Math 105 Midterm Exam Review 1. A company will sell N(x) printers after spending x thousand dollars on advertising as modeled by the function N ( x) = 60x - x 2. a) Find the average rate of change of the number of printers sold when advertising spending goes from 10 thousand dollars to 20 thousand dollars. b) Find the instantaneous rate of change in the number of printers sold when the spending on advertising is 15 thousand dollars. 2. Find the equation for the tangent line to the given curve at the indicated point. f ( x ) = x 2 - x - 30; x = 3 1 3. For the function f ( x ) = x 3 + x 2 - 4 x + 5 , find the value(s) of x where the tangent 2 line to y = f(x) is horizontal. 4. The total cost (in dollars) from producing x units of a product is given by C( x ) = 4 x + 500 a) Find the exact cost (to the nearest cent) of the 101 st item. b) Find the average cost per item if 36 items are manufactured. 5. When the admission price to a community concert was $6 per ticket, 500 tickets were sold. When the price was raised to $8 per ticket, only 400 tickets were sold. Assume that the demand function is linear and the fixed and variable costs for the stadium owners are $300 and $1.60, respectively. a) Assuming that the demand function is linear, find it. b) Find the revenue function R as a function of x. c) Give the cost function C as a function of x. 6. A company makes designer ties. The weekly cost function is C( x ) = $500 + 30x . The demand function is p = 90 - x. a) Find the Revenue function. b) Graph the Revenue and Cost functions. c) Find the break even points. 7. Find the equation for the tangent line to the given curve at the indicated point. f ( x ) = ( x 2 + 3x - 1)(2 + x );x = 2 Math 105 Midterm Exam Review 8. Past sales records for a jacuzzi manufacturer indicate that the total number of jacuzzis, N, sold during a year is given by 1990. Find N ( 4) and interpret. N(t) = t t + 5 . Let t = the number of years since 9. The total cost (in dollars) from producing x units of a product is given by C( x ) = 4 x + 500 Find the average cost per item if 36 items are manufactured. 10. Using calculus methods, find the three critical values of f and use them to find the interval(s) where f(x) is increasing, decreasing, and any local extrema. (If none, write NONE) TO RECEIVE ANY CREDIT, YOUR CALCULUS WORK MUST BE SHOWN. 1 f(x) = x x 5x + 1 4 11. The number of people N(t) (in hundreds) infected in t days after an epidemic begins is approximated by the function N(t) = 2 + 50t t When will the number of people infected start to decline? after t = ____ days 12. The number of murders in a foreign country (in hundreds) from 1988 to 2000 is approximated by f(x) = x + 6x 20x + , where x corresponds to the number of years after 1988. (e.g.: x = 1 is the year 1989) a. In what year did a minimum number of murders occur? USE CALCULUS METHODS. TO RECEIVE ANY CREDIT, YOUR WORK MUST BE SHOWN. b. How many murders occurred in that year (from part a. ) Math 105 Midterm Exam Review 13. For each function, give the equation of each vertical and horizontal asymptote and the coordinates of any x and y intercepts. If none, write NONE a. () = b. () = c. () = 14. Find the absolute maximum and minimum values of () = 2 3 12 + 24 on the interval [ -2 , 1 ] SHOW WORK FOR CREDIT abs. min. f( ____ ) = _____ abs. max. f( ____ ) = _____ 15. Given the graph of the derivative, , answer the following questions about the graph of . f is increasing _____________ f is decreasing _____________ f has local max at f has local min at -1 2 16. Find the derivative of the following functions. DO NOT SIMPLIFY. a. () = (3 + 1)2 5 b. () = (10 + 1)(3 2) c. () = d. () = 4 + 3