Take Home Test B (2.6 and Chapter 3) You don't need to print, scan, or submit page 5 (it just has images for reference for two problems) 1. Find dy/dx (or y') for the following: 4 x 2 2 y 2 + 3 x 2 y = 2 x . Show work. 2. Find the slope of the tangent line to the graph of work. 3. Determine where the function f ( x) = 3 x 3 + 5 y 2 + x = 1 at the point (-1,1). Show 1 3 x x 2 3 x + 4 is increasing/decreasing using 3 calculus. Show all work including a sign chart and clearly mark your answers. Increasing on: Decreasing on: 4. Find the relative extrema of the following function using calculus. Show all work including a sign chart. f ( x) = ( x 2 9) 4 Relative maximum occurs at the point: _________ Relative minimum occurs at the point: __________ 5. Consider the function f ( x) = x 2 8 x + 3 . a. Find the absolute maximum and minimum of the function over the interval [3,6]. Show all work. b. Find the absolute maximum and minimum of the function over the interval [0,3]. Show all work. 6. State whether true or false: If f ' (c) = 0 and f ' ' (c ) is negative, then f (c ) is a relative minimum. 7. Find the inflection points and the intervals of concavity for the function f ( x) = 2 x 3 + 5 x 1 Show your work. Concave up on:_______________________________ Concave down on:_________________________________ Inflection point(s): ___________________________ 8. Suppose that f ' (c) = 0 and f ' ( x) changes sign from negative to positive at x=c. Which of the following statements is correct? Circle your answer. a) f (x) has a relative min at x=c b) f (x) has an absolute max at x=c c) f (x) has a relative max at x=c d) f (x) has an inflection point at x=c 9. Find the point of diminishing returns for the sales function S ( x) = 112 + 1.8 x 2 0.1x 3 , where x represents thousands of dollars spent on advertising, and S is sales in thousands of dollars. Show your work. 10. Suppose that there is a function f(x) such that its second derivative is f ' ' ( x) = 6 x( x 5) 2 . Answer the following questions. Show your work below. A) What are the inflection point(s) of f(x)? B) For which x-values of f(x) concave up? Concave down? 11. Consider the function shown on the reference page. a) On which x-intervals is the function concave up? b) On which x-intervals is the function concave down? 12. Use the image on the reference page. Match the graphs. (No explanations required) The derivative of a is ______ The derivative of b is ______ The derivative of c is ______ The derivative of d is ______ 13. The cost of producing x compact refrigerators is C ( x) = 2880 + 35 x + 0.2 x 2 dollars. Find the number of refrigerators that minimizes the average cost if Applications video for a similar example) 0 x 150 . (see Chapter 3 Reference Page (don't need to print, scan or submit) For question 11: For question 12