1 - Arizona State University



a) The points [pic]and [pic]lie on the curve[pic]. Find the slope of the secant line PQ correct to two decimal places. Draw the graph of the function along with P and Q on the graph and the secant line. [10 points]

Solution: Slope = [pic]

Make the graph yourself.

b) The points [pic]and [pic]lie on the curve[pic]. Find the slope of the secant line PQ correct to two decimal places. Draw the graph of the function along with P and Q on the graph and the secant line. [10 points]

Solution: Slope = [pic]

Make the graph yourself.

1. a) Use a table of values to guess the value of [pic]. Graph the function [pic] and examine your result near zero. [10 points]

|x |y |

|-.10000 |0.99833 |

|-.01000 |0.99998 |

|-.00100 |1.00000 |

|x |y |

|.10000 |0.99833 |

|.01000 |0.99998 |

|.00100 |1.00000 |

[pic] [pic]

Limit exists and [pic]

b) Use a table of values to guess the value of [pic]. Graph the function [pic] and examine your result near zero. [10 points]

|x |y |

|-.10000 |-0.1003 |

|-.01000 |-0.0100 |

|-.00100 |-0.0010 |

|x |y |

|.10000 |0.1003 |

|.01000 |0.0100 |

|.00100 |0.0010 |

[pic] [pic]

Limit exists and [pic]

2. a) Use the squeeze Theorem to find the limit, [pic]. [10 points]

The squeeze theorem: If [pic] and [pic] then also [pic]

To solve the above limit we need to derive two such functions. It is known that [pic], since [pic], we get by multiplication [pic]

Now by Squeeze theorem [pic], where [pic]

Then [pic]

b) Use the squeeze Theorem to find the limit, [pic]. [10 points]

The squeeze theorem: If [pic] and [pic] then also [pic]

To solve the above limit we need to derive two such functions. It is known that [pic], since [pic], we get by multiplication [pic]

Now by Squeeze theorem [pic], where [pic], then [pic]

3. a) For the given function [10 points]

[pic]

a) Find [pic] if exists.

[pic] and [pic]

So limit does not exist.

b) Is the function continuous at [pic]? Circle your answer. Yes No

4. a) For the given function [10 points]

[pic]

a. Find [pic] if exists.

[pic] and [pic]

So limit does exist and [pic]

b. Is the function continuous at [pic]? Circle your answer. Yes No

Since [pic]

5. Find the limits, if exists. Show algebraic work . [15 points]

1) [pic] =DNE, since [pic] and [pic]

1) [pic] =DNE, since [pic] and [pic]

2) [pic]

3) [pic]

3) [pic]

6. a) The greatest integer function is defined by [pic]the largest integer that is less than or equal to x. Use this definition to determine [pic]. Draw the graph of [pic] near 3. [10 points]

[pic] and [pic], so [pic]

b) The greatest integer function is defined by [pic]the largest integer that is less than or equal to x. Use this definition to determine [pic]. Draw the graph of [pic] near 2. [10 points]

[pic] and [pic], so [pic]

7. a) Find the slope of the tangent line to [pic] at [pic]using the formula [pic] . [8 points]

[pic]

b) Find the equation of the tangent line to [pic] at [pic]. [7 points]

Equation of the tangent line [pic]

c) Find the slope of the tangent line to [pic] at [pic]using the formula [pic] . [8 points]

[pic]

b) Find the equation of the tangent line to [pic] at [pic]. [7 points]

Equation of the tangent line [pic]

8. Let [pic], determine [pic] . Find [pic]if [pic] [10 points]

[pic], then [pic]. For [pic], [pic], then [pic]

9. The tangent line to [pic] at [pic]passes through [pic]. Find [pic][pic]. [10 points]

The point (4, 3) lies on [pic], so [pic]. Since the tangent line is on (4, 3) on the graph and also passes through (0, 2), we have [pic]

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