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Question

An investment of $9,000 which earns 7.1% per year is growing continuously

How fast will it be growing at year 10?

Recall that the formula for continuous exponential growth with a yearly growth rate of r is 

A(t)=Noe^rt where N0 is the initial amount and of course r is written mathematically (as in 5%=0.05)

Answer 1

Your formula is this:

A(t) = N₀e^rt

The constants are:

N₀ = $9000

r = .071

Therefore we have the equation:

A(t) = 9000•e^.071t = 9000 (e^t)^.071

If we want to get how fast the money grow in year 10, we have to derive A(t). A'(t) is the derivative of the amount with respect to time t. Let's use the Chain Rule:

A'(t) = 9000(.071)(e^t)^-.929 (e^t)

A'(t) = 9000(.071)(e^t)^-.929 (e^t)

A'(t) = 639•e^.071t

Plugin t=10

A'(10) = 639•e^.071(10)= 1299.72

At year 10, the money will grow at a rate of $1299.72 / year