f(x)=16 x^2 a) Find an equation of the tangent line to f(x) at x=2. Remember to show all analytic work. b) When x=2, draw a graph of f(x) and tangent line to the curve. Label the (x,y) coordinates of the point of tangency and label the y intercept. 4. The costC =100x^2+ 800. a) Define and simplify the average cost function. Then graph the average cost function using Xmin=0, Xmax=10, Ymin=0, Ymax=1000. b) (Analytical show work to find the rate of change for the average cost function as a function of x. Then evaluate accurate to 3 decimal places the rate of change of the average cost function for values when x=2, x= 2 sq. rt 2 , x=4. (Be sure to show all analytic work.) c) Using the cost C =100x^2 + 800 and revenue R = 4x^4 , define an equation for the average profit function. Also display a graph of the average profit function with settings Xmin=0, Xmax=10, Ymin= -50, Ymax=50, and use that display to estimate the breakeven point (x value at which average profit 0 ). Then analytically evaluate the rate of change at that point.

5.y=x^33x^2 -24x+ 4 a) Find (x, y) coordinates at which the tangent line is horizontal. Use 3 decimal place accuracy as needed. Be sure to show all analytic work. b) What is the equation of the tangent line at each of the points found is part a)? c) Set up appropriate window settings and graph the original function and label the (x,y) coordinates of the points found on your graph. State your Xmin, Xmax, Ymin and Ymax settings used.

6. The Dow Jones has been volatile recently and stock closing values for the past 4 years are: Date Dow Jones close June 4, 2019 24,820 June 4, 2020 31,490 June 4, 2021 34,577 June 4, 2022 32,900 June 4, 2019 is the base with x=0. Use 3 decimal place accuracy as needed. a) Model the data with a linear function using the points in years 2019 and 2022. b) Using your model predict the Dow Jones closing value for June 4, 2023. c) 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 3 decimal places as needed. Also list the R2 value accurate to 3 decimal places? Note: You are able to use your TI calculator to compute the linear least squares results. d) (Using your least squares model predict the Dow Jones close for June 4, 2023. Also show a graph of the actual data and least squares graph superimposed on the same graph. e) Which predictor (part b or part d) for 2023 is more reliable and why?