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Management Calculus MATH.1220-031 Exam 1, 100 points

M. Stick         Spring 2021

Show all analytic work for each problem to receive credit.

1. a) (12 pts) Using the limit of the difference quotient, find f '(x) for f (x)= ½ -7x

b) (3 pts) Use the chain rule or quotient rule for the function in part a) to find the derivative.

 2.  Using appropriate differentiation techniques (not the limit process), find the derivative.  The LCD is required. Simplify results.

a. (7 pts) f (x)= ³√x + 4/x² +2 

 b) y = 2 – 8x/√4x + 3

 c) (7 pts) y=(3x 1)⁴√5x 

  3. f (x)= √2x 3 . 

1. (12 pts) Find an equation of the tangent line to f(x) at x=3.

 b) When x=3, draw a graph of f(x) and tangent line to the curve.  Label the (x, y) coordinates of the point of tangency.

4. The cost C = 3√x + 4 and revenue R = -2x²  +10.

1. Define and simplify the average cost function, the average revenue and the average profit.
2. Analytically show work to find the rate of change for the average profit function as a function of x. Then evaluate accurate to 3 decimal places the rate of change of the average profit function when x=4.
3. Graph the average profit function over an appropriate window and define your window settings.

5. f (x)−= -x³+3/2 x²+36x - 1.  

1. (6 pts) Find (x, y) coordinates of the critical points of f(x).
2.  Use the first derivative test to find the (x, y) coordinates of the extrema and indicate the intervals where the function is increasing or decreasing.  
3. Graph the function and label the (x, y) coordinates of the extrema on your graph.

 

 6. The following data represents a company’s revenue in millions of dollars. 

Year:        2014  2015  2016  2017  2018  2019  2020

Revenue:     5       7        8       10       13     14    14.5

Let the year 2013 be the base year with x=0.  Use 4 decimal place accuracy as needed.

1. (6 pts) Model the data with a linear function using the points in years 2014 and 2020.
2.  Using your model predict the revenue in the year 2021.

 

3.  Set up and exhibit a table of (x, y) values to be used for a least squares model and find a linear least squares model y = ax          b+for the data.  Express the coefficients a and b accurate to 4 decimal places.  Also list the R² value accurate to 4 decimal places? Note: You are able to use your TI calculator to compute the linear least squares results.
4. Using your least squares model predict the revenue in the year 2021.  Also show a graph of the actual data and least squares graph superimposed on the same graph.