- Return of your exams
- Three possibilities for the radius of convergence
Have you forgotten the series for
Have you forgotten how to integrate (or differentiate) series?
Have you forgotten that a power series is just an infinite polynomial? (I saw all kinds of goofy things, like powers of x in the denominator, or even sin(x) down there....).
- Limit comparison, but many of you don't see how to write the terms. If you'd write down explicitly a few terms, you'd get the idea of what the ratio of
looks like.
- See problem 1. a and b don't have to be at the limits of the interval of convergence; they have to be inside the interval of convergence, so that the equality makes sense.
- Use the fact that the alternating series has terms that eventually decrease, and the difference between the true value and the partial sum is smaller than the first neglected term.
- All you need to do is evaluate the derivatives
, then assemble them into the Taylor series.
- You have to move parallel to the axes.
- Part a: the vector should be just a scalar multiple of
. Part b: there are an infinite number of vectors perpendicular to
- Draw the vector velocity, which has components to the north and west. So she'll be moving northwest, at 13 mi/hr relative to the surface of the water.
- This is just a dot product of force and displacement vectors. The biggest problem was units (some didn't give any; some confused them).
- Three possibilities for the radius of convergence
- Quiz opportunity tomorrow will be over 10.2
- 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 have a parametric graphing mode on your calculator -- let's 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
