Page 23 - 2024-calc4e-SE proofs-4e.indd

Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.1 • Rates of Change and the Derivative 173
2
(c) f (t) − f (t 0 ) 16t − 16t 0 2 16 (t − t 0 ) (t + t 0 )
v = lim = lim = lim
t→t 0 t − t 0 t→t 0 t − t 0 t→t 0 t − t 0
= 16 lim (t + t 0 ) = 32t 0
t→t 0
At t 0 seconds, the velocity of the rock is 32t 0 ft/s.

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NOW WORK Problem 33.

4 Find the Derivative of a Function at a Number

Slope of a tangent line, rate of change of a function, and velocity are all found using the
same limit,

f (x) − f (c)
f (c) = lim
′
x→c x − c
The common underlying idea is the mathematical concept of derivative.

DEFINITION Derivative of a Function at a Number
If y = f (x) is a function defined on an open interval (a, b), and c is in the interval
(a, b), then the derivative of f at c, denoted by f (c), is the number
′
f (x) − f (c)
f (c) = lim
′
x→c x − c
provided this limit exists.

EXAMPLE 6 Finding the Derivative of a Function at a Number

2
Find the derivative of f (x) = 2x − 3x − 2 at x = 2. That is, find f (2).
′
Solution
Using the definition of the derivative, we have
2
f (x) − f (2) (2x − 3x − 2) − 0
′
f (2) = lim = lim f (2) = 2 · 4 − 3 · 2 − 2 = 0
x→2 x − 2 x→2 x − 2
(x − 2)(2x + 1)
= lim
x→2 x − 2
= lim (2x + 1) = 5
x→2
R
NOW WORK Problem 23 and AP Practice Problems 2 and 6.
So far we have given three interpretations of the derivative:

• Geometric interpretation: If y = f (x), the derivative f (c) is the slope of the
′
tangent line to the graph of f at the point (c, f (c)).
• Rate of change of a function interpretation: If y = f (x), the derivative f (c)
′
is the rate of change of f at c.
• Physical interpretation: If the signed distance s from the origin at time t of
an object moving along a line is given by the position function s = f (t), the
derivative f (t 0 ) is the velocity of the object at time t 0 .
′

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