Miss Battaglia AB Calculus. Given a point, P, we want to define and calculate the slope of the line tangent to the graph at P. Definition of Tangent Line.

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Miss Battaglia AB Calculus

Presentation transcript:

Miss Battaglia AB Calculus

Given a point, P, we want to define and calculate the slope of the line tangent to the graph at P. Definition of Tangent Line with Slope m If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c,f(c)) with slope m is the tangent line to the graph of f at the point (c,f(c)).

 Find the slope of the graph of f(x) = 2x – 3 at the point (2,1)

 Find the slope of the graph of f(x) = 3 – 5x at the point (-1,8)

 Find the slopes of the tangent lines to the graph of f(x) = x at the points (0,1) and (-1,2).

 Find the slope of the graph of f(x) = x 2 – 9 at the point (2,-5)

 The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope.

 The derivative of f at x is given by provided the limit exists. For all x for which this limit exists, f’ is a function of x. Notations: (they all mean the same thing!)

 Find the derivative of f(x) = x 3 + 2x

 Find the derivative of f(x) = 8 – (1/5)x

 Find f’(x) for. then find the slopes of the graph of f at the points (1,1) and (4,2). Discuss the behavior of f at (0,0).

(a)Find an equation of the tangent line to the graph of the equation at a given point. (b)Use a graphing utility to graph the function and its tangent line at the point (c)Use the derivative feature of a graphing utility to confirm your results. f(x) = x 2 + 3x + 4(-2,2)

 Find the derivative with respect to t for the function y=2/t.

The limit represents f’(c) for a function f and a number c. Find f and c.

 Read 2.1  Page 103 #17-31 odd, 37, 43, 45, 53-58