Subjects › Calculus AB
Calculus AB: learn limits, derivatives, and integrals by talking them through
This course covers the full AP Calculus AB curriculum: limits and continuity, the definition and rules of differentiation, applications like related rates and optimization, and integration through the Fundamental Theorem of Calculus, area and volume problems, motion, and an introduction to differential equations and slope fields. It follows the standard AB sequence, starting from a precalculus review and building one idea on the next, the same order used on the exam.
Each lesson works through a short question-and-answer exchange instead of a lecture. The tutor gives you a small piece of a problem or a prompt about a graph, asks what you notice or what you'd try next, and adjusts based on what you actually say. If you get a step wrong, the tutor asks a narrower question to find where the gap is, rather than just showing the correct steps. Later sessions circle back to earlier topics, like checking that you can still set up a related rates problem weeks after first learning the chain rule, so ideas stay usable instead of fading.
Who it is for
- Students taking or about to take AP Calculus AB and wanting practice that matches the exam's topics
- Students who've finished precalculus and want a first pass through calculus before the school year starts
- Parents looking for a way to check whether their student can explain ideas, not just recite formulas
- Teachers who want a supplementary resource organized in the same topic order as their course
What you will be able to do
- Evaluate limits algebraically, including trig limits and limits at infinity, and identify discontinuities and asymptotes
- Compute derivatives using the power, product, quotient, and chain rules, including implicit and logarithmic differentiation
- Apply derivatives to related rates, optimization, curve sketching, and linear approximation problems
- Use Riemann sums and the Fundamental Theorem of Calculus to evaluate definite integrals and set up accumulation functions
- Find areas between curves, volumes of revolution, and displacement or distance from velocity functions
- Set up and solve separable differential equations and interpret slope fields
A taste of a lesson on Related Rates
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 37 topics in Calculus AB
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Review of Functions and Graphs
Review of key precalculus ideas needed for calculus: domain/range, piecewise functions, transformations, composition of functions, and the behavior of polynomial, rational, exponential, logarithmic, and trigonometric functions. Students should be able to read and sketch graphs and identify key features like intercepts, asymptotes, and symmetry.
Introduction to Limits
The intuitive idea of a limit as the value a function approaches as input approaches a point, using tables, graphs, and numerical estimation. Includes one-sided limits and the idea that a limit may exist even where a function is undefined.
Builds on Review of Functions and Graphs
Computing Limits Algebraically
Techniques for evaluating limits algebraically: direct substitution, factoring, rationalizing, simplifying complex fractions, and the limit laws (sum, product, quotient rules). Includes recognizing indeterminate forms like 0/0.
Builds on Introduction to Limits
Special Limits Involving Trigonometric Functions
Key limits such as the limit of sin(x)/x and (1-cos(x))/x as x approaches 0, and how to use them to evaluate related trigonometric limits. Understanding why these limits are foundational for differentiating trig functions later.
Builds on Computing Limits Algebraically
Infinite Limits and Vertical Asymptotes
Understanding limits that go to infinity, identifying vertical asymptotes from function behavior, and using sign analysis near discontinuities to determine one-sided infinite limits.
Builds on Computing Limits Algebraically
Limits at Infinity and Horizontal Asymptotes
Evaluating limits as x approaches positive or negative infinity for rational, polynomial, and other functions; connecting these limits to horizontal asymptotes and end behavior.
Builds on Computing Limits Algebraically
Continuity
The formal definition of continuity at a point (limit exists, function is defined, and they are equal), classifying discontinuities (removable, jump, infinite), and the Intermediate Value Theorem with applications to root-finding.
Builds on Computing Limits Algebraically, Infinite Limits and Vertical Asymptotes
The Derivative as a Limit
Defining the derivative as the limit of the difference quotient, representing instantaneous rate of change and slope of the tangent line. Includes computing derivatives from the limit definition for simple functions and interpreting the derivative graphically and physically.
Builds on Special Limits Involving Trigonometric Functions, Continuity
Differentiability and Its Relationship to Continuity
Understanding that differentiability implies continuity but not conversely; identifying points where a function is continuous but not differentiable (corners, cusps, vertical tangents); connecting graphs of f and f'.
Builds on The Derivative as a Limit
Basic Differentiation Rules
Power rule, constant rule, sum/difference rule, and derivatives of exponential functions. Practicing quick symbolic differentiation of polynomial and exponential expressions.
Builds on The Derivative as a Limit
Product and Quotient Rules
Differentiating products and quotients of functions using the product rule and quotient rule, with practice combining these with basic rules.
Builds on Basic Differentiation Rules
Derivatives of Trigonometric Functions
Derivatives of sine, cosine, tangent, cotangent, secant, and cosecant, derived using the special trig limits and quotient rule, with practice differentiating trig expressions.
Builds on Special Limits Involving Trigonometric Functions, Product and Quotient Rules
The Chain Rule
Differentiating composite functions using the chain rule; understanding why the rule works via rates of change, and practicing with nested functions including trig, exponential, and polynomial compositions.
Builds on Product and Quotient Rules, Derivatives of Trigonometric Functions
Implicit Differentiation
Differentiating equations that are not solved explicitly for y, treating y as a function of x and applying the chain rule; finding slopes of tangent lines to implicitly defined curves.
Builds on The Chain Rule
Derivatives of Logarithmic Functions and Logarithmic Differentiation
Derivatives of natural log and general logarithmic functions; using logarithmic differentiation to differentiate products, quotients, and variable exponents efficiently.
Builds on The Chain Rule
Derivatives of Inverse Functions
The relationship between the derivative of a function and its inverse; deriving and applying derivatives of inverse trigonometric functions (arcsin, arccos, arctan) using implicit differentiation and known trig derivatives.
Builds on Derivatives of Trigonometric Functions, Implicit Differentiation
Higher-Order Derivatives
Computing second and higher derivatives; interpreting the second derivative as the rate of change of the rate of change, with application to acceleration.
Builds on The Chain Rule
Related Rates
Setting up and solving related rates problems by differentiating an equation relating two or more quantities with respect to time, using the chain rule and implicit differentiation. Includes classic problems like ladders sliding, expanding balloons, and shadows.
Builds on Implicit Differentiation, Higher-Order Derivatives
Mean Value Theorem and Rolle's Theorem
Statement and geometric meaning of Rolle's Theorem and the Mean Value Theorem; using them to justify the existence of points with specific derivative values and to reason about function behavior.
Builds on Differentiability and Its Relationship to Continuity
Extrema and Critical Points
Finding critical points, understanding the definitions of local and global extrema, and using the Extreme Value Theorem to justify the existence of absolute extrema on closed intervals.
Builds on Mean Value Theorem and Rolle's Theorem
Increasing/Decreasing Functions and the First Derivative Test
Using the sign of the first derivative to determine intervals of increase and decrease, and applying the First Derivative Test to classify local extrema.
Builds on Extrema and Critical Points
Concavity, Inflection Points, and the Second Derivative Test
Using the second derivative to determine concavity and locate inflection points; applying the Second Derivative Test to classify local extrema as an alternative to the first derivative test.
Builds on Higher-Order Derivatives, Increasing/Decreasing Functions and the First Derivative Test
Curve Sketching Using Derivatives
Combining information about domain, intercepts, asymptotes, increasing/decreasing behavior, concavity, and extrema to sketch an accurate graph of a function by hand, and to match graphs of f, f', and f''.
Builds on Limits at Infinity and Horizontal Asymptotes, Concavity, Inflection Points, and the Second Derivative Test
Applied Optimization Problems
Setting up and solving real-world optimization problems (maximizing area, minimizing cost, etc.) by writing an objective function, applying constraints, and using calculus to find absolute extrema.
Builds on Extrema and Critical Points, Curve Sketching Using Derivatives
Linear Approximation and Differentials
Using the tangent line to approximate function values near a point (linearization) and using differentials to estimate propagated error; understanding the geometric meaning of dy versus Δy.
Builds on The Derivative as a Limit
L'Hôpital's Rule
Using derivatives to evaluate limits of indeterminate forms 0/0 and ∞/∞; recognizing when the rule applies and rewriting other indeterminate forms to use it. Requires general fluency with differentiation rules to differentiate numerator and denominator of arbitrary expressions.
Builds on Limits at Infinity and Horizontal Asymptotes, Product and Quotient Rules, The Chain Rule
Antiderivatives and Indefinite Integrals
The concept of an antiderivative as the reverse of differentiation, basic antiderivative rules (power rule, exponential, trig), the constant of integration, and finding a particular antiderivative given an initial condition.
Builds on Basic Differentiation Rules, Derivatives of Trigonometric Functions
Approximating Area with Riemann Sums
Estimating the area under a curve using left, right, and midpoint Riemann sums with rectangles, and understanding sigma notation for sums; recognizing that finer partitions improve the approximation.
Builds on Review of Functions and Graphs
The Definite Integral
Defining the definite integral as the limit of Riemann sums, interpreting it as net signed area, and understanding its basic properties (linearity, additivity over intervals, and the effect of reversing limits of integration).
Builds on Introduction to Limits, Approximating Area with Riemann Sums
The Fundamental Theorem of Calculus
Both parts of the Fundamental Theorem of Calculus: how the derivative of an accumulation function relates to the integrand, and how definite integrals can be evaluated using antiderivatives. Central bridge connecting differentiation and integration.
Builds on Antiderivatives and Indefinite Integrals, The Definite Integral
Integration by Substitution
Using u-substitution to evaluate indefinite and definite integrals by reversing the chain rule, including adjusting the limits of integration for definite integrals.
Builds on The Chain Rule, The Fundamental Theorem of Calculus
Area Between Curves
Setting up and evaluating definite integrals to find the area between two curves, including cases where curves intersect multiple times and where integration with respect to y is more convenient.
Builds on The Fundamental Theorem of Calculus, Integration by Substitution
Volumes of Solids: Disks, Washers, and Known Cross-Sections
Computing volumes of solids of revolution using the disk and washer methods, and volumes of solids with known cross-sectional shapes, by setting up and evaluating integrals.
Builds on Area Between Curves
Applications of Integrals to Motion
Using integrals to find displacement, distance traveled, and position from velocity and acceleration functions; relating signed area under a velocity curve to displacement versus total distance.
Builds on The Fundamental Theorem of Calculus
Average Value of a Function
Using the definite integral to compute the average value of a function over an interval, and applying the Mean Value Theorem for Integrals to interpret this value graphically.
Builds on The Fundamental Theorem of Calculus
Differential Equations and Slope Fields
Interpreting a differential equation as describing a family of curves, sketching and interpreting slope fields, and matching a slope field to its differential equation.
Builds on Antiderivatives and Indefinite Integrals
Separable Differential Equations and Exponential Growth/Decay
Solving separable differential equations by separating variables and integrating; applying this to exponential growth and decay models, including finding particular solutions from initial conditions.
Builds on Integration by Substitution, Differential Equations and Slope Fields
Practice questions in Calculus AB
A few of the kinds of questions you will work through, with answers.
Find the limit: lim x→3 of (x² - 9)/(x - 3)
See the answer
6
Factor x² - 9 as (x-3)(x+3), cancel the (x-3) term, and substitute x = 3 into the remaining (x+3) to get 6.
Find the derivative of f(x) = 4x³ - 2x + 7
See the answer
f'(x) = 12x² - 2
Apply the power rule to each term: the derivative of 4x³ is 12x², the derivative of -2x is -2, and the derivative of the constant 7 is 0.
Differentiate y = sin(3x² + 1)
See the answer
y' = 6x cos(3x² + 1)
The derivative of sin(u) is cos(u)·u', and here u = 3x² + 1 has derivative 6x, so multiply cos(3x²+1) by 6x.
A rectangular box with a square base and open top has volume 32 cubic units. What base side length minimizes the surface area?
See the answer
Side length 4
Writing surface area S = x² + 128/x and setting S' = 2x - 128/x² = 0 gives x³ = 64, so x = 4.
Evaluate the definite integral from 0 to 2 of 3x² dx
See the answer
8
An antiderivative of 3x² is x³, and evaluating x³ at 2 minus x³ at 0 gives 8 - 0 = 8.
Solve dy/dx = 2y with initial condition y(0) = 5
See the answer
y = 5e^(2x)
Separate variables to get dy/y = 2 dx, integrate to get ln|y| = 2x + C, exponentiate, and use y(0) = 5 to find the constant, giving y = 5e^(2x).
Questions about this subject
Does this cover everything on the AP Calculus AB exam?
Yes, the 37 topics follow the AB course framework from limits through an introduction to differential equations and slope fields. It stops before BC-only topics like series, parametric and polar calculus, and advanced integration techniques.
Do I need to have seen any calculus before starting?
No. The course assumes a solid precalculus background, meaning algebra, functions, graphing, and trig identities, but starts from the idea of a limit with no prior calculus assumed.
What if I can get the answer but can't explain why a step works?
The tutor asks about both: it checks whether you can explain the reasoning behind a rule, like why the chain rule works, as well as whether you can carry out the computation, and will slow down on whichever one is shaky.
How are related rates, optimization, and other word problems handled?
The tutor walks through setting up the problem first, asking what quantities are changing and what equation relates them, before any differentiation happens, since setup is usually the harder part.
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