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AP Calculus: Unit 2 Concepts

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

๐‘š= \frac{๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘Ž)}{๐‘ฅโˆ’๐‘Ž}

We can also calculate the slope of a secant line to a function at a valueย aย by using this equation and replacing ๐‘ฅ with ๐‘Ž+โ„Ž, where โ„Ž is a value close to 0. We can then calculate the slope of the line through the pointsย (๐‘Ž,๐‘“(๐‘Ž))ย andย (๐‘Ž+โ„Ž,๐‘“(๐‘Ž+โ„Ž)). In this case, we find the secant line has a slope given by the following difference quotient with incrementย โ„Ž:

๐‘š= \frac{๐‘“(a+h)โˆ’๐‘“(๐‘Ž)}{a+hโˆ’๐‘Ž} ๐‘š= \frac{๐‘“(a+h)โˆ’๐‘“(๐‘Ž)}{h}

DEFINITION

Letย ๐‘“ย be a function defined on an intervalย containingย ๐‘Ž.ย Ifย ๐‘ฅโ‰ ๐‘Žย is on the interval,ย then

๐‘„= \frac{๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘Ž)}{๐‘ฅโˆ’๐‘Ž}

is aย difference quotient. Also, ifย โ„Ž โ‰  0ย is chosen so thatย ๐‘Ž+โ„Žย is inย the interval,ย then

๐‘„= \frac{๐‘“(๐‘Ž+โ„Ž)โˆ’๐‘“(๐‘Ž)}{โ„Ž}

is a difference quotient with incrementย โ„Ž.

Defining the Derivative

Letย ๐‘“(๐‘ฅ)ย be a function defined in an open interval containingย ๐‘Ž.ย The derivative of the functionย ๐‘“(๐‘ฅ) atย ๐‘Ž, denoted byย ๐‘“โ€ฒ(๐‘Ž), is defined by

๐‘“โ€ฒ(๐‘Ž)= \lim\limits_{๐‘ฅโ†’๐‘Ž} \frac{๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘Ž)}{๐‘ฅโˆ’๐‘Ž}

provided this limit exists. Alternatively, we may also define the derivative ofย ๐‘“(๐‘ฅ)ย atย ๐‘Žย as

๐‘“โ€ฒ(๐‘Ž)= \lim\limits_{โ„Žโ†’0} \frac{๐‘“(๐‘Ž+โ„Ž)โˆ’๐‘“(๐‘Ž)}{โ„Ž}.

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.

Notes Check

Which definitions match these images?

This figure consists of two graphs labeled a and b. Figure a shows the Cartesian coordinate plane with 0, a, and x marked on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)) and (x, f(x)). There is also a straight line that crosses these two points (a, f(a)) and (x, f(x)). At the bottom of the graph, the equation msec = (f(x) - f(a))/(x - a) is given. Figure b shows a similar graph, but this time a + h is marked on the x-axis instead of x. Consequently, the curve labeled y = f(x) passes through (a, f(a)) and (a + h, f(a + h)) as does the straight line. At the bottom of the graph, the equation msec = (f(a + h) - f(a))/h is given.

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