An investment of 25,000 earns interest at an annual rate of 8.4% compounded continuosly

a. find the instantaneous rate of change of the amount in the account after years

b. find the instantaneous rate of change of the amount in the account at the time the amount is eqaul to 30,000

So i know that the equation is A=Pe^rt. So to find the instanteous rate of change I need to find the derivative right?

Which point is located at (0.25,-0.5)

Yes, that's correct! To find the instantaneous rate of change of the amount in the account, you need to find the derivative of the equation A = Pe^rt with respect to time. The derivative will provide you with the instantaneous rate of change at any given time.

Yes, you are correct. To find the instantaneous rate of change of the amount in the account, you need to find the derivative of the equation A = Pe^rt.

Let's start with part (a) of your question, which asks for the instantaneous rate of change of the amount in the account after a certain number of years. In this case, we'll assume the number of years is denoted by t.

To find the derivative of the equation A = Pe^rt with respect to t, you can use the chain rule. The chain rule states that if you have a function inside another function, the derivative can be found by taking the derivative of the outer function and multiplying it with the derivative of the inner function.

In this case, the outer function is A, and the inner function is e^rt. The derivative of the outer function with respect to t is dA/dt, and the derivative of the inner function with respect to t is d(e^rt)/dt.

To find the derivative of e^rt, you can use the chain rule again. The derivative of e^rt with respect to t can be found by multiplying e^rt with the derivative of rt.

The derivative of rt with respect to t is simply r, as t is treated as a constant when deriving with respect to t.

So, to summarize:

dA/dt = (dA/d(e^rt)) * (d(e^rt)/dt)
= (dA/d(e^rt)) * (r)

Now, to find dA/d(e^rt), we differentiate A = Pe^rt with respect to e^rt. Since A is not a function of e^rt, dA/d(e^rt) will be equal to P, which is the initial amount.

Therefore, dA/dt = P * r

In your case, the initial investment is $25,000 and the annual interest rate is 8.4%. So, the instantaneous rate of change of the amount in the account after t years is:

dA/dt = $25,000 * 0.084

For part (b) of your question, you need to find the instantaneous rate of change of the amount in the account at the time when the amount is equal to $30,000. To do this, you'll need to first find the value of t that satisfies the equation A = $30,000.

Using the formula A = Pe^rt, we can rewrite it as:

$30,000 = $25,000 * e^(0.084t)

To find the value of t, you'll need to solve this equation. One way to do this is by taking the natural logarithm (ln) of both sides of the equation:

ln($30,000/$25,000) = ln(e^(0.084t))

ln($30,000/$25,000) = 0.084t

t = (ln($30,000/$25,000))/0.084

Once you have the value of t, you can substitute it into the equation dA/dt = P * r to find the instantaneous rate of change of the amount in the account at that time.

correct

A = 25000 e^(.084t)

dA/dt = Pr e^(rt) we know r = .084

dA/dt = .084(25000) e^(.084t)
= 2100 e^(.084t)

when A = 30,000
30,000 = 25,000 e^(.084t)
1.2 = e^(.084t)
ln 1.2 = .084t ---> I assume you know log rules
t = ln 1.2 / .084
= appr 2.17

Plug that into 2100 e^(.084t)