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# Functions : Linear Regression, Derivatives and Rate of Change

See the attached file.
1. A college calculus professor wanted to investigate the relationship between student's scores on the first exam and the overall course grades. A sample of the data is below. (All values are given in percents.)

first exam score 54 98 73 100 88 90 77 73 81
final grade % 60 93 69 95 82 87 72 71 74

a. Is the data in the table a function? Explain (justify) your answer.

b. Perform a linear regression on the data. What is the linear model that best fits the data? (Round all constants to the nearest tenth.)

c. Give the slope of the regression line in the model and explain the meaning of the slope in the context of this problem as a rate of change, using complete sentences. This explanation should specify how much the output changes compared to the input. Include units.
d. According to the linear model you constructed for part b, what would the final grade (score) be for a student in the class who had a first exam test score of 65?

e. Again, according to the linear model, what was the first test score for a student who wound up with a final grade of 80 % in the class?

2. Given that, use the definition of the derivative to find by answering questions (a) and (b) below.
a. First find the average rate of change of y with respect to x (difference quotient):

b. Find the instantaneous rate of change of y with respect to x: .
( Recall that this is the definition of the derivative.)

#### Solution Preview

Please see the attached file for the complete solution.

1. A college calculus professor wanted to investigate the relationship between student's scores on the first exam and the overall course grades. A sample of the data is below. (All values are given in percents.)

first exam score X 54 98 73 100 88 90 77 73 81
final grade % Y 60 93 69 95 82 87 72 71 74

a. Is the data in the table a function? Explain (justify) your answer.

The data in the table is NOT a function, since for the final exam score 73, corresponds to two different final grade( %): 69% and 71% .

b. Perform a linear regression on the data. What is the linear model that best fits the data? (Round all constants to the nearest ...

#### Solution Summary

The solution determines the linear regression, derivatives and rate of change are investigated and discussed in the solution.

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