12 1 finding limits numerically and graphically
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12.1 Finding Limits Numerically and Graphically PowerPoint PPT Presentation


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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. 12.1 Finding Limits Numerically and Graphically.

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12.1 Finding Limits Numerically and Graphically

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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.

12.1 Finding Limits Numerically and Graphically


12 1 finding limits numerically and graphically1

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:

12.1 Finding Limits Numerically and Graphically


12 2 limit laws

12.2 Limit Laws


12 2 limit laws1

  • Find

  • Find

  • Find

  • Find

12.2 Limit Laws


12 2 limit laws2

EX. Evaluate the limit, if it exists

12.2 Limit Laws


12 2 limit laws3

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?

12.2 Limit Laws


12 2 limit laws4

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?

12.2 Limit Laws


2 3 average rate of change aroc

Vocabulary:

secant line

AROC

constant function

constant slope

2.3 Average Rate of Change (AROC)


2 3 aroc given a graph

Wolfram Demo secant lines

2.3 AROC – given a graph


12 1 finding limits numerically and graphically

2.3 AROC – given the function

EX. 1

f(z) = 1 – 3z2; find AROC between

z = -2 and z = 0

EX. 2

g(x) = ; find AROC between

x = 0 and x = h


2 3 increasing decreasing functions

2.3 Increasing/Decreasing functions


12 3 iroc instantaneous rate of change

Wolfram Demo Secant -> Tangent lines

12.3 IROC – Instantaneous Rate of Change

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)


12 1 finding limits numerically and graphically

12.3 IROC – algebraic examples

EX. 3

f(x) = 1 + 2x – 3x2; find the equation of the tangent line at (1, 0) two different ways.

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)


12 1 finding limits numerically and graphically

12.3 IROC – algebraic examples

Let’s see if this makes sense graphically:

f(x) = 1 + 2x – 3x2; We calculated that

the equation of the tangent line at (1, 0) is y = -4x + 4


12 1 finding limits numerically and graphically

12.3 IROC – classwork practice

  • f(x) = 1/x2; 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)

  • 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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