A reliable method
Check the left limit, right limit and function value separately. For a corner, compare one-sided derivatives after continuity is established.
Worked foundation example
f(x)=x²+3 for x≠2, and f(2)=k. Which value of k makes f continuous at 2? The nearby limit equals 2²+3. Continuity requires the assigned value k to equal that limit. Set k=7.
The common trap
A continuous graph can have a corner. A matching endpoint value alone does not establish a derivative.
Connect the representations
The intermediate value theorem needs continuity on a closed interval. It guarantees a value is reached, not a unique location.
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 continuity with differentiability
The function |x| is continuous at zero because both sides approach zero and its value is zero. Its left slope is −1 and right slope is 1, so it is not differentiable there. The function x² is also continuous at zero, but its slopes approach the same value zero. A sharp corner is therefore a useful counterexample to the claim that every continuous function has a derivative.
Transfer and explain
Transfer: for f(x)=2|x−3|, state the value and both one-sided slopes at 3. The value is 0, left slope −2 and right slope 2. Matching function values does not make the slopes match.
f(x)=x²+3 for x≠2, and f(2)=k. Which value of k makes f continuous at 2?
- The nearby limit equals 2²+3.
- Continuity requires the assigned value k to equal that limit.
- Set k=7.
Compare the other answer choices
Ignore the squared input.
Ignore the constant in the nearby expression.
Treat x² as x.
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