.
1) Find the surface area of the solid obtained by revolving for about the y-axis
Surface Area
2) Find an equation relating and . Then sketch the curve C whose parametric equations are given and indicate the direction as increases
a) , for all t
My graph looks like an elongated graph which goes through (0,0) and has a horizontal tangent at (-1,-1). The direction of t is going upwards.
b) for
not since
My graph for this one is the right half of a circle of radius 1 centered on the origin. The direction of t is going clockwise.
3) Let C be the curve with parametrization for all t
Find equations of the tangent lines to C at the points corresponding to and . At what point(s) will the graph have a horizontal tangent line? Also find . Use all this information to sketch the curve C.
for , and equation of the tangent line is
for , m is undefined and the equation of the tangent line is
For the graph I have vertical tangents and x intercepts at (-16, 0), (16, 0), a horizontal tangent at (0, -4). The graph crosses itself at the y-intercept (0, 8), is concave down while above the x-axis and concave up when below. It makes a big loop from left to right.
Any help is enormously appreciated
...and just for your future benefit: When you were eliminating the paramater in 2. I would have stopped at because there is no reason whatsoever to go any further. Note that this is an excelent form to have this particular equation in because it is clear that this is nothing more than the graph of a cubic that has shifted one unit to the left and one unit down.