- Your 7.1 homework is graded.
- 2
- 34: this one was tricky, just because the "obvious" choice for u and v by parts wasn't the right choice. You can also combine this with substitution, which several of us carried out in different ways.
- 46 -- don't forget your graphs.
- We have an exam through 7.1 this Friday, next time: 6.4, 6.6, 6.8, and 7.1
- Your 8.1 homework will be due Monday, 4/20, after the exam.
- We're going to jump into chapter 10 today, but we'll begin with any questions you have about material in the four sections that will be covered on the exam.
- And now for something completely different....
A good introduction to parametric curves is given by ballistics. If we shoot a bullet into the air with speed v horizontally, and we neglect all forces but gravity, then the bullet will trace out a parabola (some bullets are larger than others):
Now how might we characterize the path of the bullet? The answer is a parametric curve, of the form C(t)=(x(t), y(t)).
or If we wish, we can solve for t in the equation for x and use that to eliminate the parameter t from the equation for y, hence getting an equation for the parabola traced out: According to this parameterization, where is the "bullet" at time t=0? In which direction is the motion occuring -- left to right, or right to left? (You probably have a parametric graphing mode on your calculator -- you might like to try it out.)
Orbits of planets in the heavens, movements of ants on a hill, a robotic arm in an assembly plant: all these can be described by parametric curves.
- A couple of famous examples involving the cycloid:
- The Brachistochrone Problem (1696, solved by Newton in a single day after he heard the challenge)
- The Tautochrone Problem (1673)
- Parametrizing a line:
- Parametrizing a circle:
- There are two distinctively different kinds of derivatives that
we're going to have to consider here:
- There are the time-derivatives, which give us the actual
speeds of the "particles" as they zip along in time. For the bullet,
the speed will be
- There is the slope of the tangent line to the curve traced
out by the particle, which one can obtain using the time-derivatives:
- There are the time-derivatives, which give us the actual
speeds of the "particles" as they zip along in time. For the bullet,
the speed will be
- One of the most important applications of parametric curves is in
the Bezier curve section. There are lots of important instances
where we need to be able to design custom curves, and this is one of
the most common ways of creating them. For example, you can use them to
create your own "sigmac", using control points. (I had all my
students create their own in numerical analysis.)
- Examples:
- #1, p. 665
- #3
- #7
- #15
