Question

With an initial deposit of $100, and an interest rate of 7%, the balance in a bank account after t years is f(t) = 100(1.07)t dollars.

(a) What are the units of the rate of change of f(t)?

(b) Find the average rate of change over [0, 0.5] and [0, 1].

(c) Estimate the instantaneous rate of change at t = 0.5 by computing the average rate of change over intervals to the left and right of t = 0.5.

Answer 1

Answer with Step-by-step explanation:

We are given that

Initial deposit=$100

Interest rat=r=7%

`f(t)=100(1.07)^t` (in dollars)

Where t(in years)

a.We have to find the rate of change of f(t).

f(t) varies with t Where f(t) in years and t in years

Therefore, the unit of rate of change of f(t) is dollar/year.

b.We have to find the average rate of change over [0,-0.5] and [0,1].

Substitute t=0

`f(0)=100(1.07)^0=100`

Substitute t=0.5

`f(0.5)=100(1.07)^(0.5)=103.44`

Substitute t=1

`f(1)=100(1.07)=107`

Average rate on interval [a,b] =
`(f(b)-f(a))/(b-a)`

Using the formula

Average rate on interval [0,0.5]=
`(103.44-100)/(0.5-0)`

Average rate on interval [0,0.5]=
`(3.44)/(0.5)=6.88`dollar/year

Average rate on interval [0,1]=
`(107-100)/(1-0)=7dollar/year`

c.Average rate on interval [0.5,1]=
`(107-103.44)/(1-0.5)=7.12`dollars/year

Instantaneous rate of change at t=0.5 =Average of the average values obtained on interval [0,0.5] and [0.5,1]=
`(6.88+7.12)/(2)=7dollar/year`