Miss Battaglia AB Calculus 2.1 The Derivative Objective: Find the slope of the tangent line to a curve at a point; Use the limit definition to find the derivative of a function. Miss Battaglia AB Calculus

The Tangent Line Problem Given a point, P, we want to define and calculate the slope of the line tangent to the graph at P. Definition of Tangent Line with Slope m If f is defined on an open interval containing c, and if the limit lim ∆𝑥→0 ∆𝑦 ∆𝑥 = lim ∆𝑥→0 𝑓 𝑐+∆𝑥 −𝑓(𝑐) ∆𝑥 =𝑚 exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c,f(c)).

The Slope of the Graph of a Linear Function Find the slope of the graph of f(x)=2x-3 at the point (2,1) lim ∆𝑥→0 𝑓 𝑐+∆𝑥 −𝑓(𝑐) ∆𝑥

Tangent Lines to the Graph of a Nonlinear Function Find the slope of the graph of 𝑓 𝑥 = 𝑥 2 +1 at the point (0,1) and (-1,2) lim ∆𝑥→0 𝑓 𝑐+∆𝑥 −𝑓(𝑐) ∆𝑥

What is the derivative? The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope.

The Derivative of a Function The derivative of f at x is given by 𝑓 ′ 𝑥 = lim ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥 provided the limit exists. For all x for which this limit exists, f’ is a function of x. Notations: f’(x), 𝑑𝑦 𝑑𝑥 , y’, 𝑑 𝑑𝑥 [𝑓 𝑥 ], 𝐷 𝑥 [𝑦] (they all mean the same thing!)

Finding the Derivative by the Limit Process Find the derivative of 𝑓 𝑥 = 𝑥 3 +2𝑥 𝑓 ′ 𝑥 = lim ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥

Using the Derivative to Find the Slope at a Point Find f’(x) for 𝑓 𝑥 = 𝑥 . Then find the slopes of the graph of f at the points (1,1) and (4,2). Discuss the behavior of f at (0,0).

Finding the Derivative of a Function Find the derivative with respect to t for the function 𝑦= 2 𝑡

Classwork/Homework Read 2.1 Page 104 #15-29 odd, 43, 44, 57, 58