Mathematics and Physics

Calculate an average rate of change from data or a function.

Estimate a limit numerically and graphically.

Derive the derivative of a simple function from the difference quotient.

Differentiate powers and linear combinations of functions.

Differentiate products and quotients of functions.

Differentiate a composite function using the chain rule.

Differentiate natural logarithmic and exponential functions.

Differentiate sine and cosine functions.

Find antiderivatives of power functions and determine the constant of integration from an initial condition.

Interpret a definite integral as signed accumulated change.

Use the Fundamental Theorem of Calculus to evaluate a definite integral.

Integrate sine and cosine functions.

Integrate exponential functions and functions of the form 1/x.

Evaluate a simple integral by substitution.

Determine velocity and acceleration by differentiating motion functions and interpreting slopes on motion graphs.

Recover velocity or position by integration with an initial condition and interpret signed areas on motion graphs.

Derive the constant-acceleration equations using differentiation and integration.

Solve a multi-stage one-dimensional motion problem.

Model projectile motion by separating horizontal and vertical components.

Relate position, velocity, and centripetal acceleration in uniform circular motion.

Construct a free-body diagram for an object or system.

Apply Newton's second law in Cartesian components.

Identify Newton's third-law force pairs without confusing forces on different objects.

Solve inclined-plane problems involving static or kinetic friction by resolving forces along suitable axes.

Solve connected-body problems by applying Newton's second law to components linked by tension forces.

Apply radial Newton's second law to circular motion.

Use Newtonian gravitation to analyze a circular orbit.

Calculate work as an integral of a variable force along one dimension.

Apply the work-energy theorem to determine a change in speed.

Relate a one-dimensional conservative force and its potential energy using ΔU = −∫F dx and F = −dU/dx.

Apply conservation of mechanical energy to a multi-stage motion.

Calculate impulse from a time-dependent force and relate it to momentum change.

Apply momentum conservation to a one-dimensional collision.

Relate angular position, velocity, and acceleration to their linear counterparts.

Calculate torque using the vector cross product.

Calculate the moment of inertia of point masses and of a simple continuous rigid body.

Apply rotational Newton's second law about a fixed axis.

Calculate the angular momentum of a particle or fixed-axis rigid body and relate its change to net torque.

Apply conservation of angular momentum to a changing rotating system.

Analyze rolling without slipping using translational and rotational energy.

Identify the order of an ordinary differential equation and the initial conditions needed for a physical solution.

Solve a separable differential equation of the form dy/dt = ky.

Calculate radioactive decay and half-life using an exponential model.

Derive and solve the simple-harmonic differential equation for an ideal mass-spring oscillator with specified initial conditions.

Derive the small-angle period of a simple pendulum.

Extract amplitude, wavelength, frequency, phase, and speed from a sinusoidal traveling wave.

Use superposition to identify nodes and antinodes in a standing wave.

Apply charge conservation and quantization to a physical process.

Calculate the vector Coulomb force between point charges.

Calculate the electric field of a discrete charge distribution by superposition.

Set up and evaluate the electric-field integral for a simple one-dimensional continuous charge distribution.

Calculate electric flux through a surface in a uniform electric field.

Apply Gauss's law to calculate the electric field of a spherically symmetric charge distribution.

Apply Gauss's law to calculate electric fields with cylindrical or planar symmetry.

Analyze the motion of a charged particle in a uniform electric field.

Calculate electric potential and potential-energy changes for point charges and relate those changes to work.

Determine a one-dimensional electric field from an electric-potential function.

Derive the capacitance of an ideal parallel-plate capacitor and use Q = CΔV.

Calculate the energy stored in a capacitor.

Use Ohm's law to analyze a single-loop DC resistor circuit.

Apply Kirchhoff's junction and loop rules to set up and solve a multi-loop DC resistor network.

Solve the differential equation for charging or discharging an RC circuit.

Calculate the magnetic force on a moving charge and use it to analyze circular motion in a uniform magnetic field.

Calculate the magnetic force on a current-carrying conductor by integrating force elements along the wire.

Calculate the magnetic field of a simple current distribution using the Biot-Savart law.

Apply Ampère's law to a highly symmetric current distribution.

Calculate magnetic flux through a surface.

Use Faraday's and Lenz's laws to determine an induced electromotive force and its direction.

Use the integral forms of Maxwell's equations to explain how electromagnetic waves arise and why their fixed speed conflicts with Galilean velocity addition.

Use a light-clock argument and the invariance of light speed to derive and apply the time-dilation relation.

Calculate relativistic energy and momentum for a massive particle.

Use photon energy to analyze the photoelectric effect.

Calculate the de Broglie wavelength of a material particle.

Interpret a one-dimensional wavefunction, normalize a simple wavefunction, and calculate a position probability.

Apply the position-momentum uncertainty relation.

Relate atomic spectral lines to transitions between quantized energy levels.

Use mass defect and E = mc² to calculate nuclear binding energy and interpret the energy basis of fission or fusion.