Unit 3 · 4 lessons
Curve Sketching
Use the first and second derivatives to find exactly where a graph rises, falls, turns and bends, then put it all together to sketch polynomial and simple rational functions, and to read a function's story from the graph of its derivative.
- 40 practice questions
- 31 flashcards
- 3 interactive lessons
Lessons
- 1Increasing and Decreasing Functions and the First Derivative TestThe sign of the derivative tells you whether a graph is rising or falling. Where the derivative is zero or undefined you find the critical numbers, and the first derivative test tells you which of them are local maxima, local minima or neither.Interactive10 questions
- 2Concavity, Points of Inflection and the Second Derivative TestThe second derivative is the rate of change of the slope. Its sign tells you which way a graph bends (concave up or down), a sign change marks a point of inflection, and at a critical number it can classify a maximum or minimum in one substitution.Interactive10 questions
- 3Sketching Polynomial and Simple Rational FunctionsPut everything together: domain, intercepts, symmetry, the first derivative for direction and turning points, the second derivative for concavity and inflection points, and asymptotes for rational functions. The result is a sketch you can trust, checked with graphing technology.10 questions
- 4Graphs of f, f′ and f″Read the graphs of the first and second derivatives straight off the graph of a function, and work backward: from information about f′ or f″, describe and sketch possible graphs of f, and explain why there are infinitely many of them.Interactive10 questions