Chapter 2(A) - Chapter 2 Differentiation Outline n...

Info iconThis preview shows pages 1–16. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 2 Differentiation
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Outline n Derivative q Definitions q Rules of Differentiation n Other Types of Differentiation q Parametric Differentiation q Implicit Differentiation q Higher Order Derivatives
Background image of page 2
n Maxima and Minima q Local and absolute extremes q Finding Extreme Values q Critical Points q Increasing and Decreasing Functions n Derivative Test q First Derivative Test for Local Extremes q Concavity and Points of Inflection q Second Derivative Test for Local Extremes n Optimization Problems q Absolute Extreme Values
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
n Indeterminate Form (Limits) q L’Hospital’s Rule q Other Indeterminate Forms
Background image of page 4
Derivative
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
n yx = 1 n dy nx dx - = () n f xx = 1 ' n f x nx - = The derivative of with respect to . The derivative of ( ) with respect to . f with respect to --- w.r.t Derivative
Background image of page 6
Some results Question: How to derive these results? Using limits
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
3. Derivative --- instantaneous rate of change of the function 2. Derivative --- (Geometrically) Gradient (Slope) of the tangent line 1. Derivative --- using the concept of limit ( ) () ' ( ) lim xa f x fa - = -
Background image of page 8
x y c y m xc =+ ------ gradient (slope) of the line m ------ intercept cy -
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
x y y m xc =+ 21 gradient (slope) yy m xx - == - 11 (,) Axy 22 Bxy Straight line --- find gradient 1 x 2 x
Background image of page 10
a x y () y fx = gradient at = gradient of tangent line at ( ) ' ( ) lim xa AA f x fa - == - ( , ( )) Aafa tangent line at A Curve --- to find gradient
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
a x y () y fx = ( , ( )) Aafa Question: Why we define ( ) ' ( ) lim xa f x fa - = - tangent line at A
Background image of page 12
a x y () y fx = ( , ( )) Aafa ( )) Bxfx x 21 gradient of line ( ) = yy AB xx f x fa xa - = - - - gradient at gradient of line A AB gradient at = gradient of tangent line at AA tangent line at A Why we define ( ) ' ( ) lim f x - = -
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
a x y () y fx = ( , ( )) Aafa ( )) Bxfx x 21 gradient of line ( ) = yy AB xx f x fa xa - = - - - If we choose B close to A, then gradient of line gradient at A BA ; Choosing closer and closer to is the same as letting approaches . Taking limit, we have, gradient at lim(gradient of ) ( ) lim A AB f x = - = - tangent line at A Why we define ( ) ' ( ) lim f x - = -
Background image of page 14
y x 4 34 yx =+ Pause and Think !!!
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 16
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 47

Chapter 2(A) - Chapter 2 Differentiation Outline n...

This preview shows document pages 1 - 16. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online