Investment Analysis: Continuous Compound Interest
a. What is the formula for A'(P)? b. How can we find and interpret A'(5000)? c. How do we compare the approximation to the actual change?
a. Formula for A'(P)
b. Finding and Interpreting A'(5000)
By substituting P = 5000 into the formula for A'(P), we can find A'(5000):
A'(5000) = e^(0.48) + 0.48(5000)e^(0.48)
Calculating the value, we get A'(5000) ≈ 1.62. This means that for every $1 increase in the principal amount of $5000, the total balance will increase by approximately $1.62.
c. Comparing Approximation to Actual Change
To compare the approximation to the actual change, we calculate the difference between A(5001) and A(5000):
A(5001) - A(5000) = 5001e^(0.04*12) - 5000e^(0.04*12)
Calculating the value, we find A(5001) - A(5000) ≈ $20.41. Therefore, the approximation of the actual change is $20.41.