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Sec 11.3 The Tangent Line Problem Definition of the slope of a graph The slope, m of the graph of f athe the point (x, f(x)) is equal to the slope of its tangent line at, (x, f(x)), and is given by m = lim msec h →0 = lim h →0 f ( x + h) − f ( x ) provided the limit exists. h Ex: Find the slope of the graph of f ( x) = x 3 at (2, 8) this question is asking us to find: lim h →0 f (2 + h) − f (2) since we know a specific value of x. h EX: Find the slope of f(x) = -3x + 5 Ex: Find the formula for the slope of the graph of f ( x= ) x 2 − 2 . What are the slopes at (-3, 7) and (1, -1)? The Derivative of a function is a fancy word for the slope of the line tangent to a function. If a function is called f(x), then its derivative is called f’(x). f’(x) is pronounced “ f prime of x” Definition of Derivative: The derivative of f at x is given by: f '( x) = lim We have been finding derivatives all year. h →0 f ( x + h) − f ( x ) h Ex: Find the derivative of f (= x) 4 x 2 − 5 x Ex: Find f’(x) for f (= x) Ex: Find f’(x) if f ( x) = EX: Use the derivative of f ( x) = x 2 − 6 x + 4 to determine any points where the line tangent to f is horizontal. x + 1 . Then find the slopes of the graph of f at the points (4, 3) and (9, 4). 1 x−2 Hint: what do we know about horizontal lines? Page 808 #{33, 35, 41, 43, 45, 51, 53, 55, 57}