Calculus vs AP Precalculus

A side-by-side look at how these two subjects compare in scope, difficulty, and content.

At a Glance

Limits
The value that a function approaches as the input approaches some value. Limits are the foundation of calculus, used to define both derivatives and integrals.
Derivatives
The derivative measures the instantaneous rate of change of a function. Geometrically, it gives the slope of the tangent line to a curve at any point.
Integrals
Integration is the reverse process of differentiation. Definite integrals compute the signed area under a curve, while indefinite integrals find antiderivatives.
Fundamental Theorem of Calculus
Links differentiation and integration as inverse processes. Part 1: integration can be reversed by differentiation. Part 2: definite integrals can be computed using antiderivatives.
Chain Rule
A formula for computing the derivative of a composition of functions. If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

Function Transformations
Moving, stretching, or flipping a graph. Changes to the input (inside) move left/right. Changes to the output (outside) move up/down. That inside/outside split trips people up—but it's the key.
Polynomial Functions
Functions built from powers of x. The highest power tells you how the ends behave: odd degree = opposite ends; even degree = same ends.
Rational Functions
One polynomial divided by another. Vertical asymptotes happen where the bottom is zero. Holes happen when top and bottom share a factor.
Exponential Growth and Decay
Multiply by the same factor over equal steps. Growth when the base > 1; decay when it's between 0 and 1. Used for populations, money, radioactivity.
Logarithms
Logs undo exponents. $\log_b(y)=x$ means "to what power do we raise b to get y?" Turns multiplication into addition.

Power Rule Forgot Factor
Misconception: A common misunderstanding about power rule forgot factor that leads to selecting incorrect answers.
Correction: The power comes down as a multiplier.
Power Rule Wrong Direction
Misconception: A common misunderstanding about power rule wrong direction that leads to selecting incorrect answers.
Correction: The Power Rule says: bring the exponent down and reduce by 1.
Derivative Vs Integral
Misconception: Believing "The slope at a point" rather than "The signed area under the curve from a to b".
Correction: Slope at a point is what the *derivative* gives you.
Integral Vs Max
Misconception: Believing "The maximum of f(x)" rather than "The signed area under the curve from a to b".
Correction: The maximum is a single point.

Horizontal shift direction
Misconception: Believes f(x - h) shifts left (when h>0) or f(x + h) shifts right.
Correction: f(x - h) shifts the graph to the right, because x must be h units larger to produce the same function input.
Inside vs Outside mean same thing
Misconception: Confuses horizontal transformations (inside) with vertical transformations (outside).
Correction: Inside changes (with x) shift the graph horizontally. Outside changes (added to f) shift vertically.
Composition substitution
Misconception: In composition, student multiplies functions or substitutes wrong.
Correction: f(g(x)) means replacing the input of f with the output of g(x).
Domain of composition
Misconception: Believes the domain of f(g(x)) is simply the domain of f.
Correction: Domain is determined by what g(x) produces that f can accept.