Unit 1 · 4 lessons
Rates of Change & Limits
How fast is something changing right now? Secants give average rates over an interval, limits shrink that interval to a single point, and the result is the derivative, the idea the rest of calculus is built on.
- 38 practice questions
- 30 flashcards
- 2 interactive lessons
Lessons
- 1Average and Instantaneous Rates of ChangeAn average rate of change compares two moments and is the slope of a secant line. An instantaneous rate of change describes a single moment and is the slope of the tangent line, which you can approximate with secants through the point that get shorter and shorter.Interactive9 questions
- 2Limits: Graphs, Tables, Sequences and ContinuityA limit is the value a function or sequence approaches, whether or not it ever gets there. Limits can be read from graphs and tables, can differ from the left and the right, describe behaviour near asymptotes and far out along a sequence, and define what it means for a function to be continuous.10 questions
- 3The Derivative from First PrinciplesThe difference quotient (f(a + h) − f(a))/h is the slope of a secant, the average rate of change from a to a + h. Its limit as h → 0 is the slope of the tangent, the instantaneous rate of change f′(a). Simplify the quotient algebraically first, and the limit becomes exact.9 questions
- 4The Derivative FunctionFind the tangent slope at every x at once and you get a new function, the derivative f′(x). Its graph can be built from the slopes of f, its equation comes from first principles with x left as a variable, its degree is one less than f's for a non-constant polynomial, and it fails to exist at corners, cusps, breaks and vertical tangents.Interactive10 questions