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2.1 Derivatives and Rates of Change

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The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of the secant through (4,16): We could get a better approximation if we move the point closer to (1,1), i.e. (3,9): Even better would be the point (2,4): The tangent problem

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The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go? The tangent problem

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slope slope at The slope of the curve at the point is: Note: This is the slope of the tangent line to the curve at the point. The tangent problem

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Consider a graph of displacement (distance traveled) vs. time. time (hours) distance (miles) Average velocity can be found by taking: A B The speedometer in your car does not measure average velocity, but instantaneous velocity. (The velocity at one moment in time.) The velocity problem

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Derivatives Definition: The derivative of a function at a number a, denoted by f ′(a), is if this limit exists. Example: Find f ′(a) for f(x)=x 2 +3.

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Equation of the tangent line The tangent line to y=f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is equal to f ′(a). Then the equation of the tangent line to the curve y=f(x) at the point (a,f(a)): Example: Find an equation of the tangent line to f(x)=x 2 +3 at (1,4). From previous slide: f ′(1)=2 1=2. Thus, the equation is y-f(1)= f ′(1)(x-1) y-4=2(x-1) or y=2x+2

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Rates of Change: Average rate of change = Instantaneous rate of change = These definitions are true for any function. ( x does not have to represent time. )

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