Mathematics and Physics

Represent a physical relationship as a formula, table, and graph.

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 a product of two functions.

Differentiate a composite function using the chain rule.

Differentiate sine and cosine functions.

Find antiderivatives of power functions.

Determine an antiderivative 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.

Evaluate a simple integral by substitution.

Approximate a definite integral from numerical data using the trapezoidal rule.

Determine velocity and acceleration by differentiating a position function.

Recover velocity or position by integrating acceleration or velocity with an initial condition.

Interpret position-time, velocity-time, and acceleration-time graphs consistently.

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 connected-body and inclined-plane problems involving tension and normal forces.

Model static and kinetic friction in a Newton's-law problem.

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.

Calculate a potential-energy change associated with a conservative force.

Determine a one-dimensional conservative force from a potential-energy function.

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 a system of point masses.

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

Calculate rotational kinetic energy.

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.

Solve the simple-harmonic differential equation for specified initial conditions.

Derive the angular frequency of an ideal mass-spring oscillator.

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.

Express the electric field of a continuous one-dimensional charge distribution as an integral.

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

Apply Gauss's law to a spherically, cylindrically, or planarly symmetric charge distribution.

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

Relate electric potential difference to work and potential-energy change.

Calculate electric potential due to a collection of point charges.

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

Calculate charge and voltage for a capacitor using its capacitance.

Calculate the energy stored in a capacitor.

Solve a DC resistor network using Ohm's law and Kirchhoff's rules.

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

Predict how dielectric insertion changes an ideal capacitor under fixed-charge or fixed-voltage conditions.

Calculate magnetic forces on moving charges and current-carrying wires.

Analyze the circular motion of a charged particle in a uniform magnetic field.

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.

Explain how Maxwell's equations predict electromagnetic waves and expose a conflict with Galilean velocity addition.

Use 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.

Calculate a position probability from a normalized one-dimensional wavefunction.

Apply the position-momentum uncertainty relation.

Relate atomic spectral lines to transitions between quantized energy levels.

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