Homework 1 1. Calculus. Homework 1 Due Date: September 26 (Wednesday) x x 280. R = x. C = 95x

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1 Homework 1 1 Calculus Homework 1 Due Date: September 26 (Wednesday) 1. A doughnut shop sells a dozen doughnuts for $4.50. Beyond the fixed cost of $220 per day, it costs $2.75 for enough materials and labor to produce each dozen doughnuts. During the month of January, the daily profit varies between $60 and $270. Between what levels (in dozens) do the daily sales vary? Revenue: R = 4.5x. Cost: C = 2.75x Profit = P = R C = 1.75x 220. So x x 280 Therefore, the daily donut sales vary between 160 dozens donuts per day and 280 dozens donuts per day. 2. The revenue for selling x units of a product is x and the cost of producing x unit is R = x C = 95x To obtain a profit, the revenue must be greater than the cost. For what values of x will this product return a profit? To make a profit, we must have R > C x > 95x x > x 36. So this product will return a profit if x 36 units.

2 2 Calculus H Fall A utility company has a fleet of vans. The annual operating cost C (in dollars) of each van is estimated to be C = 0.35m where m is the number of miles driven. What number of miles will yield an annual operating cost that is less than $13,000? C = 0.35m < 13, m < 10, 500 m < 30, 000 So, the number of miles driven must be less than 30, The acceptable weights for a 20-once cereal box are given by x 20 < 0.75 where x is measured in ounces. Determine the high and low weights for the cereal box. x 20 < < x 20 < < x < The lowest and highest acceptable weights for a 20-ounce cereal box are ounces and ounces. 5. The American Kennel Club has developed guidelines for judging the features of various breeds of dogs. To not receive a penalty, the guidelines specify that the weights for male collies must satisfy the inequality w where w is the wights (in pounds). Determine the interval on the real number line in which these wights lie? w w w 75. the guidelines specify that the weights for male collies lies between 50 pounds and 65 pounds. 6. The period of pendulum is L T = 2π 32 where T is the period (in seconds) and L is the length (in feet) of the pendulum. Find the period of a pendulum whose length is 4 feet. L 4 1 T = 2π 32 = 2π 32 = 2π 8 = π 2.22 seconds. 2

3 Homework Find all real zeros of the polynomial. (a) x 2 5x + 6 (b) 3x 2 + 5x + 2 (c) x 3 x 2 4x + 4 (d) 2x 3 + x 2 + 6x + 3 (a) x 2 5x + 6 = 0 (x 2)(x 3) = 0 x = 2, 3. (b) 3x 2 + 5x + 2 = 0 (3x + 2)(x + 1) = 0 x = 2 3, 1. (c) x 3 x 2 4x + 4 = 0 x 2 (x 1) 4(x 1) = 0 (x 1)(x 2 4) = 0 x = 1, ±2. (d) 2x 3 + x 2 + 6x + 3 = 0 x 2 (2x + 1) + 3(2x + 1) = 0 (2x + 1)(x 2 + 3) = 0 x = 1 2. Note: (x 2 + 3) has no real roots. 8. The profit P (in dollars) from sales is given by P = 200x x 3800 where x is the number of units sold per day (in hundreds). Determine the interval for x such that the profit will be greater than $1000.

4 4 Calculus H Fall x x 3800 > > 200x x > 200(x 4)(x 6). The roots are 4 and 6. By testing the points inside and outside the interval [4, 6], we found the expression is defined when 4 < x < Simplify the expression. (a) 2 t t 1 + t (b) x+1 x x x+1 2(x + 1) (c) (d) (a) (b) x 2 (2x + 3) 3/2 + 2x (2x + 3) 1/2 x 2(3 + x 2 ) + 3 3/2 (3 + x 2 ) 1/2 2 t t 1 + t = 2 t t 1 + t t t (2 t) 2(1 + t) = t = 3t t x+1 x x x+1 = = 2(x + 1) ( x + 1 x x x(x + 1) 3/2 ) x x x (x + 1)

5 Homework 1 5 (c) (d) x 2 (2x + 3) + 2x 3/2 (2x + 3) 1/2 x 2 2x(2x + 3) = + (2x + 3) 3/2 (2x + 3) 3/2 3x(x + 2) = (2x + 3) 3/2 x 2(3 + x 2 ) 3/2 + 3 (3 + x 2 ) 1/2 = x (3 + x2 ) 2(3 + x 2 ) 3/2 = 6x2 x (3 + x 2 ) 3/2 10. A retailer has determined that the cost C (in dollars) of ordering and storing x units of a product is 900, 000 C = 6x + x (a) Write the expression for cost as single fraction. (b) Which order size should the retailer place: 240 units, 387 units, or 480 units? Explaining your reasoning. (a) C = 6x + (b) When x = 240 units. 900, 000 x = 6x , 000 x = 6(x , 000). x When x = 387 units. When x = 480 units. C = 6( , 000) 240 C = 6( , 000) 387 C = 6( , 000) The retailer should place an order of 387 units, because this cost is the lowest.

6 6 Calculus H Fall Find the value(s) of y such that the distance between the points is 8. (a) (0, 0), (3, y) (b) (5, 1), (5, y) (a) (b) d = (3 0) 2 + (y 0) 2 = y2 = 8 y 2 = 55 y = ± 55. d = (5 5) 2 + (y 5) 2 = 8 (y 1)2 = 8 (y 1) 2 = 64 y = 7, The table shows the number of ear infections treated by doctors at HMO clinics of three different sizes: small, medium, and large. Number of doctors Cases per small clinic Cases per medium clinic Cases per large clinic (a) On the same coordinate plane, show the relationship between doctors and treated ear infections using three line graphs, where the number of doctors is on the horizontal axis and the number of ear infections treated is on the vertical axis. (b) Compare the three relationships. (a) aa

7 Homework 1 7 (b) The larger the clinic, the more patient a doctor can treat. 13. Sketch the graph of the equation. (a) y = x 2 3 (b) y = x + 1 (c) y = 1 x 3 (d) x = 4 y 2 (a) a (b) a

8 8 Calculus H Fall 2012 (c) a

9 Homework 1 9 (d) a

10 10 Calculus H Fall Find the points of intersection (if any) of the graphs of the equations. (a) y = x + 7, y = 1.5x 8 (b) y = x 3, y = 2x. (c) y = x 3 2x 2 + x 1, y = x 2 + 3x 1 (a) Set the two equation equal to each other. x + 7 =1.5x 8 5x = 30 x =6 Substitute x = 6 into one of the equations. y = = 1. The point of intersection is (6, 1). (b) Set the two equation equal to each other. x 3 =2x x(x 2 2) =0 x =0, ± 2.

11 Homework 1 11 The corresponding y values are 0, 2 2, and 2 2, so the points of intersection are (c) Set the two equation equal to each other. (0, 0), ( 2, 2 2), ( 2, 2 2). x 3 2x 2 + x 1 = x 2 + 3x 1 x(x + 1)(x 2) =0 x =0, 1, 2. The corresponding y values are -1, -5, and 1, so the points of intersection are (0, 1), ( 1, 5), (2, 1). 15. A 2010 Honda Accord costs $28,695 with a gasoline engine. A 2010 Toyota Camry costs $29,720 with a hybrid engine. The Accord gets 21 miles per gallon of gasoline and the Camry gets 34 miles per gallon of gasoline. Assume that the price of gasoline is $ (a) Show that the cost C g (in dollors) of driving the Honda Accord x miles is C g = 28, x 21 and the cost C h (in dollars) of driving the Toyota Camry x miles is C h = 29, x 34 (b) Find the break-even point. That is, find the mileage at which the hybrid-powered Toyota Camry becomes more economical than the gasoline-powered Honda Accord. (a) The cost C g to drive x miles is the cost of the car itself plus the cost of gasoline per mile. The cost of gasoline per gallon divided by the number of gallons per mile. C g = 28, x. 21 Similarly, the cost C h to drive x miles is the cost of the car itself plus the cost of gasoline per mile. C h = 29, x 34 (b) To find the break-even point, set the cost equations to each other. 28, x = 29, x x = 731, 850 x 20, 705 miles.

12 12 Calculus H Fall The demand and supply equation for an MP3 player are given by p = x p = x Demand Equation Supply Equation where p is the price (in dollars) and x represents the number of units (in hundreds of thousands). Find the equilibrium point for this market. Equilibrium point is (5, 115) x = x x = The graph shows the cost and revenue equations for a product. (a) For what numbers of units sold is there a loss for the company? (b) For what number of units sold does the company break even? (c) For what numbers of units sold is there a profit for the company? (a) If less than 100,000 units are sold, the company losses money. (b) If 100,000 units are sold, the company breaks even. (c) If more than 100,000 units are sold, the company make a profit.