Presentation on theme: "The Calculus of Parametric Equations"— Presentation transcript:
1The Calculus of Parametric Equations As we have done, we will do again. Parametric curves have tangent lines, rates of change, area under and above, local and absolute maxima and minima, horizontal and vertical tangents, etc.So we will find derivatives and integrals and interpret their meanings.I.O.W. – SAME CONCEPTS…DIFFERENT FUNCTIONS
2How would one go about differentiating a pair of parametric equations How would one go about differentiating a pair of parametric equations?...It is not too bad…but check out how we get to a process that is “not too bad.”
3We get the following result… Which makes sense algebraically!!!
4a.) Graph the above on the interval by hand. b.) Find the equation of the line tangent to the curve at t= 4.c.) Find each of the following and discuss their meaning
6a.) Find the value(s) of t at which the curve described above has a horizontal tangent line. Equation of tangent line? Does this match with our graph?GRAPH IN YOUR GC TO SEE!!! [-2,5,.1],[-2,30,1],[-2,10,1]b.) Find the value(s) of t at which the curve described above has a vertical tangent line. Equation of tangent line? Does this match with our graph?c.) Find the equation of the tangent line at t=3d.) Find the intervals of t during which the tangent lines have positive slope. Does this match with our graph?
7What do we need to find the intervals on which the tangent lines are decreasing? I.O.W….where the graph of the curve is ___________________.Concave down
81. Find the first derivative (dy/dx). To find the second derivative of a parametrized curve, we find the derivative of the first derivative:1. Find the first derivative (dy/dx).2. Find the derivative of dy/dx with respect to t.3. Divide by dx/dt.
9For the graph of the parametric curve described above, find the interval(s) where the curve is concave down.
10How can we use to find the locations of any relative extrema on our curve?
11Gee…it sure would be nice if we could use our GC to find some of this information we just did by hand!!!Let’s use our GC to check our answers to some of the problems we completed…We will try both in graphing mode and from the home screen!
12Notice on our graph that the curve intersects itself Notice on our graph that the curve intersects itself! Don’t worry, I will not ask you to find where this is (nor will the BC exam)…however you will be asked the following:A curve defined by x(t) =… and y(t) = … intersects itself at the point (5,0). Find the equations of the two tangent lines at (5,0)Two in the what now?!?!? – Regraph your curve until you are convinced there are two tangent lines.
13To solve such a beast:1.) Find the values of t (yes, the word is plural, but why?) where the curve crosses (5,0). Use the parametric equation that is easiest to solve!!!2.) Evaluate dy/dx at the above t-values and roll from here!
14We have covered much of the calculus interpretation of parametric curves…now a few more examples so you are in great shape!!!
15a.) the equations of the two tangent lines that occur where the graph intersects itself b.) the point(s) where the curve has a horizontal tangent on the intervalc.) using a GC, the values of t and the point(s) where the graph has a vertical tangent on the same interval as in b.
16Find the slope of the tangent line to the curve described by the following equations at t = 1. GC to the rescue!!!
17Given the curve described by the equation below, find the equation of the tangent line at t=1.
18a.) Find the equations of the tangent lines at the point where the curve described by the equations below crosses itself (you may have to find by graphing the curve on your calculator for the homework!!!)b.) Where is the graph concave up?
19Find all the points of horizontal and vertical tangency that occur on the graph described by the parametric equations below:
20The Advantage of Parametric Equations as Witnessed in Things that are Launched The following is a practical application which some of you will befamiliar with. It serves as a highlight of the advantage ofusing p-metric eqns. to describe physical events that involvetwo independent components which change w/ respect totime….you will see much more of this thought process when wedelve into vectors…YOU WILL ALSO SEE THIS ON THENEXT TEST!
21A ball is thrown with an initial velocity of 88 ft/sec A ball is thrown with an initial velocity of 88 ft/sec. at an angle of 40 degrees to the ground. The ball is released at an initial height of 6 ft above the ground. Find the following (assume the acceleration due to gravity is -32 ft/sec.):a.) A pair of parametric equations that describes the position of the ball at any time t (horizontal and vertical components).b.) The maximum height of the ball.c.) The range of the ball (maximum horizontal distance)
22Imagine a roller coaster that travels one “loop” on its tracks modeled after a parametric curve. The arc length of the p-metric curve is the distance the roller coaster travels once around the tracks.As you calculate arc length MAKE SURE YOU ARE CALCULATING ONE LOOP!!!For example – If you calculate the arc length of a circle by setting your limits so that you are travelling the circle a second time, then you will be double counting portions of the curve!!!!...which would potentially bring disaster upon the rest of the world…except on you…but the rest of the world is out of luck.
23Let’s Generate the Formula! 1.) With our original arc length formula.2.) Conceptually
24Find the length of the curve defined by the parametric equations below on the given interval. Does the curve repeat itself on the above interval? – Check with GC or graph by hand.* Note that dx/dt and dy/dt do not equal zero at the same time on the above intervalTherefore our curve is considered smooth.
25Find the length of the curve described by the following parametric equations on the given interval.
26Find the length of the curve described by the following parametric equations on the given interval.
27SURFACE AREA!!!(Curves mustSame idea as with rectangular form!!!Each cross section is a disk of which we want the SASince we are adding up a bunch of areas of these super tiny disks, we will integrate…but what does the finished integrand look like for figures rotated about the x-axis? y-axis?
28Find the area of the surface generated by revolving the region bounded by the x-axis and the curve and on the interval described below about the x – axis.Ummmm…yeah…definitely use your GC!!!ABOUT THE Y-AXIS?
29Find the surface area of the curve generated by revolving the curve about the x-axis on the given interval.GC fa sho.
30Before we say goodbye to parametric curves, let’s find the area under a parametric curve! WE KNOW:Makes sense with units!
31a.) Graph the region described by the following in GC b.) Find the area of the region.
32Let’s round the p-metric madness out with some FR!