PDF Notes

in PDFs

The following calculus notes are sorted by chapter and topic. They are in the form of PDF documents that can be printed or annotated by students for educational purposes. If you instead prefer an interactive slideshow, please click here.

  • Formula Sheet: Calculus BC

    • trigonometric identities, unit circle
    • derivative rules, derivatives of common functions
    • average/instantaneous rates of change, Mean Value Theorem, Intermediate Value Theorem
    • integration rules, integrals of common functions
    • exponential growth/decay, logistic growth
    • arc length, surface area
    • volume: solids of revolution (cross sections, shells), solid with known cross sections
    • approximation: linearization, Newton’s Method, Euler’s method, series
    • Taylor/Maclaurin series, convergence at endpoints, convergence tests

  • Lesson 1: Parametric Plotting

    • parametrically plotting curves, circles, and ellipses
    • parametric equations for curves, surfaces, and solids
    • calculus with parametrically-defined curves
    • collision versus intersection with parametrically-defined curves
    • parametric plots of surfaces of revolution

  • Lesson 2: Vectors

    • vector addition, subtraction, and scalar multiplication
    • dot products and vector projection
    • the magnitude of a vector and unit vectors
    • parametric equations for lines
    • determining when parametric lines are parallel, perpendicular, skew, or the same line twice
    • position, velocity, speed, and acceleration

  • Lesson 3: Perpendicularity

    • xyz and parametric equations for a plane
    • parallel/perpendicular planes
    • find the line of intersection between two planes
    • parametric equations for lines
    • three noncollinear points determine a plane
    • the cross product and the magnitude of the cross product

  • Lesson 3 (Continued): Linearization

    • 2D linearization: the equation of a tangent line to a curve given a point of tangency
    • 3D linearization: the equation of a tangent plane to a surface given a point of tangency
    • parametric paths on surfaces
    • mapping a parametric path on a surface onto the linearization

  • Lesson 4: Gradient Vectors

    • gradient vectors: the direction of greatest initial increase on a surface
    • level curves, level sets, and contour plots
    • parametric paths on surfaces and the chain rule in higher dimensions
    • a gradient vector is always perpendicular to the level curve through its tail
    • using the gradient to identify critical points for possible local extrema
    • The Second Derivative Test in higher dimensions (the Hessian determinant)
    • gradient vectors for a 3D level surface of a 4D hypersurface, f(x,y,z)

  • Lesson 4 (Continued): Lagrange Multipliers and Constrained Optimization

    • using Lagrange multipliers to find candidates for the extrema on a parametric path on a surface
    • critical points are where the parametric path is tangent to a level curve

  • Lesson 5: Double Integrals

    • computing double integrals over a rectangular region
    • computing double integrals over a non-rectangular region
    • using the Gauss-Green formula to compute a double integral for a base region that can be parameterized
    • a proof of the Gauss-Green formula
    • using double integrals to compute area and volume
    • changing the order of integration

  • Lesson 6: Vector Fields Acting on a Curve

    • definition of a vector field
    • examples of vector fields: gradient fields, slope fields
    • vector field trajectories
    • vector fields acting on a curve
    • the net flow of a vector field ALONG a curve
    • the net flow of a vector field ACROSS a curve

  • Lesson 7: Flow Measurements

    • path integrals
    • the net flow of a vector field along a curve
    • the net flow (flux) of a vector field across a curve
    • the net flow of a gradient field along a closed curve
    • path independence for gradient fields
    • The Gradient Test
    • The Fundamental Theorem of Line Integrals

  • Lesson 8: Sources, Sinks, and Singularities

    • Gauss-Green formula (Green’s Theorem)
    • the rotation of a vector field
    • the divergence of a vector field
    • sources and sinks
    • measuring flow in the presence of singularities
    • del, the differential operator
    • the laplacian is the divergence of the gradient field

  • Lesson 9: Using Change of Variables to Transform 2D Integrals

    • change of variables (or mapping between coordinate spaces)
    • non-area-preserving maps
    • the Jacobian determinant (area conversion factor)
    • double integrals in polar coordinates
    • Mathematica-aided change of coordinates
    • del, the differential operator
    • the laplacian is the divergence of the gradient field

  • Lesson 10: Using Change of Variables to Transform 3D Integrals

    • change of variables for triple integrals
    • the Jacobian determinant in 3D (volume conversion factor)
    • triple integrals to compute volume
    • triple integrals to compute mass of an object with non-uniform density
    • Mathematica-aided change of coordinates

  • Lesson 11: Spherical and Cylindrical Coordinates

    • parameterizing a sphere
    • the Jacobian determinant (volume conversion factor) for spherical coordinates
    • triple integrals using spherical coordinates
    • parameterizing a cylinder
    • the Jacobian determinant (volume conversion factor) for cylindrical coordinates
    • triple integrals using cylindrical coordinates

  • Lesson 12: Surface Integrals

    • flow (flux) across a surface
    • The Divergence Theorem (Gauss’ Theorem)
    • using a substitute surface when the divergence is 0
    • using a substitute surface in the presence of singularities
    • surface area

  • Lesson 13: Stokes’ Theorem

    • the curl of a vector field
    • path integrals to compute flow along a curve in 3D
    • using Stokes’ Theorem to compute flow along a closed curve in 3D
    • orientable surfaces with a simple, closed boundary curve
    • gradient fields and path independence in 3D