In the following the symbol for the first derivative is dy/dx or Y' and the symbol for the second derivative is d²y/dx² or Y", all of which are standard symbols for derivatives.

Y = X⁴ -4X³ +10

X	-1	-.25	0	.25	1	1.75	2	2.25	3	3.5	4
Y
dy/dx
d2y/dx2

A function is given above. The table gives several pre-selected values of the independent variable (X). Use all of the (X) values to have enough to answer questions for this assignment.

1. Calculate the Y value corresponding to each X value in the table. Write the Y value and its sign (+ or -) calculated in the cell below its X value. Pick any three of the X values and on a separate worksheet, show how you calculated its Y value. Use these X,Y pairs also if you chose to do the bonus option.
2. Find the critical values among the given X values by:

a)- Correctly finding the first derivative of the given function. Write dy/dx here ________________ Show how you found the derivative on the worksheet (the derivative is half the credit for this question).

b)- Using the derivative from (2a) solve it for all the X values that make the derivative equal zero. Show your work on the worksheet and also write "0" in the cell of the dy/dx row for the X value only where dy/dx = 0. There may be none, one, or two such. (the dy/dx = 0 values are the other half credit for this question).

1. Find additional critical values among the X values by:

a)- Correctly finding the second derivative of the given function. Write d²y/dx² here _____________ Show how you found this second derivative on the worksheet (the second derivative is half the credit for this question).

b)- Using the second derivative from (3a) solve it for all the X values that make d²y/dx² equal zero. Show your work on the worksheet and also write "0" in the cell of the d²y/dx² row for the X value(s) only where d²y/dx² = 0. There may be one, or two such. (the d²y/dx²= 0values are the other half credit for this question).

Read the signals given by the critical derivative values above. (NOTE: do not consider the first or the last X value as being among the critical values).

1. Are there MAXIMUMS among the X,Y pairs? If so, then the pair's dy/dx = 0 and also its d²y/dx² will be a negative (-) value for each such X,Y pair. What are the X values and also the d²y/dx² values? There may be none, one, or two maximums. Write your values below:

Value of X Value of its d²y/dx²

_________ _____________

_________ _____________

1. Are there MINIMUMS among the X,Y pairs? If so, then the pair's dy/dx = 0 and also its d²y/dx² will be a positive (+) value for each such X,Y pair. What are the X values and also the d²y/dx values? There may be none, one, or two minimums. Write your values below:

Value of X Value of its d²y/dx²

_________ _____________

_________ _____________

1. Are there INFLECTION points among the X,Y pairs? If so, then the pair's dy/dx may or may not equal zero. However, d²y/dx² at each will be zero (0) for each such X,Y pair. What are the X values and also the d²y/dx² values? There may be one, or two inflections. Write your values below:

Value of X Value of its d²y/dx²

_________ _____________

_________ _____________

References: A tutorial at the top of the Blackboard page that contains the twelve versions and Power Point lesson 13

Twenty point optional bonus opportunity. If done, submit along with the above using the same Gold submission link. Using all X,Y pairs tabled above, plot them. Scale the axis's so that the maximum(s), minimum(s) and inflection(s) are clearly obvious. Indicate numerical values on each axis. Label each maximum, minimum, and inflection point as such on the plot along with its X and Y values. EXAMPLE: Suppose at X = 3.00 and Y = 11.00, you decide this point is an inflection. Then write (3,11)INF at this point on the graph. You may plot this graph with Excel (but you must still label the various critical points). Your plot must show each maximum, minimum, and inflection point found in the six previous steps (if labeling is omitted, we will not guess which is what - you must tell us). The scoring rubric and scoring for this bonus opportunity will be shown as a separate exercise in My Grades.

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