A reliable method
For a power other than −1, raise the exponent and divide by the new exponent. For numerical sums, multiply each height by its interval width.
Worked foundation example
An antiderivative of 2x^3 is kx^4+C. Find k. Differentiate kx^4 to obtain (4)kx^3. Match its coefficient to 2. Thus k=1/2.
The common trap
A trapezoidal sum averages adjacent heights; a midpoint sum samples inside each interval. Do not mix their nodes.
Connect the representations
An integral with a negative integrand contributes negative net accumulation. Geometric area needs a separate nonnegative treatment.
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 an approximation with an exact integral
For f(x)=x² on [0,2], a two-panel midpoint sum is 1·f(1/2)+1·f(3/2)=5/2. The exact integral is 8/3. A trapezoid sum using endpoints 0,1,2 is 3. The methods use different information and need not agree. On this convex curve, midpoint rectangles underestimate while trapezoids overestimate; the sign follows from the graph shape, not from an arbitrary rule about all functions.
Transfer and explain
Transfer: compute a trapezoidal estimate from rates r(0)=2, r(1)=5, r(2)=4. The result is (2+5)/2+(5+4)/2=8. Explain why adding the three rates omits trapezoid endpoint weights and interval widths. Average each adjacent pair of rates and multiply by that interval’s width.
An antiderivative of 2x^3 is kx^4+C. Find k.
- Differentiate kx^4 to obtain (4)kx^3.
- Match its coefficient to 2.
- Thus k=1/2.
Compare the other answer choices
Differentiate instead of antidifferentiate.
Multiply by the new exponent instead of dividing.
Divide by the old exponent.
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