A reliable method
Use y=f(a)+f′(a)(x−a) near a. For extrema, inspect derivative signs or compare critical points and endpoints on the requested domain.
Worked foundation example
Use the tangent line to f(x)=x² at x=2 to estimate f(3). Give the linear approximation, not the exact value. f(2)=4 and f′(2)=4. The input change is 1, so the linear change is 4. The approximation is 8; it need not equal (3)².
The common trap
Setting the derivative equal to zero produces candidates, not automatically maxima. A normal line uses the negative reciprocal slope only when defined.
Connect the representations
The mean value theorem links an interval-average rate to an instantaneous rate under continuity and differentiability hypotheses.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare a stationary point with an extremum
Both x² and x³ have derivative zero at zero. For x² the derivative changes from negative to positive and there is a minimum. For x³ the derivative 3x² is nonnegative on both sides and there is no local extremum. Solving f′=0 gives a candidate; a sign change or another valid argument establishes its behavior. Endpoints also matter for an absolute extremum on a closed interval.
Transfer and explain
Transfer: f(x)=(x−2)² on [3,5] has its vertex outside the domain. The absolute minimum occurs at x=3, not x=2. Explain why a domain check changes the answer.
Use the tangent line to f(x)=x² at x=2 to estimate f(3). Give the linear approximation, not the exact value.
- f(2)=4 and f′(2)=4.
- The input change is 1, so the linear change is 4.
- The approximation is 8; it need not equal (3)².
Compare the other answer choices
Ignore the input change.
Use a instead of 2a as the slope.
Give the exact function value rather than the requested tangent estimate.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus