Wemazi

AP Calculus · Unit 5

Analyzing Functions

Extrema, concavity and optimization.

Topic 5.1Use the mean value theorem

  1. Check when the theorem appliesVerify continuity on the closed interval and differentiability throughout its interior.
  2. Find every matching pointCalculate the average rate, solve the derivative equation, and keep every solution inside the interval.
  3. Use a graph or table to identify a guaranteeCombine the supplied data with explicitly stated conditions; do not infer continuity from a few readings.
  4. Apply the theorem to a real situationMatch an instantaneous rate with an average rate and report the time or input with units.
  5. Distinguish existence from uniquenessSelect what the theorem guarantees. Recognize when you need more information to claim exactly one point.

Topic 5.2Extrema, critical points and guarantees

  1. Find critical numbers in the domainInclude zeros and undefined derivatives only where the function itself is defined.
  2. Distinguish local and absolute extremaIdentify whether a value is best nearby or over the whole specified domain.
  3. Check the Extreme Value Theorem conditionsVerify continuity on a closed bounded interval without requiring differentiability.
  4. Recognize when a maximum or minimum does not existDistinguish an approached bound from an actual minimum or maximum.
  5. Do not classify from a zero derivative aloneRecognize stationary points and corners that may or may not be extrema.

Topic 5.3Increasing and decreasing intervals

  1. Find where the derivative's sign can changeLocate derivative zeros, undefined derivatives, and breaks in the function's domain.
  2. Determine derivative signs on intervalsUse a formula, sign chart, or derivative graph correctly.
  3. State increasing and decreasing intervalsTranslate the derivative's sign into correctly notated intervals.
  4. Handle zeros without a sign changeRecognize that a flat point does not necessarily reverse the function's direction.

Topic 5.4Classify turning points with signs

  1. Identify candidates for local extremaFind relevant critical points and respect domain restrictions.
  2. Test the derivative on both sidesDetermine whether the sign changes from positive to negative, negative to positive, or neither.
  3. Classify a candidate using the sign changeIdentify a local maximum, local minimum, or neither.
  4. Report the requested location or pointDistinguish the input coordinate from the function value and full point.

Topic 5.5Find absolute extrema

  1. Find all candidates on the intervalInclude endpoints and every interior critical point where the function is defined.
  2. Compare function valuesEvaluate the original function at all candidates, including nondifferentiable points.
  3. Report absolute extrema in fullIf the question asks, give both the extreme value and the input where it occurs.
  4. Justify that the search is completeUse the interval conditions and candidate method or an equivalent supported argument.
  5. Recognize an unattained extremumHandle open endpoints or missing points without inventing a minimum or maximum.

Topic 5.6Concavity and inflection

  1. Determine concavity on intervalsUse the sign of the second derivative or whether the first derivative is increasing or decreasing.
  2. Find candidate inflection locationsInclude zeros and undefined second derivatives while checking the function's domain and continuity.
  3. Verify a change in concavityRequire an actual change instead of treating a zero second derivative as sufficient.
  4. Report an inflection pointIf the question asks for coordinates, find the function value as well as the input.
  5. Handle an undefined second derivativeRecognize a valid inflection point when concavity changes even though the second derivative does not exist.

Topic 5.7Use the second derivative test

  1. Check that the point is stationaryVerify that the first derivative is zero before applying the second derivative test.
  2. Evaluate the second derivativeCalculate its value at the candidate point.
  3. Classify a positive or negative resultIdentify a local minimum from a positive result and a local maximum from a negative result.
  4. Handle an inconclusive resultWhen the test gives zero or is unavailable, use first-derivative signs or nearby values to determine actual behavior.

Topic 5.8Sketch a function and its derivative

  1. Read derivative values from a function graphTranslate tangent slopes into derivative values and identify corners where the derivative is absent.
  2. Read function direction from a derivative graphUse derivative heights to find increasing and decreasing intervals and turning points.
  3. Read concavity from a derivative graphUse whether the derivative graph rises or falls, separately from whether it is above the axis.
  4. Match a function with its derivativeSelect compatible graphs or graph features from the supplied alternatives.
  5. Build a function from graph conditionsEnter a formula or point satisfying supplied slope, vertex, and height constraints in an explicitly bounded construction task.

Topic 5.9Connect f, f′ and f″

  1. Identify which function a graph representsDistinguish the roles of f, its first derivative, and its second derivative.
  2. Connect signs with direction and concavityInfer increase or decrease from the first derivative and bending from the second.
  3. Connect zeros with candidate featuresDistinguish function zeros, stationary points, and possible inflection points.
  4. Choose conclusions supported by the available informationRecognize when derivative data does not determine the function's absolute height.
  5. Combine several pieces of derivative evidenceSelect a consistent description or reconstruct a supported formula from the given conditions.

Topic 5.10Build an optimization model

  1. Identify the quantity to optimizeDistinguish the objective, such as cost or area, from the constraint.
  2. Write the constraintTranslate the geometric or contextual restrictions into an equation.
  3. Build an objective in one variableSubstitute the constraint to remove extra variables.
  4. State the feasible domainInclude physical restrictions and distinguish allowed boundary cases from impossible ones.

Topic 5.11Solve and justify an optimum

  1. Build the objective and its domainWrite a correct one-variable model for the situation.
  2. Find feasible candidatesSolve the derivative condition and include relevant endpoints or boundaries.
  3. Justify the best candidateCompare values or use a valid global sign or monotonicity argument.
  4. Answer the original questionReport the requested dimensions, input, and optimal value with units.

Topic 5.12Behavior of implicit curves

  1. Find an implicit first derivativeDifferentiate the relation and solve for the slope.
  2. Find an implicit second derivativeDifferentiate the first derivative while still treating y as a function of x.
  3. Determine concavity at a pointEvaluate the second derivative at a valid point and interpret its sign.
  4. Identify horizontal or vertical tangentsApply the appropriate conditions and check where the derivative formula is inconclusive.