This is where the derivative gets put to work. Aim it at any function and it reveals the whole story — where the curve climbs and falls, its highest and lowest points, which way it bends and where it changes its mind — then pins down the one input that makes a quantity as large, or as small, as it can possibly be. Five tools that turn a bare formula into its full behavior, and behavior into the best answer a problem has.
Before you can find a maximum or trust a tangent, two theorems promise they’re there. The Extreme Value Theorem guarantees a continuous function on a closed interval actually reaches a highest and lowest point; the Mean Value Theorem guarantees a tangent somewhere runs parallel to the secant — your instantaneous rate equalling your average. These are the quiet foundation the whole unit stands on.
The sign of f′ tells the whole story of direction: positive and the function climbs, negative and it falls, and where it flips you’ve found a peak or a valley. Build the sign chart, run the first-derivative test, and read the rises and falls straight off the slope.
f″ is the shape beneath the slope. Positive is concave up (it holds water), negative is concave down, and where it switches you get an inflection point. The second-derivative test even settles whether a critical point is a max or a min by which way the curve bends there.
Three graphs, one function — stacked and moving together. A peak on f is a zero on f′; an inflection on f is a zero on f″. Learn to read all three at once, then do the hardest thing in the unit in reverse: hand yourself only the derivative and sketch the original curve.
The grand payoff: find the best possible answer. Set up the quantity, use the derivative to locate where it stops climbing, and confirm it’s truly the maximum (or minimum). Drag the variable and watch the value trace a curve whose peak glows exactly where f′ = 0.
Fresh batches of multiple choice across all five lessons, each with a worked explanation that names every distractor’s mistake, plus a weak-spot tracker. Then its own twist — Curve Detective, a graded puzzle that hands you only the clues in f′ and f″ and asks you to rebuild the mystery function — and full free-response in the classic AP mold: the optimization problem and the graph-analysis problem the exam always brings.
Start practicing →Go in order. The existence theorems promise the extrema and tangents are there; the first-derivative test reads a function’s rises and falls off the sign of f′; concavity adds the bend from f″; connecting f, f′ & f″ fuses all three into one moving picture and teaches you to sketch in reverse; and optimization puts it all to work finding the best answer. Each tool animates the thing the textbook only describes.
Drill the two skills that lose points: reading a graph of f′ to describe f — where it climbs, peaks, and bends — and the optimization setup (name the quantity, set f′ = 0, then justify it’s the max). The cheat sheet has every test and recipe, and the exam set mixes them the way the test will.