Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Interest If $\$ 1000$ is invested in an account that pays 5$\%$ compounded annually, the total amount, $A(t),$ in the account after $t$ years is$$A(t)=1000(1.05)^{t}.$$a. Find the average rate of change per year of the total amount in the account for the first five years of the investment (from $t=0$ to $t=5 )$b. Find the average rate of change per year of the total amount in the account for the second five years of the investment (from $t=5$ to $t=10 ).$c. Estimate the instantaneous rate of change for $t=5$

a) 55.26b) 70.52c) 62.89

Calculus 1 / AB

Chapter 11

The Derivative

Section 3

Rates of Change

Derivatives

Differentiation

Applications of the Derivative

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

05:15

Interest If $\$ 1000$ is i…

11:33

Interest If $\$ 10,000$ ar…

05:21

00:40

When a deposit of $\$ 1000…

01:19

The amount of money in an …

04:24

A bank account earns $5 \%…

04:03

Interest Compounded Contin…

05:05

With an initial deposit of…

03:11

Interest A sum of $\$ 1500…

01:13

they tell us that formula A of T is equal to 1000 times 1.5 Raciti will be the amount of money we will have after two years. If we invested $1000 into account, that phase pays 5% interest that compounds annually. So they want to find the average rate of change over the first year or 1st 5 years and the 2nd 5 years and then they wants to estimate the instantaneous rate of change at five years. So let's go ahead and start with the 1st 5 years, which is gonna be 05 So we're gonna plug everything into that average rate of change equation only difference says instead of f, we're going to use a There's gonna be a of five minus a zero all over five minus zero. So let's look that in. So we have 1000 times 1.5 raise to the minus 1000 and times 1.50 which is just gonna be one. And then we would divide this by five. So first knows we could divide this five and each of these terms such as could be 200 so we need to find. So 1.5 raised to the fifth times 200. So that's about 255 minus 200. So this is somewhere in the ballpark of 50 five dollars and 26. And so this is how much interest we would earn on that 1st 5 years. Now, what about the 2nd 5 years? Well, now we're going to do a of 10 minus a five, Then be 10 minus five. So we just plug it again again. So it be 1000 1.5 race and 10th minus 1000 1.5 raised to the fifth all over 10 minus five. So 10 miles five is just going to be five. And so then we could cancel these over again, just like we did before. Start to give us 200 for each. So now let's see what we get are actually first, let's factor out that 200 sweet have 200 we could also factor out a 1.5 raised to the Biff and doing that. Would he have 1.5 raised the fifth minus one already. So just writing get like this would be a little bit easier to plug into a calculator, I think so. We have 1.5 raise to the fifth. So about 1.27 by this one. That gives us about 10.27 that we multiply that by 1.5 race to the fifth, just about 0.35 and then multiply that by 200. So this is going to be somewhere in the ballpark off about 70 dollars and 52 cents. So that's how much interest we would have burned on the 2nd 5 years now to estimate this rate of or the instantaneous story of change at T 05 so we could use the instantaneous story of change equation and then just keep taking smaller and smaller steps away from each other were we can average these two rates of change over here, So this is going to be approximately equal to to the average rate of change of 0 to 5, plus the average rate of change from 5 to 10 all over, too, because we're just averaging those two. And this might be a good one to use cause notice that this one ends at five in this one starts at five. So the slope going from 0 to 5 and we have the soap coin from 5 to 10. So you would expect it to be maybe somewhere in the middle of those, so you just go ahead and plug these values. And so this would be approximately 55.26 plus 70.52 all over, too. And so let's see what we get when we add these up. And so after we divide and add, that should give us something around $62.89. So this is what we would expect to be the instantaneous rate of change after five years, or how much money we would expect to earn after that fifth year.

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Interest If $\$ 1000$ is invested in an account that pays 5$\%$ compounded c…

Interest If $\$ 10,000$ are invested in an account that pays 2$\%$ compounde…

When a deposit of $\$ 1000$ is made into an account paying $8 \%$ interest, …

The amount of money in an account after $t$ years compounded continuously at…

A bank account earns $5 \%$ annual interest, compounded continuously. Money …

Interest Compounded Continuously. Suppose that $\$ 10,000$ is invested at an…

With an initial deposit of $\$ 100,$ the balance in a bank account after $t$…

Interest A sum of $\$ 1500$ is deposited in an account with an interest rate…

00:29

Find the derivative of each function defined as follows.$$y=3 x^{4}-…

00:54

Estimate the slope of the tangent line to each curve at the given point $(x,…

00:41

Use the product rule to find the derivative of the following. (Hint for Exer…

02:09

Insecticide The graph in the next column shows how the number of arthropod s…

01:34

Using the definition of the derivative, find $f^{\prime}(x) .$ Then find $f^…

03:21

Find the derivative of each function.$y=\frac{\ln x}{4 x+7}$

04:02

For each function, find (a) the equation of the secant line through the poin…

01:27

Find the derivative of each function defined as follows.$$g(x)=\left…

01:48

Poker Find the probabilities of the following hands at poker. Assume aces ar…

00:52

Explain why $$\lim _{x \rightarrow 2}\left(3 x^{2}+8 x\right)$$ can be evalu…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.