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# Derivatives and slope 2.1 update day1

## by Lorie Blickhan, Mathematics Teacher at Zion Benton Township High School on Dec 06, 2010

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## Derivatives and slope 2.1 update day1Presentation Transcript

• Derivatives and Slopes:Goal: identify tangent lines to a graphUse limits to find slope Use limits to find derivatives
Tangent Line to a Graph:
The slope of the line tangent to a curve at a given point determines the rate at which the graph rises or falls at that given point.
http://www.ies.co.jp/math/java/calc/index.html
• m=6
• Slope and limit:Just looking at the graph can be difficult and inaccurate.
Secant line
recall from geometry a secant is a line that passes through two points on a circle.
• Is a secant line of circle C
• Δy
Δx
Slope m= Δy/Δx, we can use this fact and the idea that f(x) = y
• This point has the
Coordinates (x + Δx, f(x + Δx))
This point has the coordinates
(x, f(x))
• (x + Δx, f(x+Δx))
(x, f(x))
As we bring this difference in closer making Δx approach 0, the secant line approaches the tangent line
• 1
• 2
• 3
• Going back to the original coordinates for the secant line:
(x + Δx, f(x + Δx)
f(x + Δx) – f(x)
(x, f(x)) Δx
Slope = m = f(x + Δx) – f(x) Difference Quotient
Δx
• Definition of the Slope of a Graph:
The slope m of the graph of ƒ at the point
(x, ƒ(x)) is equal to the slope of its tangent line at (x, ƒ(x)), and is given by
msec =
m =
Provided this limit exists.
• Use the limit process to find slope.
m =
• Ex. ƒ(x)= x2 at the point (2, 4) find the slope of the graph at this point.
lim ƒ(2 + Δx) – ƒ(2)
Δx->0 Δx
lim (2 + Δx)2 – (2)2
Δx->0 Δx
lim 4 + 4 Δx + Δx2 -4
Δx->0 Δx
• lim 4 Δx + Δx2
Δx->0 Δx
lim Δx(4 + Δx)
Δx->0 Δx
lim (4 + Δx) = 4
Δx->0
• Find the slope of the graph y = 2x + 5 using the limit process