AP Calculus AB

• Derivative definition and notation

  ≈ 2.5%1 h 45 min

  Students define the derivative through a limit of a difference quotient and use standard derivative notation. They connect derivative values to tangent-line slope and instantaneous rate of change.

  • Compute a derivative at a point from the limit definition.
  • Use standard notation such as $f'(x)$ and $\frac{dy}{dx}$ correctly.
  • Interpret the derivative as tangent-line slope and instantaneous rate of change.
  • Write the equation of a tangent line from derivative information.

• Estimating derivatives from representations

  ≈ 2%1 h 15 min

  Students estimate derivative values and derivative behavior from graphs, tables, and verbal descriptions. They connect signs and magnitudes of derivatives to function behavior.

  • Estimate $f'(a)$ from a graph or table of values.
  • Interpret positive, negative, and zero derivative values in context.
  • Compare magnitudes of derivatives to compare rates of change.
  • Infer where a derivative may fail to exist from graphical behavior.

• Differentiability and basic rules

  ≈ 2.9%1 h 45 min

  Students connect differentiability and continuity and apply the constant, constant multiple, sum, difference, and power rules. Higher-order derivatives begin here.

  • Explain the relationship between differentiability and continuity.
  • Determine where a function is differentiable or not differentiable.
  • Differentiate functions using the constant, constant multiple, sum, difference, and power rules.
  • Find higher-order derivatives such as $f''(x)$ when appropriate.

• Product, quotient, and trigonometric rules

  ≈ 3.4%2 h

  Students differentiate products, quotients, and trigonometric functions and combine rules fluently. Emphasis is on efficient symbolic differentiation and accurate notation.

  • Differentiate products using the product rule.
  • Differentiate quotients using the quotient rule.
  • Differentiate trigonometric functions using standard derivative rules.
  • Combine multiple derivative rules in a single calculation.
  • Simplify derivative expressions when appropriate.

About 6 h 45 min of study, lessons and core practice

• Chain rule and composite functions

  ≈ 2.9%1 h 45 min

  Students differentiate compositions of functions using the chain rule, including nested algebraic, trigonometric, exponential, and logarithmic expressions within AB scope.

  • Identify inner and outer functions in a composite function.
  • Differentiate composite functions using the chain rule.
  • Combine the chain rule with previously learned derivative rules.
  • Interpret a derivative of a composite function in context when appropriate.

• Implicit differentiation and tangent lines

  ≈ 2.9%1 h 45 min

  Students differentiate implicitly defined relations and use the result to find slopes and tangent-line equations. This includes equations not easily solved explicitly for $y$.

  • Differentiate an implicitly defined relation with respect to $x$.
  • Solve for $\frac{dy}{dx}$ from an implicit derivative equation.
  • Find the slope of a tangent line to an implicitly defined curve at a point.
  • Write tangent-line equations for implicit curves.
  • Recognize when implicit differentiation is more efficient than solving explicitly.

• Inverse, exponential, logarithmic, and inverse trigonometric derivatives

  ≈ 2.9%2 h

  Students differentiate inverse functions and the standard exponential, logarithmic, and inverse trigonometric functions in AP Calculus AB. They connect derivative formulas to inverse-function reasoning and chain rule structure.

  • Differentiate exponential and logarithmic functions within AB scope.
  • Differentiate inverse trigonometric functions within AB scope.
  • Use the derivative relationship for inverse functions.
  • Apply the chain rule to composite exponential, logarithmic, and inverse trigonometric expressions.
  • Interpret derivative values of inverse or logarithmic models when given context.

• Selecting derivative procedures

  ≈ 2%1 h 15 min

  Students decide which differentiation method or combination of methods is most appropriate and organize longer derivative calculations accurately. Higher-order derivatives continue in more complex settings.

  • Select an efficient derivative procedure for a given function.
  • Combine rules such as product, quotient, chain, and implicit differentiation in multistep problems.
  • Compute higher-order derivatives for functions requiring advanced derivative rules.
  • Check whether a derivative result is consistent with the original function’s structure or graph.

About 6 h 45 min of study, lessons and core practice

• Interpreting derivatives in context

  ≈ 2.9%1 h 15 min

  Students interpret derivative values, signs, and units in applied settings. Emphasis is on concise conclusions that answer the contextual question asked.

  • Interpret the meaning of a derivative in context, including units.
  • Determine whether a quantity is increasing, decreasing, or momentarily unchanged from derivative information.
  • Compare rates of change of different quantities or at different times.
  • Write a clear contextual conclusion from derivative calculations or estimates.

• Motion along a line

  ≈ 2.9%1 h 30 min

  Students analyze particle motion using position, velocity, speed, and acceleration. They interpret derivatives to describe direction of motion and speeding up or slowing down.

  • Find velocity and acceleration from a position function.
  • Interpret the sign of velocity and acceleration in motion problems.
  • Determine when a particle is moving left or right.
  • Determine when a particle is speeding up or slowing down.
  • Relate derivative information to graphs of position, velocity, and acceleration.

• Related rates modeling

  ≈ 2.9%2 h

  Students model changing quantities linked by an equation, differentiate with respect to time, and solve for an unknown rate. Accurate setup, substitution, and units are essential.

  • Translate a geometric or physical situation into an equation relating variables.
  • Differentiate the related equation with respect to time.
  • Substitute known values at the specified instant to solve for an unknown rate.
  • Interpret the sign and units of the resulting rate in context.
  • Justify the setup used in a related-rates problem.

• Local linearity and linearization

  ≈ 2.9%1 h 15 min

  Students use tangent lines as local linear approximations to a function near a point. They estimate nearby values and interpret why the approximation is reasonable.

  • Construct the linearization of a differentiable function at a point.
  • Use linearization to approximate a function value near the point of tangency.
  • Compare exact and approximated values when both are available.
  • Explain local linearity in graphical or contextual terms.

About 6 h of study, lessons and core practice

• Extrema and critical points

  ≈ 2.9%1 h 30 min

  Students identify critical points and find absolute and relative extrema, especially on closed intervals. They use derivative information and endpoint checks to justify answers.

  • Find critical points by solving $f'(x)=0$ or identifying where $f'(x)$ does not exist.
  • Determine absolute extrema on a closed interval.
  • Distinguish between relative and absolute extrema.
  • Use endpoint values and critical-point values to justify a closed-interval answer.
  • Apply the Extreme Value Theorem to justify that a continuous function on a closed interval must attain an absolute maximum and an absolute minimum.
  • Explain when the hypotheses of the Extreme Value Theorem are not satisfied, so existence of absolute extrema is not guaranteed.

• Mean Value Theorem and related results

  ≈ 2%1 h 15 min

  Students verify conditions for theorem use and apply the Mean Value Theorem, including common special-case reasoning. They connect secant slopes to tangent slopes.

  • State the conditions required for the Mean Value Theorem.
  • Verify whether a function satisfies the hypotheses on a given interval.
  • Find value(s) $c$ guaranteed by the Mean Value Theorem.
  • Interpret the theorem in graphical or contextual language.
  • Use related theorem-based reasoning such as Rolle’s Theorem when appropriate.

• First-derivative analysis

  ≈ 3.9%1 h 45 min

  Students use first-derivative information to determine intervals of increase and decrease and classify local extrema. They work from formulas and from graphs of $f$ or $f'$ alike.

  • Determine intervals where a function is increasing or decreasing.
  • Use sign changes in $f'(x)$ to identify local maxima and minima.
  • Apply the first derivative test.
  • Infer properties of $f$ from a graph or table of $f'(x)$.
  • Support conclusions with interval notation and concise justification.

• Second-derivative analysis and curve sketching

  ≈ 3.9%2 h

  Students analyze concavity and inflection points, apply the second derivative test, and assemble derivative evidence into a coherent sketch of a function.

  • Determine intervals of concavity from $f''(x)$ or from the behavior of $f'(x)$.
  • Identify possible points of inflection and justify whether they occur.
  • Apply the second derivative test when valid.
  • Infer properties of $f$ from graphs of $f'$ or $f''$.
  • Sketch a function consistent with derivative, concavity, and asymptotic information.

• Optimization with derivatives

  ≈ 3.9%2 h

  Students formulate and solve optimization problems by defining variables, writing an objective function, applying constraints, and justifying extrema in context.

  • Define variables and write an objective function for an optimization problem.
  • Use a constraint to express the objective in one variable.
  • Differentiate the objective and locate critical points.
  • Justify that the selected value gives the required maximum or minimum.
  • State the result in context with appropriate units.

About 8 h 30 min of study, lessons and core practice

• Riemann sums and accumulation

  ≈ 3.9%1 h 45 min

  Students approximate accumulation with left, right, and midpoint sums and interpret signed area as net accumulation. Graphical, tabular, and contextual setups are all included.

  • Write and evaluate left, right, or midpoint Riemann sums.
  • Interpret a Riemann sum in terms of accumulation from a rate.
  • Determine whether a numerical approximation overestimates or underestimates in simple settings.
  • Use units correctly when interpreting an accumulation quantity.

• Definite integrals and net change

  ≈ 3.9%1 h 45 min

  Students interpret the definite integral, evaluate integrals using geometry when possible, and connect integrals of rates to net change. They distinguish net change from total accumulation when sign matters.

  • Interpret $\int_a^b f(x)\,dx$ as a net accumulation.
  • Evaluate definite integrals from geometric regions when appropriate.
  • Use properties of definite integrals to rewrite or simplify expressions.
  • Apply the net change theorem to a quantity defined by a rate.
  • Distinguish net change from total change or total distance when signs vary.

• Fundamental Theorem of Calculus and accumulation functions

  ≈ 3.9%2 h

  Students use both parts of the Fundamental Theorem of Calculus to connect derivatives and integrals. They differentiate accumulation functions and evaluate definite integrals by antiderivatives.

  • Evaluate a definite integral using an antiderivative.
  • Differentiate an accumulation function with a variable limit of integration.
  • Apply the chain rule when the limit of integration is a function of $x$.
  • Interpret the derivative of an accumulation function graphically or contextually.
  • Use the Fundamental Theorem of Calculus to connect accumulated quantity and rate.

• Antiderivatives and indefinite integrals

  ≈ 3.9%1 h 45 min

  Students find antiderivatives using basic integration rules and use initial conditions to determine particular solutions. This includes algebraic, exponential, trigonometric, and simple composite forms within AB scope.

  • Find an antiderivative using power and constant multiple rules.
  • Integrate standard exponential and trigonometric forms within AB scope.
  • Write the family of antiderivatives using $+C$.
  • Use an initial condition to determine a particular antiderivative.
  • Check an antiderivative by differentiating it.
  • Find antiderivatives involving $$\frac{1}{x}$$ using $$\int \frac{1}{x}\,dx=\ln|x|+C$$.
  • Recognize logarithmic antiderivatives that arise after algebraic rewriting or substitution, such as forms equivalent to $$\frac{f'(x)}{f(x)}$$.

• Substitution for integrals

  ≈ 2%1 h 30 min

  Students use $u$-substitution to evaluate indefinite and definite integrals when a composite structure is present. They choose substitutions that simplify the integrand and bounds.

  • Identify when $u$-substitution is an appropriate method.
  • Choose a substitution that simplifies a composite integrand.
  • Evaluate indefinite integrals using $u$-substitution.
  • Evaluate definite integrals using $u$-substitution, including updated bounds or back-substitution.
  • Check that the result is reasonable by differentiating or interpreting the original integrand.

• Graphing calculator methods for numerical calculus

  ≈ 1%1 h

  Students use a graphing calculator on calculator-allowed parts to estimate roots, intersections, numerical derivatives, and definite integrals and to interpret tables and graphs accurately.

  • Use a graphing calculator to estimate a derivative numerically at a specified point.
  • Use a graphing calculator to evaluate a definite integral numerically.
  • Find roots or intersections needed for a calculus setup.
  • Interpret calculator-generated graphs and tables in support of a calculus conclusion.

About 9 h 45 min of study, lessons and core practice

• Slope fields and solution curves

  ≈ 2%1 h 15 min

  Students read and sketch slope fields and use them to reason about solution behavior. They connect a differential equation to families of solution curves and particular initial-value solutions.

  • Match a differential equation to its slope field.
  • Sketch a solution curve through a given point on a slope field.
  • Estimate qualitative behavior such as increasing or decreasing from a slope field.
  • Use a slope field to compare possible solutions or long-term trends.

• Modeling and verifying differential equations

  ≈ 2%1 h 15 min

  Students write differential equations from verbal or contextual descriptions and verify whether a proposed function is a solution. The focus is on interpreting rates and model structure.

  • Write a differential equation that models a described rate relationship.
  • Verify that a given function is a solution to a differential equation by substitution and differentiation.
  • Interpret the meaning of terms in a differential equation in context.
  • Use an initial condition to identify a particular solution conceptually or algebraically.

• Solving separable differential equations

  ≈ 2%1 h 45 min

  Students solve separable differential equations by separating variables and integrating, then apply initial conditions. They interpret the resulting solution in context when relevant.

  • Separate variables in a differential equation when possible.
  • Integrate both sides to obtain a general solution.
  • Use an initial condition to find a particular solution.
  • Interpret the solution or its domain in context.
  • Differentiate the solution to check that it satisfies the original differential equation.

• Exponential and logistic models

  ≈ 2%1 h 15 min

  Students analyze standard growth and decay models and logistic differential equations within AB scope. Emphasis is on interpretation of parameters, long-term behavior, and contextual meaning.

  • Model exponential growth or decay with a differential equation or solution form.
  • Use an initial condition to determine a particular exponential model.
  • Interpret parameters and long-term behavior of exponential or logistic models.
  • Identify carrying capacity and equilibrium behavior in a logistic model.

• Euler’s method approximations

  ≈ 2%45 min

  Approximate solution values to an initial-value problem by stepping forward with tangent-line estimates from the differential equation.

  • Approximate solution values for an initial-value problem using Euler’s method with a given step size.
  • Compute successive Euler updates from $$\frac{dy}{dx}=f(x,y)$$ and organize the work in a table or sequence of steps.
  • Interpret each Euler step as a local linear or tangent-line approximation to the true solution curve.
  • Judge whether an Euler approximation is reasonable by comparing it with a slope field, graph, or contextual behavior.

About 6 h 15 min of study, lessons and core practice

• Area between curves

  ≈ 2.9%1 h 30 min

  Students find the area of a region bounded by two curves using definite integrals with respect to $x$. They set correct intersections and top-minus-bottom structure.

  • Determine intersection points that bound a region.
  • Set up an integral for area between two curves with respect to $x$.
  • Evaluate the area exactly or approximately as appropriate.
  • Interpret the difference between signed integral and geometric area.

• Volumes with known cross sections

  ≈ 2.9%1 h 45 min

  Students set up and evaluate volumes of solids with known cross-sectional areas. They translate geometric descriptions into area formulas as functions of $x$.

  • Express the cross-sectional area in terms of the independent variable.
  • Set up a definite integral for the volume of a solid with known cross sections.
  • Use appropriate geometric formulas for common cross sections.
  • Evaluate the resulting integral and report volume with correct units.

• Disk and washer methods

  ≈ 2.9%1 h 45 min

  Students compute volumes of solids of revolution using the disk and washer methods within AB scope. They focus on radius expressions, bounds, and correct subtraction of inner from outer radius.

  • Identify when the disk method or washer method applies.
  • Write radius expressions from graphs or formulas.
  • Set up a volume integral using $\pi[R(x)]^2$ or $\pi([R(x)]^2-[r(x)]^2)$.
  • Evaluate a volume of revolution and interpret units.
  • Check that the chosen bounds and radii match the described region.

• Average value and motion from integrals

  ≈ 2%1 h 30 min

  Students use integrals to compute the average value of a function and to analyze displacement and total distance traveled from velocity. They interpret results carefully in context.

  • Compute the average value of a function on an interval.
  • Interpret the average value in context.
  • Find displacement from a velocity function using a definite integral.
  • Find total distance traveled by integrating speed or absolute velocity structure.
  • Explain the difference between displacement and total distance traveled.

About 6 h 30 min of study, lessons and core practice