College · Aligned to AP Calculus AB
Calculus I Foundations (MATH 151) — study guide & practice questions
Limits, derivatives, integrals, differential equations and volume: full AP Calculus AB scope.
Study guide
Limits: Foundations & Techniques
- lim(x→a) f(x) = L means f(x) gets arbitrarily close to L as x approaches a, regardless of f(a) itself; both one-sided limits must agree.
- Factor/cancel or rationalize to resolve a 0/0 form; for rational functions, compare degrees to find limits at infinity.
Continuity & the Intermediate Value Theorem
- f is continuous at x = a when f(a) is defined, the limit exists, and the limit equals f(a); discontinuities can be removable, jump, or infinite.
- The IVT guarantees a continuous function hits every value between f(a) and f(b) somewhere in [a,b].
The Derivative: Definition & Basic Rules
- f'(x) = lim(h→0) [f(x+h) − f(x)] / h — the instantaneous rate of change, or the slope of the tangent line.
- Power, constant multiple, sum/difference, product and quotient rules cover most differentiation; sin/cos/tan have their own derivative rules.
The Chain Rule, Implicit Differentiation & Higher-Order Derivatives
- Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x) — "outside times inside."
- Implicit differentiation finds dy/dx for equations not solved for y; the second derivative of position is acceleration.
Applications: Extrema & Curve Sketching
- Critical points occur where f'(x) = 0 or is undefined.
- Classify them with the first derivative test (sign change of f') or the second derivative test (sign of f''); inflection points are where concavity changes.
Motion, Related Rates & Optimization
- Position, velocity (s') and acceleration (s'') describe motion; related rates connect quantities changing together via implicit differentiation with respect to time.
- Optimization finds a max/min by setting up an equation, then using critical points; linear approximation uses the tangent line to estimate nearby values.
The Definite Integral & the Fundamental Theorem of Calculus
- ∫[a,b] f(x) dx is the signed (net) area between f(x) and the x-axis, defined as the limit of Riemann sums.
- FTC Part 1: the derivative of an integral gives back the integrand. FTC Part 2: if F is an antiderivative of f, then ∫[a,b] f(x) dx = F(b) − F(a).
Techniques of Integration & Applications
- The power rule for integration and u-substitution (reversing the chain rule) find most antiderivatives; every indefinite integral needs a "+C.".
- Area between curves is ∫[top − bottom]; average value of f on [a,b] is (1/(b−a)) times its definite integral.
Course modules
8 foundation modules free to read here; 3 AP-level and advanced modules with Pro. Every module ends with a quiz in the app.
1. Limits: Foundations & Techniques Free
What a limit means, and the algebraic techniques for evaluating one when substitution fails.
Read the 4 lessons →2. Continuity & the Intermediate Value Theorem Free
The precise definition of continuity, the ways a function can fail it, and a powerful existence theorem.
Read the 4 lessons →3. The Derivative: Definition & Basic Rules Free
The formal definition of the derivative, and the shortcut rules that make differentiating fast.
Read the 4 lessons →4. The Chain Rule, Implicit Differentiation & Higher-Order Derivatives Free
Differentiating functions built from other functions, and equations that never isolate y.
Read the 4 lessons →5. Applications: Extrema & Curve Sketching Free
Using a function's derivatives to find its peaks, valleys, and overall shape.
Read the 4 lessons →6. Motion, Related Rates & Optimization Free
Calculus in action: describing movement, relating changing quantities, and finding the best possible outcome.
Read the 4 lessons →7. The Definite Integral & the Fundamental Theorem of Calculus Free
How summing infinitely many rectangles connects to antiderivatives.
Read the 4 lessons →8. Techniques of Integration & Applications Free
How to actually find antiderivatives, and what integrals are used for beyond area.
Read the 4 lessons →9. Key Theorems & Local Linearity Pro · AP Calc AB · Units 4–5
The Mean Value Theorem, Extreme Value Theorem, and linear approximation, including how to justify them the way AP free-response graders expect.
10. Differential Equations & Slope Fields Pro · AP Calc AB · Unit 7
Model change with differential equations, read and sketch slope fields, and solve separable equations, including exponential growth and decay.
11. Applications of Integration: Accumulation & Volume Pro · AP Calc AB · Unit 8
Net change and particle motion, and volumes by discs, washers, and known cross sections.
Sample practice questions
From module 1, Limits: Foundations & Techniques. Pick your answers, then check them in the app, with an explanation for every question.
1. Evaluate lim(x→∞) (2x + 1)/(4x − 3).
2. What does a vertical asymptote at x = a typically indicate about the limit there?
Sample flashcards
4 of the cards in this course's deck. Tap a card to flip it.
LimitsWhat does it mean for lim(x→a) f(x) = L?
As x gets arbitrarily close to a (from either side), f(x) gets arbitrarily close to L, regardless of the actual value (or existence) of f(a).
Limits & ContinuityWhat three conditions make f continuous at x = a?
f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a).
DerivativesWhat is the definition of the derivative f'(x)?
f'(x) = lim(h→0) [f(x+h) − f(x)] / h — the instantaneous rate of change, or slope of the tangent line at x.
Derivative RulesState the power rule and the chain rule.
Power rule: d/dx[xⁿ] = n·xⁿ⁻¹. Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x).
The full course is in LectureMate's Course Library: every module quiz with explanations, the complete flashcard deck with spaced repetition, and a 6-step strategy for AP Calculus AB questions. Foundation modules are free; AP-level and advanced modules need Pro.