Rates of change, the definition of the derivative and differentiability

Learning objectives

Calculate the average rate of change of a function over an interval as a difference quotient and interpret it as the slope of a secant line, then define the instantaneous rate of change at a point as the limit of the difference quotient and interpret it as the slope of the tangent line

Define the derivative as a function using the limit definition in either form, lim (f(x + h) − f(x))/h as h → 0 or lim (f(x) − f(a))/(x − a) as x → a, and use the notations f′(x), y′, dy/dx and d/dx[f(x)] interchangeably

Recognize a given limit expression as the derivative of a specific function at a specific point and evaluate it using derivative rules

Estimate the derivative at a point from a table of values using a difference quotient over the smallest available interval, and from a graph by sketching a tangent line, and sketch the graph of f′ from the graph of f

Explain why differentiability at a point implies continuity there but not the converse, and identify from a graph or formula where a derivative fails to exist: at a discontinuity, a corner, a cusp or a vertical tangent

Decide whether a piecewise-defined function is differentiable at the join by checking continuity and comparing the one-sided derivatives, including finding parameters that make it differentiable