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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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