AP® Calculus AB-BC
Modeling with Rate Functions
Modeling with Rate Functions
AB and BC
AB and BC
Problems involving rate functions (like velocity or growth rates) require setting up and solving differential equations by separating variables, using initial conditions, and finding particular solutions to model real-world situations.
Frog Pond (Calculator)
Frog Pond (Calculator)
Honey in a Beehive (Calculator)
Honey in a Beehive (Calculator)
BC Only
BC Only
Modeling with Rate Functions involves analyzing the behavior of functions representing changing rates to solve complex problems in contexts such as motion and growth.
Ocean Litter (Calculator) (BC)
Ocean Litter (Calculator) (BC)
Charts and Riemann Sums
Charts and Riemann Sums
AB and BC
AB and BC
Riemann sums estimate the area under a curve by dividing a region into rectangles and summing their areas, with height determined by evaluating the function at left endpoints, right endpoints, or midpoints.
Table of Temperature Values (No Calculator)
Table of Temperature Values (No Calculator)
Physics Table Question Given Velocity (No Calculator)
Physics Table Question Given Velocity (No Calculator)
Grain in a Silo (No Calculator)
Grain in a Silo (No Calculator)
Cooling Coffee Pot (Calculator)
Cooling Coffee Pot (Calculator)
Comparing and Analyzing $f$, $f’$, and $f’’$
Comparing and Analyzing $f$, $f’$, and $f’’$
AB and BC
AB and BC
Functions, their derivatives, and second derivatives are analyzed together to understand relationships between their values, slopes, concavity, and key features like maximum/minimum points and inflection points.
Determine Characteristics of $f(x)$ (No Calculator)
Determine Characteristics of $f(x)$ (No Calculator)
Graphical Interpretation with $g(x)=f'(x)$ (No Calculator)
Graphical Interpretation with $g(x)=f'(x)$ (No Calculator)
Graph of $f^{\prime}$ (No Calculator)
Graph of $f^{\prime}$ (No Calculator)
Analyzing a Given Piecewise Function (No Calculator)
Analyzing a Given Piecewise Function (No Calculator)
Particle Motion
Particle Motion
AB and BC
AB and BC
The position, velocity, and acceleration of a moving particle can be analyzed using derivatives and integrals, with positive/negative signs indicating direction of motion and points of inflection revealing key changes in behavior.
Rectilinear Motion (Calculator)
Rectilinear Motion (Calculator)
Movement of a Toy Train (Calculator)
Movement of a Toy Train (Calculator)
Particle Motion (Calculator)
Particle Motion (Calculator)
Motion Along the $x$-axis (Calculator)
Motion Along the $x$-axis (Calculator)
Slope Fields and Differential Equations
Slope Fields and Differential Equations
AB and BC
AB and BC
A slope field is a visual representation showing the slope of a differential equation's solution curves at many points in the plane, with each small line segment indicating the direction a solution curve would follow through that point.
Differential Equation: Tangent Lines (No Calculator)
Differential Equation: Tangent Lines (No Calculator)
Differential Equation Slope Field (No Calculator)
Differential Equation Slope Field (No Calculator)
Growth Curve (No Calculator)
Growth Curve (No Calculator)
BC Only
BC Only
Slope fields and differential equations encompass techniques like Euler's method, logistic growth models, and integration approaches to analyze and solve first-order differential equations graphically and analytically.
Differential Equation Solutions (Calculator) (BC)
Differential Equation Solutions (Calculator) (BC)
Volume and Area
Volume and Area
AB and BC
AB and BC
Area and volume can be calculated for regions and solids by integrating cross-sectional slices, shells, or layers, including scenarios with known cross sections and those formed by rotating functions around axes.
Volumes and Area from a Region (Calculator)
Volumes and Area from a Region (Calculator)
Polar Bounded Region (Calculator)
Polar Bounded Region (Calculator)
Area and Volume Based on a Given Region (No Calculator)
Area and Volume Based on a Given Region (No Calculator)
BC Only
BC Only
Arc length and surface area of revolving functions can be found using formulas involving integrals with square roots, while work and differential equations related to force use advanced integration techniques.
Analyze a Graph Defined With Polar Coordinates (BC) (Calculator)
Analyze a Graph Defined With Polar Coordinates (BC) (Calculator)
Convergent and Divergent Functions
Convergent and Divergent Functions
BC Only
BC Only
Topics include advanced concepts such as series convergence tests, power series representations, and using Taylor and Maclaurin series for function approximations.
Family Function (No Calculator) (BC)
Family Function (No Calculator) (BC)
Taylor Polynomials and Maclaurin Series
Taylor Polynomials and Maclaurin Series
BC Only
BC Only
Taylor polynomials and Maclaurin series provide approximations for functions near a given point using calculated derivatives, helping to represent complex functions algebraically for analysis and solution derivation.
Taylor/Maclaurin Series (BC) (No Calculator)
Taylor/Maclaurin Series (BC) (No Calculator)
Power Series Ratio Test and Error Estimation (BC) (No Calculator)
Power Series Ratio Test and Error Estimation (BC) (No Calculator)
Miscellaneous (AB/BC)
Miscellaneous (AB/BC)
Continuity and Differentiability
Continuity and Differentiability
A continuous function has no gaps, jumps, or holes in its graph, while a differentiable function must be both continuous and have a defined derivative (slope) at every point.
Continuity and Differentiability of Piecewise (No Calculator)
Continuity and Differentiability of Piecewise (No Calculator)