MATH 6 ELAC FALL 7 TEST NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the integral using integration by parts. ) 9x ln x dx ) ) x 5 - x dx ) ) The supply and demand equations for a certain product are s =,5p - 4,5 and d =,, where p is the price in dollars. Find the price where supply equals demand. p ) Use point-by-point plotting to sketch the graph of the equation. 4) f(x) = x x - 4) 5) The point at which a company's costs equals its revenue is the break-even. C represents cost, in dollars, of x units of a product. R represents the revenue, in dollars, for the sale of x units. Find the number of units that must be produced and sold in order to break even. C = 5x +, R = 8x - 6 5) Graph the linear equation and determine its slope, if it exists. 6) x + 5y = 6) 7) A small company that makes hand-sewn leather shoes has fixed costs of $ a day, and total costs of $ per day at an output of pairs of shoes per day. Assume that total cost C is linearly related to output x. Find an equation of the line relating output to cost. Write the final answer in the form C = mx + b. 7) 8) Find the derivative of y = x5-7x - 4 x. 8) 9) Find the vertical asymptote(s) of the graph of the given function. f(x) = x - (x - 9)(x + ) 9) ) Simplify and express using positive exponents: 4x / x -/4 - )
Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. ) ) 5 + 7 ) The expression may be factored as shown. Find the missing factor. 7x 4/5 + x -/5 = x -/5 ( ) ) ) Solve the inequality and express the answer in interval notation: x - 4x x + 5 >. ) 4) In a certain city, the cost of a taxi ride is computed as follows: There is a fixed charge of $.5 as soon as you get in the taxi, to which a charge of $.45 per mile is added. Find a linear equation that can be used to determine the cost, C, of an x-mile taxi ride. 4) 5) Find the equation of the following line: Enter your equation in the simplest possible form., -4 and -, -4 on the line (in the xy -plane). 5) 6) Find the equation of the following line: Perpendicular to y - 5 x = ; y-intercept is. 6) Enter your answer in slope-intercept form. 7) The financial department of a company that produces digital cameras arrived at the following price-demand function and the corresponding revenue function: 7) p(x) = 95.4-6x price-demand R(x) = x p(x) = x(95.4-6x) revenue function The function p(x) is the wholesale price per camera at which x million cameras can be sold and R(x) is the corresponding revenue (in million dollars). Both functions have domain x 5. They also found the cost function to be C(x) = 5 + 5.x (in million dollars) for manufacturing and selling x cameras. Find the profit function and determine the approximate number of cameras, rounded to the nearest hundredths, that should be sold for maximum profit. 8) Suppose the cost of producing x items is given by C(x) = 4 - x and the revenue made on the sale of x items is R(x) = 4x - x. Find the number of items which serves as a break-even point. 8)
9) The financial department of a company that manufactures portable MP players arrived at the following daily cost equation for manufacturing x MP players per day: C(x) = 5 + 5x + x. The average cost per unit at a production level of players per day is C(x) = C(x) x. (A) Find the rational function C. (B) Graph the average cost function on a graphing utility for x. (C) Use the appropriate command on a graphing utility to find the daily production level (to the nearest integer) at which the average cost per player is a minimum. What is the minimum average cost (to the nearest cent)? 9) ) Write x (x -) - x (x - ) 4 reduced to lowest terms. (x - ) with positive exponents only, and as a single fraction ) Find the standard form of the equation of the line passing through the two points. (, - 6) and (- 9, 6) ) ) ) Express as a simple fraction reduced to lowest terms: a b - - b a a b - + b a ) Find the vertex form for the quadratic function. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range ) f(x) = x + 4x - 5 ) 4) Suppose an object moves along the y-axis so that its location is y = f(x) = x + x at time x (y is in meters and x is in seconds). Find the instantaneous velocity at x = 4 seconds. 4) 5) Let C(x) be the cost function and R(x) the revenue function. Compute the marginal cost, marginal revenue, and the marginal profit functions. C(x) =.4x -.6x + x + 4, R(x) = 45x 5) 6) The total cost in dollars of producing x lawn mowers is given by C(x) = 4, + 9x - x. 6) Find the marginal average cost at x =, C'() and interpret the result.
Sketch a possible graph of a function that satisfies the given conditions. 7) f(-) = -7 ; lim f(x) = -; x (-) - lim f(x) x (-) + = -7 7) 8) If f(t) =, then lim x/ h f(-8 + h) - f(-8) h equals 8) 9) Use limits to compute f'() where f(x) = - 5x. Enter just a fraction in lowest terms or an integer. 9) ) If f(x) = 4x + 4x + 5 and g(x) = 4x - 7, find g(f(x)). ) Given functions f and g, determine the domain of f + g. ) f(x) = x x - 4, g(x) = 4 x + ) Find the domain of the composite function f g. ) f(x) = 6 x + 6, g(x) = 8 x ) ) Suppose that the cost C of removing p% of the pollutants from a chemical dumping site is given by C(p) = $4, - p. Can a company afford to remove % of the pollutants? Explain. ) Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. 6x + 4) lim x x - 7 4) Find the limit, if it exists. 5) Let f(x) = x - x -. Find lim f(x). 5) x + x - 6) Let f(x) = x - 6 x + 4 if x > x - 6 x - 4 if x < 6) Find lim f(x). x 4
7) Find: lim x 5 x - 5 x - 5 7) 8) Find: d dx 4 x 4-4 5 x 8) 9) Find the second derivative of f(x) = x/ - 4 x /4 + 5x - 9) Evaluate the integral using integration by parts. 4) (x + 5) e -x dx 4) 4) 6x sin x dx 4) 4) xe x dx 4) 4) x e x dx 4) 44) xe 8x dx 44) 45) Does this integral x ln x dx = x ln x - x 4 + C? 45) Enter "yes" or "no". Evaluate the integral. 46) x x + 4 dx 46) 47) - e/x dx 47) x Divide the interval into n subintervals and list the value of x and the midpoints x,..., xn of the subintervals. 48) -8 x -5; n = 6 48) 5
Approximate the integral by the midpoint rule. 49) Approximate x dx; n = 4 Enter just a real number rounded to two decimal places. 49) Approximate the integral by the trapezoidal rule. 5) (ln x) 6 dx; n = 4 Express your answer to five decimal places. 5) Use n = 4 to approximate the value of the integral by the trapezoidal rule. 5) 9 dx 5) + x Round your answer, if appropriate. 5) The following table shows the rate of water flow (in gal/min) from a stream into a pond during a -minute period after a thunderstorm. Use the trapezoidal rule to estimate the total amount of water flowing into the pond during this period. 5) Time (min) Rate (gal/min) 5 5 5 5 5 5 5) Approximate x dx ; n = 4, by (a) the trapezoidal rule, (b) the midpoint rule, and (c) then find the exact value of the integral. Enter just a, b, c where a and b are real numbers to two decimal places (rounded off), and c is an integer, all separated by commas and answering (a), (b), (c) in order but unlabeled. 5) 5 54) Approximate dx ; n = 8, by (a) the trapezoidal rule, (b) the midpoint rule, and (c) x then find the exact value of the integral. Enter just a, b, c as real numbers all rounded to two decimal places. Do not label, but answer in the above order using commas to separate. 54) 55) Approximate x 4 dx; n = 4, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma. 55) 6
Use n = 4 to approximate the value of the integral by Simpson's rule. 5 56) 5x x - dx 56) / 57) Consider f(x)dx, where f(x) = e -x. Find a number A such that f (x) A for all x satisfying x. Use this A to obtain a bound on the error of using Simpson's rule with 57) n = 5 to approximate the definite integral. 58) A money market fund has a continuous flow of money at a rate of.9x + 7 dollars for years. Find the present value of this flow if interest is earned at 4% compounded continuously. 58) 59) The rate of a continuous money flow is e -.4 dollars per year for years. Find the present value if interest is earned at % compounded continuously. 59) Evaluate the improper integral whenever it is convergent. If it is divergent, state this. 6) dx 6) (x + ) 6) dx x(ln x) 6) 6) (x +) -/ dx 8/ Enter your answer as just a reduced fraction or the word "divergent". 6) 6) x(x + ) - dx Enter your answer as a reduced fraction or the word "divergent". 6) 64) Find the area under the graph of y = xe -x for x. Enter just a reduced fraction. 64) 65) Find the area under the graph of y = 4e -x for x. Enter your answer as ae b with any fractions reduced of form a b. 65) 66) Find the expected value of the random variable whose density function is f(x) = 8 x, x. 66) 7
67) Find the expected value and variance for the random variable whose probability density function is f(x) = e x, x. (You may use the fact that lim be b =.) b Enter just two integers (unlabeled) in the order E(X), separated by a comma. and range Sketch the surface z = f(x,y). Graph the level curve for z=k=,,, 4. State the domain in set-builder notation 68) f(x, y) = -x - y 68) 67) 69) f(x, y) = 4 - x - y 69) 8