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Calculating Tangent Line Slopes - Derivative Definition

Learn how to find the slope of a tangent line to a curve at a given point using the limit definition of the derivative. Practice finding slopes of linear and nonlinear functions and use the derivative to find tangent line equations.

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Calculating Tangent Line Slopes - Derivative Definition

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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 BCCalculus

  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 at the point (2,1)

  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 slope of the graph of at the point (0,1) and (-1,2)

  6. Tangent Lines to the Graph of a Nonlinear Function • Find the slope of the graph of 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: f’(x), , y’, , (they all mean the same thing!)

  9. Finding the Derivative by the Limit Process • Find the derivative of

  10. Finding the Derivative by the Limit Process • Find the derivative of

  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 • (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 and (c) Use the derivative feature of a graphing utility to confirm your results (-2,2)

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

  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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