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The Slope of a Curve or How Secant Lines become Tangent Lines

# The Slope of a Curve or How Secant Lines become Tangent Lines

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Derivatives

The Slope of a CurveorHow Secant Lines become Tangent Lines1

What is the slope of the line through these two points?(0, 0)(2, 4)This is called a secant line through the curve y = x2 because it passes through two points on the curve.

Finding the slope of a line is easyWhat about finding the slope of a curve?2

What is the slope of the line through this point?(1, 1)This is called a tangent line through the curve y = x2 at the point (1, 1).

Since the slope of a curve like this one is always changing, we can only talk about slope in terms of specific points or intervals on the curve3

(0, 0)(1, 1)Lets start by drawing a secant line through the point (1, 1) and some other point close to it.

This is clearly not the slope at (1, 1)So now what do we do?Since the slope of a curve like this one is always changing, we can only talk about slope in terms of specific points or intervals on the curveTry a point even closer4

(1, 1)Lets start by drawing a secant line through the point (1, 1) and some other point close to it.

This is a lot closer to the slope at (1, 1)How much closer can we get?Since the slope of a curve like this one is always changing, we can only talk about slope in terms of specific points or intervals on the curveIf we use limits, we can get as close to the point as we want.

5

What is the slope of the line at (1, 1)?(1, 1)And the first thing we do when taking a limit is

Since the slope of a curve like this one is always changing, we can only talk about slope in terms of specific points or intervals on the curve

Plug it inNow what?6

What is the slope of the line through this point?(1, 1)So the slope of the tangent line through the curve y = x2 at the point (1, 1) is 2Since the slope of a curve like this one is always changing, we can only talk about slope in terms of specific points or intervals on the curve

7

f(x)x

But we need to be able to write and read this formula in text book terms (which means replacing y with f (x))8For all of the enrichment provided by problem-based learning, some students still find comfort in being able to wrap it all up in one or two formulas.

f(x)x

9For all of the enrichment provided by problem-based learning, some students still find comfort in being able to wrap it all up in one or two formulas.

is called the derivative of f at aWe write:

The derivative of f with respect to x is There are two formulas for the derivative of

10There are many ways to write the derivative of

The Definition of the Derivative:The Derivative at a Point:

y2y1x2x1x2x1(Also called the Numerical Derivative in your text)11

f prime xorthe derivative of f with respect to x

y prime

or12

Heres how to find the general derivative of a polynomial function.

13

Heres how to find the general derivative of a polynomial function.

So in this case the derivative of f (x) isAnd what this means isThe slope of y = x2 3 atx = 1 is

x = 2 is

x = 2 is

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