Calculus question

Factual Questions
1

Disclaimer: This is school related, but it not, I repeat, NOT homework. My calculus exam is in two days and I have been studying like crazy, rereading the textbook, reviewing my notes, doing questions on previous exams. I generally haven’t had much problem, and I’ve been able to figure the questions out with some thinking, but there’s one I can’t get and it’s driving me crazy. I’ve searched my textbook and my notes for some indication on how to do it, bt alas, nothing. I’ve asked my brother for help, but he’s forgotten how to do it, so the 'Dope is my absolute last resort. With this long disclaimer said, here it goes:

There are two parts to this question, which is:
a) Beth has just but a $500 deposit into a new savings account at the Calculus Credit Union. The nominal annual interest rate is 4%, compounded continuously. How long will it take for her deposit to double?

I get this part of the question.

The forumal for continuous growth is A=Pe^(kt)
Pluck everything in, and it’s 1000=500e^(0.04t) which turns into 2=e^(0.04t) which turns into ln2=0.04t and the answer is t=(ln2)/(0.04) We’re supposed to leave it in a “calulator ready expression” since we’re not allowed to have calculators in the final exam.

It’s the second part that has me tearing my hair out:

b) At the moment that the deposit has just doubled, at what instantaneous rate (in dollars per year) will the amount of money in the account be growing? Please give a numerical answer.

I’ve tried everything. I took derivatives here and there. I did logarithims to bring down k and t. I’ve even done things that were mathematically impossible just to see how deranged this question will make me. If someone, anyone, can give me a clue on how to tackle this question, I’d really appreciate it. You don’t even have to give me a solution/answer, in fact, I already know that it’s $40 (given to me on the course webpage). It’s just that not knowing how to get $40 is driving me crazy. If anyone can steer me in a direction, it would really help me a lot.

Mods, if you think it’s still inappropriate, feel free to close it, but this is truly my last resort and if I can’t get help here, I’m screwed.

2

Well, I don’t know exactly what your problem is, so here’s how you do it.

The rate of change is taking the derivative:

dA/dt = Pke[sup]kt[/sup] = 20e[sup]0.04t[/sup]

Now, you already found in part (a) that at the moment that the deposit doubles, 0.04t = ln(2). And then e[sup]ln(2)[/sup] = 2. So dA/dt = 40.

3

You’re making this way too complicated. She has $1000 (twice her original amount) in the bank, and it’s always growing at a rate of 4% per year (given in our problem statement). 4% of $1000 is $40, so at that moment, it’s growing at a rate of $40 per year.