Algebra with Calculus for Business: Review (Summer of 07)

Transcription

1 Algebra with Calculus for Business: Review (Summer of 07) 1. Simplify (5 1 m 2 ) 3 (5m 2 ) Simplify (cd) 3 2 (c 3 ) 1 4 (d 1 4 ) Simplify (x y 1 2 )(x 1 2 y 1 2 ) 4. Solve the equation 4 3 (x 2) 1 2 = 2( 3 4 x 1). 5. Solve the equation m 2 1 m = 6m Solve the equation 4m 3 = Solve the equation x 2 4x + 1 = Solve the equation x 2 5x 6 = Solve the equation x 2 5x 6 = The total resources T (in billions of dollars) of the Pension Benefit Guaranty Corporation, the government agency that insures pensions, can be approximated by the equation T =.26x x , where x is the number of years after When will the Corporation be out of money? 11. Find the x-intercept and the y-intercept of the graph of 2y + 3x = Find the x-intercept and the y-intercept of the graph of y = 3x 2 +4x Find the slope of the line that passes (9, 0) and (2, 1). 14. Find the equation of the line with the y-intercept 6 and slope Find the slope m and the y-intercept of 4x + 3y = Find the equation of the line that passes (1, 2) with slope Find the equation of the line that passes (1, 2) and is perpendicular to 2x + 3y = 4. 1

2 18. Over the next 25 years, the number of job in the San Antonio metropolitan area (in thousands) is projected to be given by: y = 45.23x How many jobs are there in 2008? 19. Solve 3x 7 < 11x Solve 5+2m < 2. Projections from the US Department of Commence show that the number of Internet users (in millions) is approximately 16.4x During what years was the number of Internet users between 203 and 285 million? 21. Find f(4) if f(x) = 5 x. x 2, if x < 2, 22. item Find f(4) if f(x) = 25x 7, if x The Mississippi state income tax for a single person in 2004 was determined.03x, if 0 x 5000, by the rule T (x) = (x 5000), if 5000 < x 10000, (x 10000), if x > where x is the person s taxable income. Find the tax due on each of these incomes (a)$4750 (b) $27950 (c)$ Graph the function f(x) = x 2, if x < 2, 4 x, if x Graph the function f(x) = 3 2x Assume the cost function can be approximated by a linear model. If the fixed cost is $2000, and 50 item cost $20000 to produce. What is the cost function? 27. A Honda Civic LX sedan is worth $15350 now and is expected to be worth $9910 in 4 years. (a) Find a linear depreciation function for this car. (b) Estimate the value of the car in 6 years. 2

3 (c) At what rate is the car depreciating? 28. The supply and demand for a particular material follow by the model p = 120 3q, where p is the price per unit and q is the quantity demanded. (a) Find the demand at a price of $30. (b) Find the price if the demand is 25 units. (c) At what rate is the price is decreasing? 29. Let f(x) = 3(x 3) 2 2. Find the vertex and the x- and y-intercept. 30. Find the vertex, the axis, and the x-and y-intercept of the graph of x 2 5x + 6. Sketch the graph. 31. Find the quadratic function with the vertex (1,-1) that passes (5, 6). 32. The manager in a bicycle shop has found that, at a price (in dollars) of p(x) = 150 x per bicycle, x bicycles will be sold. 4 (a) Find the Total Revenue function from the sale of x bicycles. (b) Find the number of bicycle sales that leads to maximum revenue. (c) Find the maximum revenue 33. Find the equilibrium quantity and equilibrium price for the commodity whose supply and demand functions are given by p = q 2 + q + 10 and p = 10p respectively. 34. Graph the functions f(x) = 2e 2,, g(x) = 3 x. 35. Problem 37 on page Problem 1-4 on page Without using a calculator, evaluate (i) log 1000, (ii) log 3 81, (iii) log 2 1 4, (iv)ln e 38. Write each expression as the logarithm of a single number or expression. (1) ln 3 + ln 5, (2) ln(x + 1) + ln(x 1), (3) 2ln 3 ln 9 3

4 39. Write each expression as a sum and/or a difference of logarithms with all variables to the first degree: (1) ln(7x 3 y), (2) ln x 2 1, (3) ln 3x 5y Solve log 3 (3x 1) = Solve log(3x 1) log x = Solve 3 x 1 = Solve 2e x 1 = Solve 2e x 1 = Solve e x 1 = e 3x. 46. The US gross domestic product, in trillions of dollars, is approximated by f(x) = ln(x + 10). where x = 0 corresponds to According to this model, when will the GDP reach $12 trillion? 47. What the three elementary row operations? 48. Solve the system of equations: 3x + 3y = Solve the system of equations: x 2y = Solve the system of equations: 3x + 3y = 5 3x 2 + z = Solve the system of equations: 3x + y z = 4 x + 2y 2z = Kate borrows $10,000, some from her friend at 8% annual interest, twice as much as that from her bank at 9%, and the remainder from her insurance company at 5%. She pays a total of $830 in interest for the first year. How much did she borrow from each source? 4

5 53. Let M = 2M N. 54. Let A = [ 1 3 ] and N = [ ] and B = Find AB Example 4 on p Graph the linear inequality: 2x + 3y Graph the linear inequality: 4x 3y Graph the feasible region for x + y 4, 59. Problem 51 and 52 on p [ ] Find (i) M + N; (ii) x y Find value of x 0 and y 0 that minimize z = 3x + 2y, subject to the constraints: 6x + 5y 25, 61. Problem 8-10 on page x + 6y 15. 5