Difficulty: Introductory | Prerequisites: Unit circle basics, exponent rules, Section 1.2 Part 1 (polynomials and power functions)
This is the second half of the "Modeling with Functions" section in Calculus I. Where the first half covered polynomials and power functions (algebraic families), this half covers the transcendental families: trig, exponential, and logarithmic functions. These come up constantly in calculus, from derivatives of sin(x) and eˣ to integration techniques that use logarithms. If you are not comfortable with the unit circle or exponent rules, review those before working through these notes.
Sine and cosine are periodic functions that repeat every 2π and stay between -1 and 1. Exponential functions (bˣ) model growth and decay depending on whether the base b is above or below 1. Logarithms are the inverse of exponentials: log_b(x) undoes bˣ, and log_b(1) always equals 0.
Periodic function
A function that repeats its values at regular intervals. The length of one complete cycle is the period.
Think of it as a pattern on repeat, like the wave on an oscilloscope.
Period
The smallest positive number T such that f(x + T) = f(x) for all x.
For sin(x) and cos(x), the period is 2π.
Exponential function
A function of the form f(x) = bˣ, where b > 0 and b ≠ 1. The variable is in the exponent, not the base.
In simple terms, the output multiplies by a fixed factor every time x increases by 1.
Exponential growth
Occurs when the base b > 1. The function increases rapidly as x increases.
Exponential decay
Occurs when 0 < b < 1. The function decreases towards zero as x increases.
Logarithmic function
The function f(x) = log_b(x), defined as the inverse of f(x) = bˣ. It answers the question: "To what power must I raise b to get x?"
Think of it as the "undo" button for an exponential.
Inverse function
A function that reverses another function. If f takes a to b, then f⁻¹ takes b back to a. The graph of an inverse is the reflection of the original over the line y = x.
In simple terms, it runs the process backwards.
Sine and cosine
Both sin(x) and cos(x) are periodic with period 2π
This means sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x) for all x
Range for both: [-1, 1], so the output never exceeds 1 or drops below -1
Zeros of sine
sin(x) = 0 at x = kπ, where k is any integer
So sin(0) = 0, sin(π) = 0, sin(2π) = 0, sin(-π) = 0, and so on
Zeros of cosine
cos(x) = 0 at x = kπ + π/2, where k is any integer
So cos(π/2) = 0, cos(3π/2) = 0, cos(-π/2) = 0, and so on
Graph shapes
The sine graph starts at the origin, rises to 1 at π/2, returns to 0 at π, falls to -1 at 3π/2, and completes the cycle at 2π. The cosine graph has the same shape but is shifted left by π/2, starting at 1 when x = 0.
Exponential functions: f(x) = bˣ
The base b must be positive (b > 0)
When b > 1, the function models exponential growth: it starts small and increases ever more steeply
When 0 < b < 1, the function models exponential decay: it starts large and decreases towards zero
Every exponential function passes through (0, 1) because b⁰ = 1 for any positive b
Domain: (-∞, ∞). Range: (0, ∞). The output is always positive.
Logarithmic functions: f(x) = log_b(x)
The logarithm is the inverse of the exponential: if bˣ = y, then log_b(y) = x
The graph of log_b(x) is the graph of bˣ reflected over the line y = x
Key property: log_b(1) = 0, because b⁰ = 1
Domain: (0, ∞). Range: (-∞, ∞). You cannot take the log of zero or a negative number.
When b > 1, log_b(x) is an increasing function (grows slowly)
When 0 < b < 1, log_b(x) is a decreasing function
The exponential-logarithm relationship
These two functions undo each other. log_b(bˣ) = x and b^(log_b(x)) = x. This inverse relationship is used constantly in calculus to solve equations and simplify expressions.
\sin(x + 2\pi) = \sin(x), \quad \cos(x + 2\pi) = \cos(x)Periodicity of sine and cosine. Both repeat every 2π.
-1 \leq \sin(x) \leq 1, \quad -1 \leq \cos(x) \leq 1Range of sine and cosine.
\sin(x) = 0 \text{ at } x = k\pi, \quad \cos(x) = 0 \text{ at } x = k\pi + \frac{\pi}{2}Zeros of sine and cosine, where k is any integer.
f(x) = b^x, \quad b > 0Exponential function. b > 1 for growth, 0 < b < 1 for decay.
f(x) = \log_b(x), \quad b > 0Logarithmic function, the inverse of bˣ.
b^0 = 1 \implies \log_b(1) = 0Fundamental property linking exponentials and logarithms.
Trigonometric functions model anything that oscillates: sound waves, alternating current in electrical circuits, the motion of a spring, tidal patterns. Any time you see a repeating cycle, sin or cos is likely the right model.
Exponential growth models population growth, compound interest, and viral spread. Exponential decay models radioactive half-life, cooling of a hot object, and depreciation of an asset.
Logarithms are used in the Richter scale (earthquake magnitude), the decibel scale (sound intensity), and pH (acidity). Each of these compresses a wide range of values into a manageable scale, which is what logarithms do.
Students often think sin(x) = 0 at x = π/2. It does not. sin(π/2) = 1. The zeros of sine are at multiples of π (0, π, 2π, ...), not π/2.
Students sometimes confuse the zeros of sine and cosine. Remember: sin is zero at kπ, cos is zero at kπ + π/2. They are offset by π/2.
A common error is treating log_b(x + y) as log_b(x) + log_b(y). That is false. The log of a sum is not the sum of the logs. The rule is log_b(xy) = log_b(x) + log_b(y), which applies to products, not sums.
Students often forget that the domain of log_b(x) excludes zero and all negative numbers. You cannot take log(0) or log(-3).
⚠️ Know the period, range, and zeros of sin(x) and cos(x). These are tested directly and are assumed knowledge for later derivative and integral problems.
⚠️ Be able to distinguish exponential growth from exponential decay by looking at the base. If b > 1, growth. If 0 < b < 1, decay.
⚠️ The fact that log_b(1) = 0 is tested repeatedly. It follows from b⁰ = 1 and is the starting point for many log problems.
⚠️ Expect questions that ask you to identify the inverse relationship between exponential and logarithmic functions, often by asking you to reflect a graph over y = x.
True or False: sin(x) has a period of π.
Fill in the blank: cos(x) = 0 when x = ___.
True or False: If b = 0.5, then f(x) = 0.5ˣ models exponential growth.
Fill in the blank: log_b(1) = ___.
True or False: The domain of log_b(x) is all real numbers.
Answers: 1. False (the period is 2π), 2. kπ + π/2 for any integer k, 3. False (0 < b < 1 is decay), 4. 0, 5. False (domain is (0, ∞))
Q: What is the period of sin(x)? What does "period" mean in this context?
A: The period is 2π. It means sin(x) completes one full cycle (from 0 up to 1, down to -1, and back to 0) every 2π units along the x-axis.
Q: At what values of x does cos(x) = 0?
A: cos(x) = 0 at x = kπ + π/2, where k is any integer. The first few positive values are π/2, 3π/2, 5π/2.
Q: A population doubles every year, starting at 100. Write the function and identify the base.
A: f(t) = 100 · 2ᵗ. The base is 2, which is greater than 1, so this is exponential growth.
Q: Why does log_b(1) = 0 for any valid base b?
A: Because b⁰ = 1 for any positive b. The logarithm asks "what exponent gives me this output?" and the exponent that gives 1 is always 0.
Q: Is the graph of log_b(x) a reflection of bˣ? Over what line?
A: Yes. The graph of log_b(x) is the graph of bˣ reflected over the line y = x. This is true for any function and its inverse.
The derivatives of sin(x) and cos(x) are among the first derivative rules you will learn: d/dx sin(x) = cos(x) and d/dx cos(x) = -sin(x). These are used throughout the course.
Exponential functions lead directly to the number e (Euler's number), which is the base that makes the derivative of eˣ equal to eˣ itself. This is one of the most important results in calculus.
Logarithmic differentiation is a technique you will use later to differentiate complicated products and quotients. The natural log (ln, base e) will become your default logarithm for the rest of the course.
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