Transcript 2.1 The Derivative Objective: Find the slope of the

Miss Battaglia
BC Calculus
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
∆𝑦
𝑓 𝑐 + ∆𝑥 − 𝑓(𝑐)
lim
= lim
=𝑚
∆𝑥→0 ∆𝑥
∆𝑥→0
∆𝑥
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)).
Find the slope of the graph of f(x)=2x-3 at
the point (2,1)
𝑓 𝑐 + ∆𝑥 − 𝑓(𝑐)
lim
∆𝑥→0
∆𝑥

Find the slope of the graph of f(x)=3-5x at
the point (-1,8)
𝑓 𝑐 + ∆𝑥 − 𝑓(𝑐)
lim
∆𝑥→0
∆𝑥


Find the slope of the graph of 𝑓 𝑥 = 𝑥 2 + 1 at the point
(0,1) and (-1,2)
𝑓 𝑐 + ∆𝑥 − 𝑓(𝑐)
lim
∆𝑥→0
∆𝑥

Find the slope of the graph of 𝑓 𝑥 = 𝑥 2 − 9 at the point
(2,-5)

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 f at x is given by
𝑓 𝑥 + ∆𝑥 − 𝑓(𝑥)
𝑥 = lim
∆𝑥→0
∆𝑥
provided the limit exists. For all x for which this limit
exists, f’ is a function of x.
𝑓′
Notations: f’(x),
𝑑𝑦
,
𝑑𝑥
y’,
𝑑
[𝑓
𝑑𝑥
𝑥 ], 𝐷𝑥 [𝑦]
(they all mean the same thing!)

𝑓′
Find the derivative of 𝑓 𝑥 = 𝑥 3 + 2𝑥
𝑓 𝑥 + ∆𝑥 − 𝑓(𝑥)
𝑥 = lim
∆𝑥→0
∆𝑥

𝑓′
Find the derivative of 𝑓 𝑥 = 8
𝑓 𝑥 + ∆𝑥 − 𝑓(𝑥)
𝑥 = lim
∆𝑥→0
∆𝑥
1
− 𝑥
5

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).

𝑓
(a) Find an equation of the tangent line to the graph of the equation at a
given point. (b) Use a graphing utility to graph the function and its tangent
line at the point and (c) Use the derivative feature of a graphing utility to
confirm your results
𝑥 = 𝑥 2 + 3𝑥 + 4 (-2,2)

Find the derivative with respect to t for the function 𝑦 =
2
𝑡

The limit represents f’(c) for a function f and a number c.
Find f and c.
5 − 3 1 + ∆𝑥
lim
∆𝑥→0
∆𝑥
−2
Read 2.1
 Page 103 #17-31 odd, 37, 43, 45,
53-58
