Wemazi

AP Calculus · Unit 7

Differential Equations

Slope fields, separation of variables and growth models.

Topic 7.1Model a differential equation

  1. Translate a rate statement into an equationExpress the named derivative in terms of the relevant changing quantities.
  2. Model proportional growth or decayDistinguish proportionality to the amount from proportionality to a difference or another variable.
  3. Combine competing ratesUse inflow minus outflow or another correctly signed net-rate model.
  4. Use initial conditions and unitsIdentify what an initial value supplies and check the model's dimensional meaning.

Topic 7.2Verify a proposed solution

  1. Differentiate a proposed solutionFind the derivative needed by the differential equation.
  2. Check the equation throughout an intervalSubstitute the function and its derivative rather than checking only one point.
  3. Check the initial conditionEvaluate the proposed function at the stated starting input.
  4. Determine a parameter and valid domainFind constants or restrictions that make both the equation and initial condition hold.

Topic 7.3Draw a slope field

  1. Calculate slopes at supplied pointsEvaluate the differential equation at each ordered pair.
  2. Match a slope segment to a pointSelect the appropriate orientation and steepness.
  3. Locate zero-slope setsDetermine where the derivative is zero and distinguish a set of zero slopes from a constant solution.
  4. Recognize patterns across a fieldMatch rows, columns, or other slope patterns with a candidate differential equation.

Topic 7.4Reason from a slope field

  1. Follow a solution from its initial pointSelect a path consistent with the field and starting value.
  2. Determine increasing and decreasing behaviorRead the sign of the slopes along the solution.
  3. Identify equilibrium solutionsFind constant solutions and distinguish them from isolated horizontal tangents.
  4. Predict long-term behaviorUse the nearby slope directions to determine approach toward or movement away from equilibrium.
  5. Determine concavity when supportedRelate changing slopes along the solution to its bending.

Topic 7.5Euler approximation

BC only
  1. Take one Euler stepAdd step width times the slope at the current point.
  2. Update both coordinates after every stepRecalculate the next slope using the new approximate point.
  3. Reach the requested inputUse the stated widths and direction, including backward or unequal steps.
  4. Interpret the approximation's errorCompare with an exact value or use relevant concavity information when supplied.

Topic 7.6Separate variables

  1. Recognize a separable equationChoose a rearrangement with each variable on its own side.
  2. Integrate both sidesForm an implicit solution family with a constant.
  3. Recover an explicit family when possibleSolve for the dependent variable while preserving valid signs and branches.
  4. Keep solutions lost through divisionCheck constant or equilibrium solutions before dividing by a variable expression.
  5. State relevant domain restrictionsExclude poles or invalid branches from the claimed solution interval.

Topic 7.7Apply an initial condition

  1. Find a general solutionSeparate and integrate the differential equation or use the appropriate given family.
  2. Apply the initial conditionSubstitute both the initial input and output to determine the constant.
  3. Choose the valid branch and intervalKeep the solution passing through the initial point and stop at relevant singularities.
  4. Use and verify the particular solutionEvaluate the requested quantity and check the equation and starting value.

Topic 7.8Exponential growth and decay

  1. Connect an exponential model with its rate lawRecognize that the derivative is proportional to the amount or relevant difference.
  2. Determine the model constantsUse initial and later observations to find the starting amount and growth or decay constant.
  3. Predict an amount or a timeSolve the exponential model for the requested unknown with units.
  4. Use doubling time or half-lifeTranslate the repeated factor into the continuous rate constant and vice versa.

Topic 7.9Logistic growth

BC only
  1. Identify carrying capacity and equilibriaRead the logistic equation and find its constant solutions.
  2. Determine growth or declineUse the population relative to the equilibria to interpret the rate sign.
  3. Find the fastest growthLocate half the carrying capacity and calculate the maximum rate when requested.
  4. Describe concavity and long-term behaviorDistinguish increasing population from increasing growth rate, including starts above capacity.
  5. Use a logistic solution or observationsDetermine a population, parameter, or time from the supplied model and data.