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Miss Battaglia AB Calculus

2.1 The Derivative Objective: Find the slope of the tangent line to a curve at a point; Use the limit definition to find the derivative of a function. Miss Battaglia AB Calculus. The Tangent Line Problem. Given a point, P, we want to define and calculate the slope of the line

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Miss Battaglia AB Calculus

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  1. 2.1 The DerivativeObjective: Find the slope of the tangent line to a curve at a point; Use the limit definition to find the derivative of a function. Miss Battaglia AB Calculus

  2. The Tangent Line Problem 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)).

  3. The Slope of the Graph of a Linear Function • Find the slope of the graph of f(x) = 2x - 3

  4. The Slope of the Graph of a Linear Function • Find the slope of the graph of f(x) = 3 – 5x at the point (-1,8)

  5. Tangent Lines to the Graph of a Nonlinear Function • Find the slopes of the tangent lines to the graph of f(x) = x2 + 1 at the points (0,1) and (-1,2).

  6. Tangent Lines to the Graph of a Nonlinear Function • Find the slope of the graph of f(x) = x2 – 9 at the point (2,-5)

  7. What is the derivative? • The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope.

  8. The Derivative of a Function • 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!)

  9. Finding the Derivative by the Limit Process • Find the derivative of f(x) = x3 + 2x

  10. Finding the Derivative by the Limit Process • Find the derivative of f(x) = 8 – (1/5)x

  11. Using the Derivative to Find the Slope at a Point • 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).

  12. Using the Derivative to Find the Slope at a Point Find an equation of the tangent line to the graph of the equation at a given point. Use a graphing utility to graph the function and its tangent line at the point Use the derivative feature of a graphing utility to confirm your results. f(x) = x2 + 3x + 4 (-2,2)

  13. Finding the Derivative of a Function • Find the derivative with respect to t for the function y=2/t.

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

  15. Classwork/Homework • Read 2.1 • Page 103 #17-31 odd, 37, 43, 45, 53-58

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