Definition of the Derivative Lesson 3.4 Tangent Line Recall from geometry Tangent is a line that touches the circle at only one point Let us generalize the concept to functions A tangent will just "touch" the line but not pass through it Which of the above lines are tangent? 2
A Secant Line Crosses the curve twice x=a h The slope of the secant will be y f (a h) f (a ) f (a h) f (a ) x ah a h
3 Tangent Line Now let h get smaller and smaller x=a h Try Trythis thiswith with Geogebra Geogebra
f ( a h) f ( a ) lim The slope of the tangent line h 0 h 4 Tangent Line Try it out given f ( x ) 6 x 2 4 x Determine the slope of the tangent line at x = 0 Evaluate
f (0 h) f (0) lim h 0 h 5 Tangent Line Use the limit command on your calculator 2 to determine the slope of f ( x) 6 x 4 x for x = 1 Once you have the slope and the point You can determine the equation of the line y y1 m ( x x1 )
6 The Derivative We will define the derivative of f(x) as f ( a h) f ( a ) f '( x) lim h 0 h Note The derivative is the rate of change function for f(x) The derivative is also a function of x The limit must exist
7 Comparison Difference Quotient f (a ) f (b) a b Derivative lim h 0 f ( a h) f (a ) h Slope of secant
Slope of tangent Average rate of change Average velocity Instantaneous rate of change Instantaneous velocity8 Finding f'(x) from Definition 1. 2. Strategy
Find f(x + h) Find and simplify f(x + h) f(x) 3. Divide by h to get 4. Let h 0 f ( x h) f ( x ) h f ( x h) f ( x ) f '( x) lim
h 0 h 9 Try It Out Use the strategy to find the derivatives of these functions 5 f ( x) x 3 f ( x) 4 x 2 x
w( x) x 12 Hint: Hint:rationalize rationalizethe the numerator numerator 10 Equation of the Tangent Line We stated previously that once we determine the slope of the tangent y y1 m ( x x1 ) We can "cut to the chase" and state it as
y y1 f '( x) ( x x1 ) 11 Warning Our definition of derivative included the phrase "if the limit exists" Derivatives do not exist at "corners" or "sharp points" on the graph The slope is different on each side of the point f(x) = | x 3 | The limit does not exist 12 Assignment Lesson 3.4
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