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(a) If $\$ 1000$ is borrowed at 8$\%$ interest, find the amountsdue at the end of 3 years if the interest is compounded(i) annually, (ii) quarterly, (iii) monthly, (iv) weekly,(v) daily, (vi) hourly, and (vii) continuously.(b) Suppose $\$ 1000$ is borrowed and the interest is compounded continuously. If $A(t)$ is the amount due after$t$ years, where $0 \leqslant t \leqslant 3,$ graph $A(t)$ for each of theinterest rates $6 \%, 8 \%,$ and 10$\%$ on a common screen.

a)i.$$\$ 1259.71$$ii. $$\$ 1268.24$$iii. $$\$ 1270.24$$iv. $$\$ 1271.02$$v. $$\$ 1271.22$$vi. $$\$ 1271.25$$vii. $$\$ 1271.25$$b) See video for graph

Calculus 1 / AB

Chapter 3

INVERSE FUNCTIONS

Section 4

Exponential Growth and Decay

Derivatives

Differentiation

Applications of the Derivative

Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

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Hey, guys, welcome back. This problem is and interest from $1000 borrowed 8% interests. And there were Newsome, different formulas based on how much the interest is compound. So we know that we'll use the formula. A is equal to P Times one over or the end times nt You know that teams air time of year's time in years. She'll be three years. You know that P is how much money start with or principle in this case at his $1000 here that are is our interest rate. This case is 8% which, if you move that in no decimal, be 0.8 and then we know that end is how many times number of times bring your that your compound ing money times per year. And that is what will change a bunch and all the different parts of this problem. So first we want to see what happens if we compound it annually. So if you do it annually, you know that and will be one time per year. So we'll say that a This eagle two 1000 one plus R points your way over end. It won raised to the NT, and it's just one. He has three. And if that, you know, value of 1000 200 59 in 71 cents, and that is our answer to part a part B or asked what happens if a compound it quarterly. If its quarterly that be four times a year, we will do the same equation plugging in and as you were before, So we have a 2 1000 one plus 10 over four raised to the fore Times three. And we knew that you get a value of 1200 and 68 24 Next dress. What happens if you do it monthly Its monthly? An end to 12. We'll have a 0 to 1000 one plus point. Oh, wait over 12 12 times three. That would come out to 1000 270 point 24 Next rest. It happens if you do it weekly. Do it weekly. There are 52 weeks in a year. We have a 0 to 1000 on plus point away over 52 52 times three. That comes out to $1271 two cents. Now ask what happens if we do it daily? Yes, 365 days in a year. We have a zeal to 1000 one plus point. Oh, wait over 365. Raise to the 365 times three. And that comes out to 1000 270 $1. 20. 22 cents. 20 Then you're asked what happens if we do it? Hourly? Do it hourly. That would be 365 days in a year. Times 24 hours in a day Ejected 8000 760 hours in a year. 7000. It's our 8760 times get compound. If a is equal to 1000 one plus point, no wait over 8000 160 raised to the 8000 760 times three. That comes out to 1000 200 71.25 Then rest. What happens if you do it continuously? Well, if you do it continuously, that's basically saying what happens if an goes to infinity? If n goes to infinity, that are, formula becomes a different simplifies down and it's just it a is equal to P. Yeah, the rt fill it in. It would be equal to 1000 e the point. Oh, wait. Times three, we get the exact same value. We had our last part, which is 1271.25 and then finally using our formula A is equal to P E to the rt. You want a graph zero to T years ago. Three zero from T is equal to three different interest rates to on the graph are you 2.6 are 0.8 and our issue 2.1 I want had and I grabbed this on Dismas and that graph can be shown here. I don't exercise not cancelled Carry over. This was acceptable to zero Xeloda one exit time. Is he going to I'm is equal to three in this obviously in years and her wise and dollars you can see how the Green line when are you would appoint one is slightly higher. Our blue line When are you gonna 0.0 waits in the middle and then our lowest is when already lit a 0.6 And that concludes this problem. Thank you for watching

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