- Applied Mathematics for the Managerial, Life, and Social Sciences
- Applied mathematics for the managerial life and social sciences 6th e…
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The range of f is the set of all elements f x whenever x is an element in its domain. An independent variable is a variable in the domain of a function f. The dependent variable is y 2. The graph of a function f is the set of all ordered pairs x y such that y domain of f. Use the vertical line test to determine if every vertical line intersects the curve in at most one point. If so, then the curve is the graph of a function.
Yes, every vertical line intersects the curve in at most one point. No, a vertical line intersects the curve at more than one point. Because the point 1 5 lies on the graph of f it satisfies the equation defining f. Thus, f 1. Because the point 2 4 lies on the graph of f it satisfies the equation defining f. Thus, f 2. Because the square root of a number is defined for all real numbers greater than or equal to zero, we have x or x. Because the square root of a number is defined for all real numbers greater than or equal to zero, we have 5 or x of f is.
Therefore, the domain is the set of all real numbers in The numerator is defined when x the domain is [1 3 and 3. If x 1 the graph of f is the half-line y x 1.
Fo x 1, the graph consists of the line segment y 0. For x 1, the graph is the half-line y x 1. Each vertical line cuts the given graph at exactly one point, and so the graph represents y as a function of x. Because the y-axis, which is a vertical line, intersects the graph at two points, the graph does not represent y as a function of x.
Applied Mathematics for the Managerial, Life, and Social Sciences
Because there is a vertical line that intersects the graph at three points, the graph does not represent y as a function of x. Each vertical line intersects the graph of f at exactly one point, and so the graph represents y as a function of x. The y-axis intersects the circle at two points, and this shows that the circle is not the graph of a function of x.
A vertical line containing a line segment comprising the graph cuts it at infinitely many points and so the graph does not define y as a function of x. The circumference of a circle with a 5-inch radius is given by C 5. Next, the slope of the straight line passing through 10 0 59 and 20 0 60 is 0 60 0 59 m2 0 , and so an equation of the straight line passing through the two points is 20 10 y 0 59 0 t 10 or y 0 t 0 The slope of the straight line passing through 20 0 60 and 0 66 0 60 30 0 66 is m 3 0 , and so an equation of the straight line passing through the two points is 30 20 y 0 60 0 t 20 or y 0 t 0 The slope of the straight line passing through 30 0 66 and 0 78 0 66 40 0 0 78 is m 4 0 , and so an equation of the straight line passing through the two points 40 30 0 t 0 61 if 0 t The gender gap was expanding between and and shrinking between and The gender gap was expanding at the rate of 0 yr between and , shrinking at the rate of 0 yr between and , shrinking at the rate of 0 yr between and , and shrinking at the rate of 0 yr between and Next, the slope of the straight line passing through the points 20 0 95 1 1 0 95 and 30 1 1 is m 2 0 , so an equation of the straight line passing through 30 20 the two points is y 0 95 0 t 20 or y 0 t 0 Therefore, a rule for f is.
The ratios were changing at the rates of 0 yr from through and 0 yr from through The ratio was 1 when t 20 3.
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The projected number in is P 20 approximately 8 million. The projected number in is P 40 approximately 13 3 million. When the proportion of popular votes won by the Democratic presidential candidate is 0 60, the proportion of seats in the House of Representatives won by Democratic candidates is given by 06 3 0 0 0 The amount spent in was S 0 S 4. The median age of the U. In , it was f 6 13 88, or approximately.
Let A g. F t represents the total revenue for the two restaurants at time t. F t represents the net rate of growth of the species of whales in year t. Between and , the percentage of motorcyclists wearing helmets had dropped from 64 to 51, and as a consequence, the mortality rate of motorcyclists had increased from 26 million miles traveled to 42 million miles traveled. So in , the percentage of reported serious crimes that end in arrests or in the identification of suspects was Between and , the total number of detectives had dropped from to and as a result, the percentage of reported serious crimes that ended in arrests or in the identification of suspects dropped from 23 to Yes: R 3 Let f , g, and h define the revenue in dollars in week t of three branches of a store.
Then its total revenue in dollars in week t is s t Let t denote time. Suppose f gives the number of people at time t in a town, g gives the number of cars as a function of the number of people in the town, and H gives the amount of carbon monoxide in the atmosphere.
Applied mathematics for the managerial life and social sciences 6th e…
Then h g f t h g f t gives the amount of carbon monoxide in the atmosphere at time t. Substituting this value of x into the first equation yields y y Thus, the point of intersection is 2 We solve the system y 4x. Substituting this value of x into the first equation, we have y Therefore, the point of intersection is 4 6.
We solve the system y.
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We solve the equation R x C x , or 15x 5x 10,, obtaining 10x 10,, or x Substituting this value of x into the equation R x 15x , we find R 15, Therefore, the break-even point is We solve the equation R x C x , or 21x 15x 12,, obtaining 6x 12,, or x Substituting this value of x into the equation R x 21x , we find R 42, We solve the equation R x C x , or 0 4x 0 2x , obtaining 0 2x , or x Substituting this value of x into the equation R x 0 4x , we find R We solve the equation R x C x or x x 20,, obtaining x 20, or x Therefore, the break-even.
Similarly, when t 50, V 0, so the line passes through 50 0. Then the slope of the line is 0 1,, given by m Using the point-slope form of the equation of a line with the point 50 0 0 , we have V 1,, 20, t 0 , or V 20,t 1,, Let V be the book value of the automobile after 5 years. Since V 34, when t 0, and V 0 when t 5, the 0 34, slope of the line L is m Using the point-slope form of an equation of a line with the point 5 0 0 5 , we have V 0 t 5 , or V t 34, If t 3, V 3 34, 13, In , t. Using the point-slope form of the equation of a line with the point 4 , we have V 12, 12, t 4 , or V 12,t 60, When t.
The slope of the line passing through the points 0 C and N S is m point-slope form of an equation of a line with the point 0 C , we have V The formula given in Exercise 32 is V S. D 0 4 , or approximately 65 mg. If we think of D as having the form D t mt b, then m 24 24 24 D is a linear function of t. In , t 10 and V 1,, 20, The graph of f passes through the points P1 0 17 5 and P2 10 10 3.
Its slope is An equation of the line is y f t 0 72t 17 5. The percentage of high school students who drink and drive at the beginning of is projected to be f The function is linear with y-intercept 1 44 and slope 0 , so we have f t b. The projected spending in will be f 9. The median age was changing at the rate of 0 3 years year.
The median age in was M The slope of the graph of f is a line with slope line is y 13 2 t 0 or y 13 2t b. The emissions cap is projected to be f 2 dioxide equivalent. The graph of f is a line through the points P1 0 0 7 and P2 20 1 2 , so it has slope equation is y. The projected annual rate of growth is the slope of the graph of f , that is, 0 billion per year, or 25 million per year. The projected number of boardings per year in is f 10 boardings per year. Since the relationship is linear, we can write F C. Using 40 the points 70 and 80 , we find that the slope of the line joining these points is 4.
Therefore, N 4T This means that the revenue is equal to the cost when units are produced and consequently the company breaks even at this point. R x 8x and C x 25, 3x , so P x when P x 0, that is, 5x 25, 0, or x Let x denote the number of units sold. Then, the revenue function R is given by R x 9x. To find the break-even point, we set R x C x , obtaining 9x 3 6x 50,, 5 4x R x 9x gives R Substituting this value of x into the equation 9 83, Thus, for a break-even operation, the firm should manufacture. The cost function associated with renting a truck from the Ace Truck Leasing Company is C1 x 25 0 5x.
The cost function associated with renting a truck from the Acme Truck Leasing Company is C2 x 20 0 6x. The cost of renting a truck from the Acme Truck Leasing Company for one day and driving it 30 miles is C 2 This answer may also be obtained by inspecting the graph of the two functions and noting that the graph of C2 x lies below that of C1 x for x Because C1 60 customer should rent the truck from Ace Trucking Company in this case. Therefore, machine II should be used in this case. Next, comparing the costs of producing units on each machine, we find. Therefore, machine II should be used in this instance.
Therefore, machine I should be used in this case. We use the equation P x. First, we find the point of intersection of the two straight lines. This gives the time when the sales of both companies are the same. Substituting the first equation into the second gives 2 3 0 4t 1 2 0 6t , so 1 1.
From the observation that the sales of Cambridge Pharmacy are increasing at a faster rate than that of the Crimson Pharmacy its trend line has the greater slope , we conclude that the sales of the Cambridge Pharmacy will surpass the annual sales of the Crimson Pharmacy in 5 12 years.
We solve the two equations simultaneously, obtaining 18t 13 4 12t 88, 30t 74 6, and t 2 , or approximately 2 5 years. The number of digital cameras sold in is given by f 0 3 05 0 6 85 6 85, or 6 85 million. The number of film cameras sold in is given by g 0 1 85 0 16 58, or 16 58 million. Therefore, more film cameras than digital cameras were sold in The sales are equal when 3 05t. A demand function defined by p f x expresses the relationship between the unit price p and the quantity demanded x. It is a decreasing function of x.
A supply function defined by p f x expresses the relationship between the unit price p and the quantity supplied x. It is an increasing function of x. Market equilibrium occurs when the quantity produced is equal to the quantity demanded. The equilibrium quantity is the quantity produced at market equilibrium. The equilibrium price is the price corresponding to the equilibrium quantity. These quantities are found by finding the point at which the demand curve and the supply curve intersect.
Next, observe that the discriminant of the quadratic equation 3x 2 4x 2 0 is 4 2 4 3 2 16 24 8 0 and so it has no real roots. Therefore, the parabola has no x -intercepts. We find b 0 upon multiplying by 2a 2a so b 0. Therefore, the points of intersection are 3 2 and 7 2 Rewriting, we have x 2 2 or 3.
Therefore, the points of intersection are 2 2 and 3 3. Factoring the left-hand side, we have x 15 x 5 the negative root, so x 5 and the corresponding value of p is p 8 5. Thus, x 5 which we discard or x 2. The corresponding value of p is The time at which the stone reaches the highest point is given by the t -coordinate of the vertex of the parabola.