Difficulty: Introductory. Prerequisites: basic algebra (exponent rules, graphing functions, factoring).
Exponential functions are one of the core building blocks of calculus, sitting alongside polynomials, logarithms, and trigonometric functions. They model anything that grows or decays at a rate proportional to its current size: populations, radioactive isotopes, compound interest, viral spread.
You need to be comfortable with basic exponent rules from algebra (how to multiply and divide powers, what a negative exponent means) before this section will click. Everything here feeds directly into differentiation and integration of exponentials later in the course, so it is worth getting the foundations solid now.
An exponential function has the form f(x) = bˣ, where the base b is a positive constant. When b > 1 the function grows; when 0 < b < 1 it decays. The special base e ≈ 2.718 shows up everywhere because eˣ is the unique exponential whose slope at x = 0 equals 1, and it is the base used in continuous compound interest, f(x) = Aeᵖˣ.
Exponential function
A function of the form f(x) = bˣ where b > 0 and b ≠ 1. The variable sits in the exponent, not the base. Think of it as: the output multiplies by the same factor every time x increases by 1.
Base (b)
The constant being raised to the power x in bˣ. In simple terms, the base controls how fast the function grows or shrinks.
Exponential growth
When b > 1, the function increases as x increases. The curve rises steeply to the right and flattens toward zero on the left.
Exponential decay
When 0 < b < 1, the function decreases as x increases. The curve drops from the left and flattens toward zero on the right. Think of it as: the same shape as growth, but mirrored across the y-axis.
Natural exponent (e)
The irrational number e ≈ 2.71828, defined as the base for which the slope of bˣ at x = 0 is exactly 1. In simple terms, e is the "sweet spot" base that makes calculus with exponentials clean.
Principal amount (A)
The initial deposit or starting value in a compound interest formula. Think of it as: whatever you start with before interest kicks in.
Compound interest
Interest calculated on the current balance (principal plus previously earned interest), not just the original deposit. Each compounding period, the balance grows by a fixed percentage of itself.
Continuous compounding
The limiting case where interest is compounded infinitely often. The formula shifts from A(1 + p)ˣ to Aeᵖˣ, and this is where the natural exponent enters the picture.
The basic form: f(x) = bˣ
The base b must be positive (b > 0).
When b > 1, the graph rises from left to right (exponential growth).
When b = 1, bˣ = 1 for all x, so the graph is just the horizontal line y = 1.
When 0 < b < 1, the graph falls from left to right (exponential decay).
Every exponential passes through (0, 1)
No matter what positive base you pick, b⁰ = 1, so the y-intercept is always at y = 1.
Growth and decay are reflections of each other
The graph of (1/b)ˣ is the graph of bˣ reflected over the y-axis.
Algebraically: 1/bˣ = b⁻ˣ. If bˣ = f(x), then (1/b)ˣ = f(−x), which is the definition of a y-axis reflection.
Laws of exponents (b > 0, a > 0)
Product rule: bˣ⁺ʸ = bˣ · bʸ
Quotient rule: bˣ⁻ʸ = bˣ / bʸ
Power rule: (bˣ)ʸ = bˣʸ
Product of bases: (ab)ˣ = aˣ · bˣ
These rules are not new, they come straight from algebra, but you will use them constantly when differentiating and integrating exponentials.
Compound interest as an exponential
Suppose you deposit A dollars at an annual interest rate p (expressed as a decimal). Interest is calculated on the current balance, not just the original deposit.
Year 0: balance = A
Year 1: balance = A + pA = A(1 + p)
Year 2: balance = A(1 + p) + p · A(1 + p) = A(1 + p)²
Year 3: balance = A(1 + p)³
Year x: balance = A(1 + p)ˣ
The pattern emerges because each year you factor out the previous balance and multiply by (1 + p) one more time.
Example from the notes
Deposit $100 at 5% annual interest. After x years: f(x) = 100(1.05)ˣ.
Continuous compounding
When interest is compounded continuously (not just once per year), the formula becomes f(x) = Aeᵖˣ, where e ≈ 2.71828 is the natural exponent.
Why e matters in calculus
Among all exponential functions bˣ, the function eˣ is the one whose tangent line at x = 0 has slope exactly 1.
This property makes eˣ its own derivative, which is why e dominates in calculus.
The fact that continuous compounding naturally produces eᵖˣ is one of the first real-world reasons students encounter e.
Exponential function (general form)
f(x) = b^x, \quad b > 0,\; b \neq 1Laws of exponents
b^{x+y} = b^x \cdot b^y \qquad b^{x-y} = \frac{b^x}{b^y} \qquad (b^x)^y = b^{xy} \qquad (ab)^x = a^x \cdot b^xCompound interest (annual compounding)
f(x) = A(1 + p)^xA = principal (initial deposit), p = annual interest rate as a decimal, x = number of years.
Continuous compounding
f(x) = A e^{px}e ≈ 2.71828. Used when interest accrues continuously rather than at fixed intervals.
Graph features to remember
y-intercept is always (0, 1) for bˣ, or (0, A) when a coefficient A is in front.
No x-intercept: the curve approaches the x-axis but never touches it (horizontal asymptote at y = 0).
The graph of b⁻ˣ is the reflection of bˣ over the y-axis.
Compound interest is the textbook example, but exponential functions appear anywhere a quantity changes at a rate proportional to its current value. Bacterial colonies double at a fixed interval, so population after t hours is modelled by P(t) = P₀ · 2^(t/d), where d is the doubling time. Radioactive decay works the same way in reverse: the amount of a substance halves every fixed interval, giving an exponential with 0 < b < 1.
In finance, the shift from annual to continuous compounding is not just a mathematical curiosity. Bond pricing, option valuation, and discounted cash flow models all use eʳᵗ because continuous-time models are cleaner to differentiate and integrate.
Students often think bˣ can be negative. It cannot, as long as b > 0. The output is always positive, no matter what x is.
Confusing the base and the exponent: xⁿ (a power function, base varies) is a completely different animal from bˣ (an exponential, exponent varies). The distinction matters for differentiation.
Assuming compound interest is calculated on the original deposit each year. It is calculated on the current balance, which is why the formula involves (1 + p) raised to a power rather than a simple multiplication.
Thinking e is just a rounded number you can substitute 2.7 for in exact work. e is irrational, and in calculus you almost always leave it as e rather than approximating.
⚠️ Know the shape of bˣ for b > 1 versus 0 < b < 1. You may be asked to sketch or identify which graph corresponds to a given base.
⚠️ Be able to apply all five exponent laws fluently. These are used constantly in simplification problems throughout the course.
⚠️ Understand how to derive the compound interest formula from first principles (the year-by-year factoring argument). This is a common "show your work" exam question.
⚠️ Know what makes e special: the slope of eˣ at x = 0 is 1. This fact underpins the derivative of eˣ, which is the next major topic.
True or false: The graph of 2ˣ crosses the x-axis at some point.
False. 2ˣ is always positive; the x-axis is a horizontal asymptote.
Fill in the blank: b⁰ = ___ for any positive base b.
True or false: (1/3)ˣ is an example of exponential growth.
False. Since 1/3 is between 0 and 1, this is exponential decay.
Fill in the blank: If you deposit $200 at 4% annual interest compounded yearly, after x years you have f(x) = ___.
200(1.04)ˣ.
True or false: The slope of eˣ at x = 0 is e.
False. The slope of eˣ at x = 0 is 1. That is the defining property of e.
Q: Simplify 5³ · 5⁴ using the laws of exponents.
A: 5³⁺⁴ = 5⁷ = 78,125.
Q: You invest $1,000 at 3% annual interest, compounded yearly. Write the function for the balance after t years, and calculate the balance after 10 years.
A: f(t) = 1000(1.03)ᵗ. After 10 years: 1000(1.03)¹⁰ ≈ $1,343.92.
Q: Explain why the graph of (1/2)ˣ is a reflection of the graph of 2ˣ over the y-axis.
A: (1/2)ˣ = 2⁻ˣ. Replacing x with −x in f(x) = 2ˣ gives f(−x) = 2⁻ˣ, which is the definition of a reflection across the y-axis.
Q: What is the y-intercept of f(x) = 7 · 3ˣ?
A: f(0) = 7 · 3⁰ = 7 · 1 = 7. The y-intercept is (0, 7).
Q: A radioactive substance decays so that only half remains every 6 hours. If you start with 80 grams, write an exponential function for the amount remaining after t hours.
A: f(t) = 80 · (1/2)^(t/6). After t hours, the substance has halved t/6 times.
This connects to logarithmic functions because logarithms are the inverse of exponentials: if bˣ = y, then log_b(y) = x. You will need this relationship heavily in the next section.
The derivative of eˣ (coming up in differentiation) is eˣ itself, which is the reason e is so central to calculus. Understanding exponential growth and decay here also sets up later work on differential equations, where solutions frequently take the form Ce^(kt).
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