Problem

Interest If $\$ 10,000$ are invested in an accoun…

05:21

Problem 28 Medium Difficulty

Interest If $\$ 10,000$ are invested in an account that pays 2$\%$ compounded annually, the total amount, $A(t),$ in the account after $t$ years is.
$A(t)=10,000(1.02)^{t}$
(a) Find the average rate of change per year of the total amount in the account for the first ten years of the investment (from $t=0$ to $t=10 )$
(b) Find the average rate of change per year of the total amount in the account for the second ten years of the investment $\quad($ from $t=10$ to $t=20)$
(c) Estimate the instantaneous rate of change for $t=10$

Answer

$$
\$ 1159.27
$$

Related Courses

Calculus 1 / AB

Calculus with Applications

Chapter 3

The Derivative

Section 3

Rates of Change

Discussion

You must be signed in to discuss.

Top Calculus 1 / AB Educators

Watch More Solved Questions in Chapter 3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Video Transcript

So for this problem, we given the equation for the amount of money that would be left in the account after a certain amount of time in time is measured and years. So don't worry about the details about the and interest rate and stuff, because we're really not gonna use that. It's already included in the equation. So let's just focus on the definition of the average change. A change of great weight and the general definition for that has changed. And a which is changing the amount of money in the account over change in time. So it's a basic definition. So now, because we're in part a were asked if eyeing the average change of great when time goes from zero Teo, it's time go through ten years. Okay, I'm not writing the units here, but you can. So now I'm just going to substitute okay, Before I substitute, let me just make it clear what we're doing. We're doing a time team equals ten minus a. But time, people zero divided by change in time, which is ten years minus zero years. So now if you plug in the numbers what your heart is ten thousand Marius. It gives a ten thousand this Coleman for bows on just going toe. Put it outside of the bracket one point O too. Raise it to ten minus one point. Oh, shoo. Home time, Ciro, Reach to here. All right. Divided by ten. Okay, so if you run the numbers for this on your calculator, you point the way to a hey. No. So that's, uh, the average rate of change for a year. Okay, so that's for a now this trip part, where is the same? The only change is the time. Time goes from C equals head, too. T equals twenty. Okay, So what we're gonna do is changing, eh? Just want money divided by change in time. This will be a twenty minus ten. What if I find my name? Okay, so we're doing the same thing that we did earlier. For eight. Sometimes one going all to wrist you twenty minus one point two. Yes, he can divided by. So for this one, when you're forgiven members, you should call for later. I own to be two hundred sixty six, knowing all to me. That's the average rate of change going from T ten to twenty years service. This is for years. Remember, the unit here is near. So that's why they keep writing for a year. And, um so this is dollars for a year. You want to put a dollar sign or you can put a dollar sign here? Not there. You're so home. Note that in part A the time difference was ten and part B to the time difference is ten. But the change in the amount of money is actually slightly different rate. So here it's two hundred, about two hundred eighteen. And, um sorry. I don't know where that my and import be. It's about two hundred sixty six, so that's significant. But remember, this is what people call usually compounded interest, the power of compounded interest. But we're not getting into that. But okay, let's Newport Sea Sports. He is asking us to kwai the instantaneous weight of change, okay? Or to say instead, rate of change in on time. But this one is stay wine. A T equals ten. Yeah, see? So remember that the definition for instantaneous rate of change iss using limit right now. So we're going to say this is limit of and C h my news here too. Fighting for our lives. The difference here. Age an age. We're doing them. Limit us. A school's here. So now again, we're doing the same thing. We're just plugging him the numbers. So a different color in housing. So the limit is out here. I'm not waiting that they met right now. It's just focus on the equation. You have one week all too. Two plus age minus form old too. Team provided for H, right? Okay. But we know t equal stand so And we can take out team one point this fort in this both sides have the same numbers so we can pull it out to the side. Right. So just we write that nice you're a hero to to the T and t is ten, right? And then what? You're like, where's is one point two bridge to the age miners. One right, provided by beach. Okay, so this is a constant rate. The only thing that's changing as h So now let's make a table for a small increments of age. One morning out of space years always threw it on the next page, so it's safe for H. But start was large. Wally's of H Great. Okay, so, comport. See us. This is the equation. So age over here, it's too at H equals one, for instance, h People's zero point one at age people's zero point zero wine age people's zero point zero zero one. You can keep going and going so and now the follies from equation that we found here. If you just plug in the volume H for different while he's of age where this, for instance, for each people's one bomb. Two for three point seven, eh? Or this one two for one for you fix three. Two no children, Hugh for six. And then I absolutely didn't do thiss one. What a dead wass. I just like, made this one really training a murder just to see what happens, right? So I put an bites sirus one, too. Or why e got a two for one for me, really there. So it's safe to say that of the average, not the average, that instantaneous change AT T equals ten. It's about two hundred forty one Maureen, four dollars for a year, so that is at all this water

Get More Help with this Textbook