Short on time? This page is the recap. For explanations, graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: g(x) = ∫ (a to x) f(t) dt, with f continuous.
Recap
- The graph you are given is g′. Heights of f are slopes of g. Areas under f are changes in g. Slopes of f are the concavity of g.
- g(a) = 0. Every other value of g is a running total of signed area from a. Area to the left of a counts with the opposite sign.
- Behaviour comes from signs; values come from areas. You rarely need a formula for g.
- The reasoning is the same whether f is a graph, a table, a formula or a verbal rate. With a table, use the Intermediate Value Theorem to locate zeros and a Riemann or trapezoidal sum to estimate values of g.
Key relationships
| Feature of f | Feature of g |
|---|---|
| f > 0 | g increasing |
| f < 0 | g decreasing |
| f changes + to − at c | relative maximum of g at c |
| f changes − to + at c | relative minimum of g at c |
| f = 0 but no sign change | no extremum of g |
| f increasing | g concave up |
| f decreasing | g concave down |
| f has a relative max or min (including at a corner) | point of inflection of g |
| ∫ (p to q) f(t) dt | g(q) − g(p) |
Absolute extrema of g on [p, q]: list the endpoints and the zeros of f with a sign change, find g at each with areas, and compare.
Assumptions behind the method
- f is continuous, so g′ = f holds everywhere on the interval.
- A table alone shows signs only at the listed points. You need extra facts (for example, “f is increasing”) to rule out extra zeros.
Mistakes to avoid
- “f is largest, so g is largest.” f largest means g rises fastest.
- Calling every zero of f an extremum. Check for a sign change.
- Using f(c) as g(c). One is a slope, the other an area.
- Forgetting endpoints in an absolute extremum question.
- Losing the sign of areas to the left of a.
- Giving reasons about g itself (“g is positive”) instead of about g′ = f.
Quick self-check
- f changes from negative to positive at t = 4. What does g have at x = 4? (A relative minimum, since g′ = f changes from − to +.)
- f has a relative maximum at t = 2. What does g have at x = 2? (A point of inflection: g″ = f′ changes from + to −.)
- g(x) = ∫ (2 to x) f(t) dt and ∫ (0 to 2) f(t) dt = 5. Find g(0). (−5)
Next: practice questions.