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12.1 Finding Limits Numerically and Graphically We are asking “What numeric value does this function approach as it gets very close to the given value of x?” Numeric approach: Complete the table of values to estimate the value of the limit. x1.91.991.9992.12.012.001 f(x)

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12.1 Finding Limits Numerically and Graphically We are asking “What numeric value does this function approach as it gets very close to the given value of x?” Graphic approach: Consider the graph and the different types of limits:

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12.2 Limit Laws

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a)Find b)Find c)Find d)Find

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12.2 Limit Laws EX. Evaluate the limit, if it exists

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12.2 Limit Laws EX. Evaluate the limit, if it exists First check – does subbing in the value of x take the denominator to 0? EX. Evaluate the limit, if it exists First check – does subbing in the value of x take the denominator to 0?

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12.2 Limit Laws EX. Evaluate the limit, if it exists First check – does subbing in the value of x take the denominator to 0? EX. Evaluate the limit, if it exists First check – does subbing in the value of x take the denominator to 0?

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2.3 Average Rate of Change (AROC) Vocabulary: secant line AROC constant function constant slope

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2.3 AROC – given a graph Wolfram DemoWolfram Demo secant lines

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2.3 AROC – given the function EX. 1 f(z) = 1 – 3z 2 ; find AROC between z = -2 and z = 0 EX. 2 g(x) = ; find AROC between x = 0 and x = h

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2.3 Increasing/Decreasing functions

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12.3 IROC – Instantaneous Rate of Change Wolfram Demo Wolfram Demo Secant -> Tangent lines Note that in these problems we will use the letter ‘h’ to represent the distance away from the point where we are considering the tangent line ( x)

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12.3 IROC – algebraic examples EX. 3 f(x) = 1 + 2x – 3x 2 ; find the equation of the tangent line at (1, 0) two different ways. a.Method 1: consider the limit as x approaches 1 b. Method 2: consider the limit as h approaches 0 Now use the point slope formula to find the line’s equation: slope = -4, goes through (1, 0)

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12.3 IROC – algebraic examples Let’s see if this makes sense graphically: f(x) = 1 + 2x – 3x 2 ; We calculated that the equation of the tangent line at (1, 0) is y = -4x + 4

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12.3 IROC – classwork practice a.f(x) = 1/x 2 ; find the equation of the tangent line at (-1, 1) and graph the function and the tangent line requested. Use method 1 (look at the limit as x approaches -1) b.f(x) = ; find the equation of the tangent line at (4, 3) and graph the function and the tangent line requested. Use method 2 (look at the limit as h approaches 0)

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Finding the Derivative The Limit Process. What is the derivative of something? The derivative of a function f(x) is, mathematically speaking, the slope.

Finding the Derivative The Limit Process. What is the derivative of something? The derivative of a function f(x) is, mathematically speaking, the slope.

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