Recall that the compound interest formula for continuous compounding is A(P, r, t) = Pert, where A is the future value of an investment of P dollars after t years at an interest rate of r.
(a) Calculate ∂A/∂P, ∂A/∂r, and ∂A/∂t, all evaluated at (80, 0.2, 9). (Round your answers to two decimal places.)
∂A/∂P = e^(0.2*9) ≈ 1.96
∂A/∂r = 80 * 9 * e^(0.2*9) ≈ 141.38
∂A/∂t = 80 * 0.2 * e^(0.2*9) ≈ 31.36
Interpret your answers.
For a $80 investment at 0.2% interest invested for 9 years and compounded continuously, the accumulated amount is increasing at a rate of $1.96 per $1 of principal, at a rate of $141.38 per increase of 1 in r, and at a rate of $31.36 per year.
(b) What does the function ∂A/∂P(80, 0.2, t) tell about your investment?
∂A/∂P(80, 0.2, t) tells you the rate at which the accumulated amount in an account bearing 0.2% interest, compounded continuously, with a principal of $80, is growing per $1 increase in t years after the investment.
