AP Calculus: Unit 2 Concepts

The slope of this line is given by an equation in the form of a difference quotient:

We can also calculate the slope of a secant line to a function at a value a by using this equation and replacing 𝑥 with 𝑎+ℎ, where ℎ is a value close to 0. We can then calculate the slope of the line through the points (𝑎,𝑓(𝑎)) and (𝑎+ℎ,𝑓(𝑎+ℎ)). In this case, we find the secant line has a slope given by the following difference quotient with increment ℎ:

DEFINITION

Let 𝑓 be a function defined on an interval containing 𝑎. If 𝑥≠𝑎 is on the interval, then

is a difference quotient. Also, if ℎ ≠ 0 is chosen so that 𝑎+ℎ is in the interval, then

is a difference quotient with increment ℎ.

Defining the Derivative

Let 𝑓(𝑥) be a function defined in an open interval containing 𝑎. The derivative of the function 𝑓(𝑥) at 𝑎, denoted by 𝑓′(𝑎), is defined by

provided this limit exists. Alternatively, we may also define the derivative of 𝑓(𝑥) at 𝑎 as

Video Introduction

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Want to see these numbers in action? This tool from Wolfram uses a “snowball” to show the rate of change for different functions.

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