Investment Growth: Derivatives & Integrals

Understanding how calculus concepts apply to a $1000 investment with compound interest

Our Investment Scenario

Initial Investment: $1,000 | Annual Interest Rate: 5% | Time Period: 10 years

We'll use compound interest to demonstrate how derivatives and integrals work with financial growth.

Compound Interest Formula: A(t) = P × (1 + r)^t
Where: P = $1000, r = 0.05, t = time in years

The Derivative: Instantaneous Growth Rate

The derivative tells us how fast our investment is growing at any specific moment.

Investment Interpretation

If A(t) represents your account balance at time t, then:

A'(t) = the instantaneous rate at which your money is growing at time t

A(t) = 1000 × (1.05)^t
A'(t) = 1000 × (1.05)^t × ln(1.05)

Practical Examples:

After 5 years, your account has $1,276.28. The derivative tells us it's growing at:

A'(5) = 1000 × (1.05)^5 × ln(1.05) ≈ $60.77 per year

After 10 years, with $1,628.89 in your account, the growth rate is:

A'(10) = 1000 × (1.05)^10 × ln(1.05) ≈ $77.57 per year

Key Insight: Even though the percentage rate is constant at 5%, the dollar growth rate increases over time because it's 5% of a larger balance.

The Integral: Total Accumulated Growth

The integral measures the total accumulation of growth over a period of time.

Investment Interpretation

If A'(t) is the growth rate at time t, then:

∫ A'(t) dt from a to b = total growth between year a and year b

Total Growth from year 0 to year T = ∫[0 to T] A'(t) dt = A(T) - A(0)

Practical Examples:

Total growth over the first 5 years:

A(5) - A(0) = $1,276.28 - $1,000 = $276.28

Total growth from year 5 to year 10:

A(10) - A(5) = $1,628.89 - $1,276.28 = $352.61

Another Perspective: Continuous Contributions

If you added money continuously at a rate of $50/month ($600/year):

Total additional investment = ∫[0 to 10] 600 dt = 600 × 10 = $6,000

But with interest, the accumulated value would be more!

Key Insight: The integral gives us the "big picture" - the total result of all the small growth moments added together over time.

Visualizing Investment Growth

This graph shows how both the account balance and its growth rate change over time:

Derivative vs. Integral: Investment Perspective

Concept	Investment Meaning	Question It Answers	At Year 5	At Year 10
Derivative	Instantaneous growth rate	How fast is my money growing RIGHT NOW?	$60.77/year	$77.57/year
Integral	Total accumulated growth	How much has my investment grown OVER THE PAST 5 YEARS?	$276.28 (years 0-5)	$352.61 (years 5-10)

Key Takeaway

The derivative (A'(t)) is like checking your investment's current performance - it tells you how fast it's growing at this moment.

The integral (∫ A'(t)dt) is like reviewing your investment statement - it shows the total growth over a period.

And the Fundamental Theorem of Calculus connects them: The total growth (integral) is the result of accumulating all the momentary growth rates (derivative).