Conditions before computation
In a scaled difference quotient, replace the actual input change by a new variable and adjust the denominator. For tabular estimates, divide by the full distance between the selected samples and label the result as an estimate. For a joining point, verify continuity before comparing derivative limits. When differentiating an implicit function, work on a branch where the dependent variable is differentiable; for the upper circle branch, y is positive away from its endpoints.
Worked example with a method check
f(x)=x² for x≤3, and f(x)=m(x−3)+9 for x>3. Find m so f is differentiable at 3. Both branches approach 9, so continuity already holds for every m. The left derivative is 2(3)=6; the right derivative is m. Matching one-sided derivatives gives m=6.
Compare two methods
Compare [f(a+3h)−f(a)]/h with [f(a+3h)−f(a)]/(3h). Their limits are 3f′(a) and f′(a), respectively. The same numerator does not make the denominators interchangeable. Likewise, differentiating yy′ gives (y′)²+yy″, not just yy″: the first factor also varies with x.
Transfer: solve before reading
Suppose f′(2)=−5. Evaluate lim h→0 [f(2−2h)−f(2)]/h. Then let x²+y²=25 at (3,4) on the upper branch: explain why y″ is negative without appealing to a memorized circle formula.
Transfer reasoning and answer
The scaled quotient is (−2)(−5)=10. On the circle, y′=−x/y=−3/4 and 1+(y′)²+yy″=0. Since y>0, y″=−[1+(y′)²]/y=−25/64<0. The sign comes from the implicit equation and the branch condition.
Practice the next representation
Work all four task families linked to this lesson. For the illustrated items, state which numerical labels and relationships you used. For a written extension, compare each line of your solution with its original self-assessment criteria; these criteria are practice feedback, not an official AP score.
f(x)=x² for x≤3, and f(x)=m(x−3)+9 for x>3. Find m so f is differentiable at 3.
- Both branches approach 9, so continuity already holds for every m.
- The left derivative is 2(3)=6; the right derivative is m.
- Matching one-sided derivatives gives m=6.
Compare the other answer choices
Use the input as the derivative of x².
Match the slope to the common function value.
Negate the right derivative just because it is a one-sided derivative.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
(a) State the relationship or theorem that makes this calculation valid, including its relevant conditions. (b) Show the intermediate computation and give the requested result. (c) Explain why one plausible competing method would fail for this problem.
This draft stays on this page only. Use paper if you want to keep your working.
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 and BC Course and Exam DescriptionCalculus Volume 1: Fundamental Theorem of Calculus