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Copyright © 2011 Pearson Education, Inc. Slide 12.4-1 12.4 Tangent Lines and Derivatives A tangent line just touches a curve at a single point, without passing through any other nearby points, and indicates the direction of the curve. Tangent to a circle

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-2 12.4 Tangent Lines and Derivatives P 1 and P 3 are tangent lines P 2, P 4 and P 5 are not

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-3 12.4 The Tangent Line as a Limit of Secant Lines A secant line is a line through an arbitrary point (x, f(x)) and a given point (a, f(a)) on a curve.

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-4 12.4 The Tangent Line as a Limit of Secant Lines As x approaches a the secant lines approach the tangent line at (a, f(a)).

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-5 12.4 The Tangent Line as a Limit of Secant Lines Tangent Line The tangent line of the graph of y = f(x) at the point (a, f(a)) is the line through this point having slope provided this limit exists. If the limit does not exist, there is no tangent line at this point.

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-6 12.4 Finding the Equation of a Tangent Line Example Find the equation of the tangent line to the graph of the function f(x) = x 2 + 2 when x = -1. Solution

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-7 12.4 Finding the Equation of a Tangent Line Solution Use the point-slope formula with m = -2 and the point (-1, 3).

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-8 12.4 Finding the Equation of a Tangent Line Solution (Graphing calculator)

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-9 12.4 Derivative of a Function Derivative If a is in the domain of f, then the derivative of f at a is defined by provided this limit exists.

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-10 12.4 Finding a Derivative Example Find f ´(4), where Solution

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-11 12.4 Interpretation of the Derivative as a Rate of Change The function s(t) gives the number of feet traveled by a car after t seconds. s´(2)=20 says that after 2 seconds the car is traveling at a velocity of 20 feet per second.

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-12 12.4 Interpretation of the Derivative as a Rate of Change The average velocity during the time interval from 2 to t is The instantaneous velocity at time 2 is

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-13 12.4 Interpretation of the Derivative as a Rate of Change Velocity as a Derivative If an object is moving along a straight line and its position on the line at time t is s(t), then the velocity at time a is

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-14 12.4 Interpretation of the Derivative as a Rate of Change If an object is thrown straight up into the air from a height h 0 with initial velocity v 0, then the height (in feet) of the object above ground after t seconds is given by

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-15 12.4 Interpretation of the Derivative as a Rate of Change Example A ball is thrown straight up into the air has an initial height of 112 feet and an initial velocity of 96 feet per second. (a)Find s(t), the height of the ball after t seconds. (b)How high above the ground is the ball after 5 seconds? (c)What is the velocity of the ball at 5 seconds?

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-16 12.4 Interpretation of the Derivative as a Rate of Change Solution (a) (b) The height at 5 seconds is

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-17 12.4 Interpretation of the Derivative as a Rate of Change Solution (c) The velocity at 5 seconds is

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-18 12.4 Marginal Concept in Economics In economics, derivatives are described by the adjective “marginal.” If C(x) is the cost function (the cost of producing x units of a commodity), then C´(a) is the marginal cost at the production level of a units. The marginal cost can be interpreted as the approximate cost of producing one additional unit of goods.

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Copyright © 2011 Pearson Education, Inc. Slide 12.4-19 12.4 Summary of Ideas Summary of Ideas Related to f ´(x) 1.Geometric definition of f ´(a) f ´(a) is the slope of the tangent line to the graph of f (x) at x = a. 2.Algebraic definition of f ´(a) 3.Interpretations of f ´(a) f ´(a) represents velocity when f(x) defines a position function and marginal change when f(x) defines an economics function.

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