1. ## Derivative help? Positive/Negative.

I have 3 questions!

QUESTION 1:

Suppose P(t) is the monthly payment, in dollars, on a mortgage which will take t years to pay off.

What are the units of P '(t)?
a. years
b. dollars
c. dollars/year
d. years/dollar

I think the answer is dollars/year.

What is the practical meaning of P '(t)?
a. The rate at which the monthly payments change as the duration of the mortgage increases.
b. The rate at which the amount of money owed changes as the duration of the mortgage increases.
c. The amount left to pay at time t.
d. The amount already paid at time t.

It's (a). The rate at which the monthly payments change as the duration of the mortgage increases.

What is the sign of P '(t)?
a. positive
b. negative

*I know it's negative. But could you give me an explanation as to why? I'm not really sure.

QUESTION 2:

Let f(x) be the elevation in feet of the Mississippi river x miles from its source.

(a) What are the units of f '(x)?

I know it's feet/mile.

(b) What is the sign of f '(x)?

*I know it's negative as well. But I want an explanation please.

QUESTION 3:

Let f(t) be the number of centimeters of rainfall that has fallen since midnight, where t is the time in hours. Interpret the following in practical terms, giving units.

(a) f(2) = 3.0
a. 2 cm of rain fall at a rate of 3.0 cm per hour.
b. A total of 5 cm of rain falls on the ground.
c. When t = 2, 3.0 cm of rain has fallen.
d. When t = 3.0, 2 cm of rain has fallen.

I know it's (c). When t = 2, 3.0 cm of rain has fallen.

(b) f^-1(2.0) = 17
a. When 17 cm of rain have fallen, 2.0 hours have passed.
b. When 2.0 cm of rain have fallen, 17 hours have passed.
c. When t = 2.0, then the rate of rainfall is 17 cm per hour.
d. When t = 17, then the rate of rainfall is 2.0 cm per hour.

I know this is (b). When 2.0 cm of rain have fallen, 17 hours have passed.

(c) f '(2) = 0.6
a. When 0.6 hours have passed, the rate of rainfall is 2 cm per hour.
b. At t = 2 hours, the rate of rainfall is 6 times as great as at t = 1.
c. At t = 0.6 hours, 2 cm of rain have accumulated.
d. When t = 2, the rate at which rain is falling is 0.6 cm per hour.

*I don't know this one!

(d) (f^-1) '(2.0) = 6
a. When t = 6 hours, the rate of rainfall is increasing by 2.0 cm/hr^2
b. When 2.0 cm of rain have fallen, the rain is falling at a rate such that it will take 6 additional hours for another centimeter to fall.
c. At t = 2.0 hours, the rain is falling at a rate of 6 cm per hour.
d. When t = 2.0 hours, the rate of rainfall is increasing by 6 cm/hr^2

I know this is (b). When 2.0 cm of rain have fallen, the rain is falling at a rate such that it will take 6 additional hours for another centimeter to fall.

Thanks so much!

2. What is the sign of P'(t)?
Unless I'm missing something, the sign would be positive, since t is describing the number of months, and the months aren't negative.

3. Originally Posted by melliep
I think the answer is dollars/year.

Correct

Originally Posted by melliep

It's (a). The rate at which the monthly payments change as the duration of the mortgage increases.

Yep.

Originally Posted by melliep

What is the sign of P '(t)?
a. positive
b. negative

*I know it's negative. But could you give me an explanation as to why? I'm not really sure.
What does this mean? The amount of principal remaining?

4. Originally Posted by melliep
I have 3 questions!
The length of your post is a little intimidating! But then one notices it's a bunch of multiple choice..

1.

(a) yes, might help to think of secant (instead of dy/dx think $\displaystyle \Delta$y/$\displaystyle \Delta$x)

(b) yes

(c) explanation: we assume the total mortgage is constant (not counting interest), then of course the longer you draw it out, the lower your monthly payment will be.

2.

(a) yes

(b) explanation: water goes down with gravity

3.

(a) yes

(b) yes

(c) derivative gives instantaneous rate of change so it's choice (d)

(d) yes. but how did you know this and not part (c)?

5. Thank. You. SO. Much.

Your explanations were flawless. I really appreciate it.