Comprehensive Practice Handout MATH 1325 entire semester

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1 1 Comprehensive Practice Handout MATH 1325 entire semester Test 1 material Use the graph of f(x) below to answer the following 6 questions Find the value of lim x + f(x) 2. Find the value of lim x 7 f(x) 3. Find the value of lim x f(x) Find the value of lim x f(x) 5. The function is not continuous at x = 7. Mathematically explain. 6. The function is not continuous at x =. Mathematically explain. 7. Find lim x 3 2x 2 + x 15 x x Find lim x 4 x 2 7x + 12 x Find lim x 5 x x 2 12x 5x x Find lim x x 2 + x Find where g(x) is continuous, given x 2 + 4, x 3 g(x) = 5x + 2, < x < 3 9x 3, x 12. Use the limit definition of derivative to find f (x) if f(x) = 8x For what interval(s) is the function f(x) continuous: 3x 2 4, x 2 f(x) = 6 2x 2, x < 2 x 1, x < 14. For what interval(s) is the function f(x) continuous: 4x + 4, x < 0 f(x) = x 2 2x + 4, 0 x 3 5 x, x > 3

2 2 Test 2 material 1. Find dy if y = 4x 9 5 6x 2. Let y = 1 4 e4x3 x 2. Find dy 3. Find the derivative of g(x) = 3x 2 e 4x 4. Find the derivative of y = 12 (3e x 2x) 6 5. Find the derivative of y = (3x 4 2x + 9) 5 6. Find f (x) and simplify completely, if f(x) = 2x x 7. Find the derivative of y = 4(x 3 + 1) 8/3 8. Find the derivative of f(x) = 4x 3 ln x 9. The financial department of a company that produces bouncy balls arrived at the following Revenue function: R(x) = 2x 2 + 5x where x is the number of bouncy balls produced and sold. Find the Marginal Revenue when 400 bouncy balls are produced and sold. 10. Find the derivative of y = 8x 3 4x + 15 x 11. Let P (x) =.05x x + 20 be the profit function for a certain type of scooter, where x is the number of scooters and P (x) is the profit in dollars. Find the marginal profit when 50 scooters are produced and sold? 12. The growth of the world population (in millions) is approximated by the exponential function A(t) = 3100e 0.66t, where t is the number of years since What was the rate of change of the population in 25? 13. It has been observed that there has been an increase in the proportion of medical research papers that use the word novel in the title or abstract, and that this proportion can be accurately modeled by the function p(x) = e x, where x is the number of years since Estimate the rate of change in the proportion of medical papers using this word in the year Find the derivative of y = 7x 2 e x 15. Find the derivative of f(x) = 3x ln(5x + 4)

3 3 Test 3 material 1. If f (x) = 2x 2 13x 24, where would f(x) have a relative minimum occur? 2. Let f(x) = 6x 3 4x 2 + 7x 8. On what interval is f(x) concave up? Use the graph to the right to answer the next 3 questions. 3. If the graph to the right was of f (x), where would f(x) be increasing? If the graph to the right was of f(x), where would f(x) be concave down? If the graph to the right was of f (x), where would f(x) be concave down? 6. Let the derivative of f(x) be given by f (x) = x(x 3) 2 (x + 4). On what intervals is f(x) increasing? 7. The revenue, in thousands of dollars, from selling x units of an item is given by R(x) = x x x. Find the maximum revenue the company can earn on sales of this item. 8. Find all relative extrema for f(x) = x 2 e x. 9. Find all hole(s) for the function: f(x) = x3 + 4x 2 32x 3x 2 18x If f (x) = 6x 4 2x 3, where would the original function, f(x) be decreasing? 11. If f (x) = 6x 4 2x 3, where would the original function, f(x) be concave up? 12. Find all inflection points (both x and y) for f(x) = x 3 + 6x 2 + 6x Find where this functions is inc/dec, concave up/down, relative extrema, and inflection points: f(x) = x 3 9x x If the graph above was of f (x), where would f(x) be concave down? 15. If the graph above was of f (x), where would f(x) be increasing? 16. If the graph above was of f (x), where would f(x) be have a relative minimum?

4 4 Test 4 material 1. Find all absolute extrema for f(x) = 1 4 x4 3x 3 18x on [ 4, 1]. 2. The revenue, in thousands of dollars, from selling x units of an item is given by R(x) = x x x. Find the maximum revenue the company can earn on sales of this item. 3. Given 7x 2 = 5y 2 + 4xy + 1, find the rate of change, dy 4. Find the equation of the tangent line to xy + 2x 5y = 2 at x = Given 3x 3 y 4 5x 2 = 8y + 6 find the rate of change, dy 6. Find the equation of the tangent line for y 3 + 2x 2 y 8y = x at x = 2 7. The price, p, in dollars of x items is related by the demand equation 9x + 2p + xp = 96. Find the rate of change in the price with respect to the quantity, dp 8. A sports store determines that the monthly revenue for a new type of running shoe is given by R = 250q 2 5 q2 Where q is the number of pairs of shoes produced and sold monthly and R is the revenue in dollars. The current production and sales is 100 pairs of shoes monthly, and the number of pairs of shoes being sold is increasing at a rate of 200 pairs of shoes per month. At what rate is the revenue changing with respect to time? 9. The cost C (in dollars) of manufacturing x number of high-quality computer laser printers is C(x) = 15x 4/3 + 54x 2/ , 000 Currently, the level of production is 1728 printers and that level is increasing at the rate of 350 printers each month. Find the rate at which the cost is increasing each month. Test 5 material and section If z = (2x 8 y 2 ) 3, find f x (x, y) and simplify. 2. Evaluate the definite integral 3. Evaluate 4. Evaluate x 5 x x 3x 2 5 a (2x + 6) 5. Given the graph below, how would you JUST SET UP the integral used to find the area bounded between these two curves on the interval [a, c]?

5 5 6. Find the cost function for x items with marginal cost function given the fixed costs are $ Evaluate the definite integral a C (x) = 0.5e 0.1x + 4 (2x + 10) 8. Find the area bounded between f(x) = 5 2 x and g(x) = 3 x on the interval [, 2]. Round to 2 decimals, if necessary. 9. If z = (3x 4 y 7 ) 2, find f y (x, y) and simplify. 10. Find the cost function for x items with marginal cost function give the cost of producing 2 items is $400. C (x) = 2x + 4 x Let z = g(x, y) = 3x 2 2xy + 8yx 4, find g x (x, y) and g y (x, y) 2x 3 3x Evaluate (3x 4 6x 3 ) Find the area between y = x 3 5x 2 + 3x + 4 and the x-axis on the interval from x = 0 to x = 3 Sketch a graph, shading the area. Set up the integral and solve. Use 2 decimal places everywhere (if necessary). 14. Find the cost function, C(x) if the marginal cost function, in dollars, is given by C (x) = 4x + 6e 3x, where x represents the number of items made and C(x) is the total cost. The fixed costs are $ Suppose the number of gadgets produced at The Gadget Company is increasing at a rate given by G(t) = 4t + 6, where t is in hours since the factory opened. Assuming the 3t + 2 company never closes, how many gadgets (whole gadgets, no fractions) are produced by The Gadget Company during their first two days of operation? 16. What is the approximate area between the graph of y = 3 4 (x + 3)2 + 2 bounded by the x-axis between -4 and 0? (round to 2 decimals everywhere) 4x (2x 2 3x) Evaluate (8x + 7) ln x 19. Evaluate 4x ln(x) 20. Evaluate (x + 6)e 3x 21. If z = f(x, y) = 6x 2 y 2xy 2 + 3x 9y find f x (x, y) and f y (x, y)

Math 142 Week-in-Review #7 (Exam 2 Review: Sections and )

Math 142 WIR, copyright Angie Allen, Spring 2013 1 Math 142 Week-in-Review #7 (Exam 2 Review: Sections 4.1-4.5 and 5.1-5.6) Note: This collection of questions is intended to be a brief overview of the