(Math 253)Homework 3(Due 3/4)

12.3: 24,26,27,29(only compute the direction cosines, not the angles),32,34,36,37,40,45

12.4: 1,3,7,10,13,17,18,19,20,27

12.5: 2~5

Remind you: Extra credit problems in section 10.4 is due 2/26! You are super welcome to come and discuss them with me, review Riemann sum, write all the problems as the limit of Riemann sums first!


(Math 253)Sum up of parametric equations and polar coordinates

Hi all,

Now we are done with Chapter 10, we will move on to analytical geometry. Chapter 10 is mostly introducing a tool, something you can use to simplify your computation, work and therefore life.

If you are given F(x,y)=0, a curve in the normal Cartesian coordinate system, you might want to write it in terms of a new parameter, say t. There are a couple of reasons for this: it is sometimes easier to see the property of the curve (x^2+y^2=1 with x=cos(t) and y=sin(t) clearly tells you the periodicity of the curve), sometime you want to obtain a more dynamic view of the curve (long exposure photo vs. a video), etc. You can, of course, choose to use whichever one in a particular problem.

Every curve y=f(x) always have a trivial parametric equation: x=t and y=f(t). You have a lot of freedom in terms of how to choose your parametric equations x=a(t) and y=b(t). As long as F(a(t),b(t))=0, the parametric equation is legitimate. This is the same idea as substitution in integral calculus, so just like before, make sure to use the chain rule whenever you need to integrate or differentiate.

The idea of polar coordinate system is like parametric equations, just with two parameters r and theta: r measures how far the point is to the origin, and theta is the angle measured in the standard position.  The conversion between Cartesian and Polar is always the same: from polar to Cartesian: x=r*cos(theta) and y=r*sin(theta); or from Cartesian to polar: r=sqrt{x^2+y^2}, tan(theta)=y/x. Same idea as the parametric equations, make sure to apply chain rule whenever you use differentiation or integration.

Use either coordinates, as long as it simplifies your work.


(Math 152)Sum up of Riemann sum

Hi all,

By now we are mostly done with the theory part of definite integration. Before we dive into all the techniques, here is a sum up of the Riemann sum and the idea of cutting things into pieces then add up.

You have seen in Math 151 (or any Cal 1 class you took) how to define a tangent line at point P(x0,y0): you start off computing a secant line at point P with another point Q(x,y) on the curve. You let the point Q vary, and as Q gets closer and closer to P, we define the tangent line as the limiting result. This kind of “approximate then take the limit” idea is the same essential idea when we define definite integrals.

The first questions we see in Math 152 is computing the area under curve. We will again start with approximation, then take the limit. We cut the region under curve into n vertical strips (n being a large integer), approximate the area of each strip by a rectangle (of course, you can choose to use trapezoid, or whatever else whose area is easy to compute), and add up all the areas to get the approximate area of the whole region. By now, the approximation step is over. Taking the limit in this case is making n really large–the larger n is, the thinner the strips are, the more accurate the approximation is. The limit is then defined to be the area under curve.

This kind of “cut things up, approximate piece by piece and take the limit” works for the definition of all cases, as long as the object you consider is a product. An example is defining the displacement. We know if the velocity is constant, then the displacement between time t1 and t2 is simply velocity*(t2-t1). What if the velocity is not constant? We zoom in in time  until the time interval is so small, it looks like the velocity is almost constant (like Neo in the Matrix. Remember how he dodged all the bullets? Apply that skill here), we approximate the displacement in that small time interval by the product of a sample velocity (say, the velocity at the end of the small time interval) and the length of the interval. Add all of these small displacements up to obtain an approximation to the total displacement between t1 and t2, then we take the limit: limit is taken so that each approximation is good, which means the small time intervals are really small. Finally, we define this result to be the displacement.

This would be a consistent theme throughout the course. Later we will learn how to compute volumes and work, one from analytical geometry and one from physics. How did people come up with these fancy formulas and computations? It is all the same idea we have talked about in Section 5.1: cut things up into small pieces, approximate it then take the limit.