Since This Is a Mortgage, It Would Be Reasonable to Assume the Following

1) Consider a function I(m; t; p; d) which describes the total amount of interest paid over the whole course of a mortgage, given that m is the original loan in dollars, t is the ammortization period in years, p is the interest rate in percentage (compounded yearly) and d is the monthly payment rate. Considering this a function of four variables, answer the following questions: (8)

(a) What is the domain of the function. (You should imply reasonable practical limits on domain, as well as mathematically necessary limits).

Since this is a mortgage, it would be reasonable to assume the following:

5,000 ≤ m ≤ 10,000,000 (mortgage ranging from a backwoods cabin to a luxury home.)

3 ≤ p ≤ 16

100 ≤ d ≤ 10,000

Thus a good domain would be

Note: This is not a well-defined function. m, t, p, and d are not independent variables because d is itself a function of m, t, and p.

(b) What is the range of the function?

I will range from 400 to 40,000,000

I will increase as any of m, t, p increase. Any increase in amount being financed, interest rate, or length of loan will naturally increase the cost of the mortgage.

I will decrease as d increases. Increasing payments will pay off the mortgage faster, hence will decrease the cost of financing, thus lowering the overall interest.

2) Which if the following sets are open, closed, both or neither?

[Notes for understanding:

The following metaphor only works for points in Cartesian space, but that’s what we’re looking at here. Think of a “open neighbor of P” as the inside of a circle (in R2) or a sphere (in R3) with P at the center.

Now a set is “closed” if each point not in the set can be isolated from the set by some open neighbor; the set includes its own boundaries. More formally, a set is closed if it contains all its limit points.

And a set is “open” if each point in it is contained in an open set.]

[Every point inside the cube or making up its surface is in the set we’re talking about. Any point not in the cube or on its surface can be isolated from the cube by creating a small enough sphere around the point (radius is half of the distance the point is from the cube).]

(d) The domain of the function f(x, y) = 1/((x^2)+(y^2 )) is the entire Cartesian plane except (0,0). Thus it is an open set.

[Every point in the set is contained in a neighbor so it is open. However,the function tries to reach 0 making 0 an unreachable limit, so therefore it is not closed.]

(e) The domain of the function f(x, y, z) = xyz are all triples from which is an open and closed set.

[Almost the same as (d), except it does include 0.]

(f) The hollow unit sphere in R3 is a closed set consisting of only the points that define the sphere.

[Any point within the sphere can find a neighbor also with the sphere.]

(g) The set of all points that are more than 3 units away from (0, 3, 0) in R3 defines a sphere, hence that set is also closed.

[All points within the sphere can be isolated in neighbors, but not the points on its surface. For any point on the surface, every neighbor of that point will grab some points outside the sphere.]

3. Find the set of points where the following functions are discontinuous. Are any of these sets open or closed?

f(x, y, z) = abs[x + y + z] has no discontinuities because both addition and absolute value are well-defined over all real numbers.

4. Let a and b be non-negative integers., can you construct a general criterion for when the function f(x, y) = (x^a)(y^b)/ ((x^2) + (y^2)) has a well defined limit at (0, 0)?

f should have a well-defined limit at (0,0) when a≥2, b≥2

[Notes: I am using the idea that the numerator must equal or exceed the denominator exponentially in order for the function to have the asymptote as its limit rather than infinity. The only possibility I have not been able to deal with yet is if a+b=2, which suggests a possible asymptote. I will think more about this overnight.]

5.Consider this function: f(x, y) = [(x^2)-(y^2)]/(x-y) This function is not defined on the line x = y. What values would you add to the function on that line to make the function continuous?

f(x,y) = x+y when x = y.

[Notes: Just factor the numerator

So the limit should be .]