Calculus and Above

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1) Consider a function I(m...

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).

(b) What is the range of the function?

(c) Does the function increase or decrease in each of the variables?

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

(c) The unit cube in R3 including its boundary.

(d) The domain of the function f(x, y) = 1/((x^2)+(y^2 )).

(e) The domain of the function f(x, y, z) = xyz.

(f) The hollow unit sphere in R3.

(g) The set of all points that are more than 3 units away from (0, 3, 0) in R3.

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]

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)?

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?

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).

(b) What is the range of the function?

(c) Does the function increase or decrease in each of the variables?

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

(c) The unit cube in R3 including its boundary.

(d) The domain of the function f(x, y) = 1/((x^2)+(y^2 )).

(e) The domain of the function f(x, y, z) = xyz.

(f) The hollow unit sphere in R3.

(g) The set of all points that are more than 3 units away from (0, 3, 0) in R3.

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]

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)?

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?

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