Review for Exam 2. item to the quantity sold BÞ For which value of B will the corresponding revenue be a maximum?

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1 Review for Exam 2.) Suppose we are given the demand function :œ& % ß where : is the unit price and B is the number of units sold. Recall that the revenue function VÐBÑ œ B:Þ (a) Find the revenue function corresponding to the given demand function. (b) Graph VÐBÑÞ (c) Find the revenue when B œ ß B œ ) and when B œ %Þ (d) According to your graph in part (b), which B-value yields the greatest revenue? What is the greatest revenue? What is the corresponding unit price? 2.) Suppose that the function, :œ B %!, relates the selling price :Ðin dollars) of an item to the quantity sold BÞ For which value of B will the corresponding revenue be a maximum? B 3.) A quadratic function is given. (i) Express the quadratic function in graphing form by completing the square. (ii) Sketch its graph. (iii) Find its maximum or minimum value (a) 0ÐBÑ œ B B (b) 0ÐBÑ œ B B (c) 0ÐBÑ œ 'B B.) Solve the following inequalities. Express the solution in interval form. %B (a) B B )Ÿ! (b) B B (c) B (d) ÐB Ñ! (e) Ÿ ÐB ÑÐB Ñ B B B 5.) Sketch the graph of the following polynomial functions by finding the C-intercept, all B-intercepts, classifying the roots and using a table of sign(sign chart). & (a) :ÐBÑ œ % ÐB Ñ ÐB Ñ (b) :ÐBÑ œ B *B (c) :ÐBÑ œ B %B (d) :ÐBÑ œ B B )B * (e) :ÐBÑ œ Ð BÑÐB Ñ ÐB Ñ 6.) Sketch the graph of the following rational functions: B B B B B (a) <ÐBÑ œ (b) <ÐBÑ œ (c) <ÐBÑ œ B B B B B ' B %B & %B B B (d) <ÐBÑ œ (e) <ÐBÑ œ (f) <ÐBÑ œ B B B B B

2 .)A bank loaned out 2,000, part of it at the rate of 8% per year and the rest at the rate of 8% per year. If the interest received totaled 00, how much was loaned at 8%? 8.) A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price set at, the average attendance at recent games has been 2,000. A market survey indicates that for every dollar the ticket price is lowered, attendance will increase by How should the ticket price be set to maximize revenue? 9.)A farmer has %!! feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. Let B be the side parallel to the river and C be the other side. (a) Express the area as function of CÞ (b) What are the dimensions of the field that has the largest area?.) The cost G (in millions of dollars) of removing :% of the industrial and municipal pollutants discharged into a river is given by &&: Gœ!! : ß!Ÿ:!! (a) Graph G (b) Find the costs of removing 50% and 90% of pollutants. (c) According to this model would it be possible to remove 0% of the pollutants?.) An open box with a rectangular base is to be made from a sheet of copper, with dimensions of 6 inches by 8 inches, by cutting out a square from each corner and turning up the sides. Express the volume Z of the box as a function of the length of the side of the square cut from each corner. [You may leave final answer in factored form] x x x 2.) At a price of 2.50 per bushel, the annual US supply and demand for corn are 8.5 and 9.8 million bushels, respectively. When the price rises to 3.30, the supply increases to.5 million bushels while the demand decreases to.8 million bushels. (a) Assuming the price-demand and price-supply equations are linear, find equations for each. (b) Find the equilibrium point for the US corn market.

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4 SOLUTIONS.) Suppose we are given the demand function :œ& % ß where : is the unit price and B is the number of units sold and!ÿbÿ!. Recall that the revenue function VÐBÑœB:Þ (a) VÐBÑ œ B: œ B ˆ B & œ &B B % % B (b) (c) VÐÑ œ *ß VÐ)Ñ œ %ß VÐ%Ñ œ (d) According to the graph, revenue is greatest when B œ!þ To find the maximum! revenue, evaluate VÐ!Ñ œ &Þ The corresponding unit price is : œ & œ Þ& 0 % 2.) We first obtain the revenue function VÐBÑÞ Thus VÐBÑ œ Bˆ B %! œ B %!BÞ This is a quadratic function that opens down since +!Þ Thus the vertex will be the highest point on the graph. The B value of the, %! vertex is B œ œ œ '!Þ Thus selling '! units will maximize revenue. + ˆ 3.) (a) 0ÐBÑ œ B B œ B B œ ÐB BÑ œ ÐB B Ñ œ ÐB Ñ

5 Maximum Value is (b) 0ÐBÑ œ B B œðb %BÑ œðb %B %Ñ œðb Ñ Minimum value is (c) 0ÐBÑ œ 'B B œ B 'B œ ÐB 'B *Ñ * œ ÐB Ñ!

6 Maximum value is..) (a) Ò ß'Ó (b) Ð ß!Ñ Ðß_Ñ (c) Ð _ß Ñ (d) Ð _ß Ñ Ð ß Ñ Ðß _Ñ (e) Ð ß Ñ Ò ß!Ñ 5.) Sketch the graph of the following polynomial functions by finding the C-intercept, all B-intercepts, classifying the roots and using a table of sign(sign chart). (a) :ÐBÑ œ % ÐB Ñ ÐB ÑÞ C-intercept Ð!ß % Ñß B-intercepts a ß! bß aß! b is a triple root and 3 is a single root & (b) :ÐBÑ œ B *B œ B ÐB ÑÐB ÑÞ C -intercept a!ß! b, B-intercepts a!ß! bßa ß! bß aß! bþ and are single roots and! is a triple root

7 (c) :ÐBÑ œ B %B œ BÐB ÑÐB ÑÞ C-intercept Ð!ß!Ñ, B-intercepts Ð!ß!Ñß Ð ß!Ñß Ðß!ÑÞ All roots are single roots (d) :ÐBÑœB B )B *œðb ÑÐB ÑÐB Ñ C-intercept Ð!ß *ÑÞ B-intercepts Ð ß!Ñß Ðß!Ñß Ð ß!ÑÞ All roots are single roots

8 (e):ðbñ œ Ð BÑÐB Ñ ÐB Ñ. C-intercept Ð!ß Ñ, B-intercepts Ðß!Ñß Ð ß!Ñß Ðß!ÑÞ is a single root. is a double root Þ is a triple root ) Sketch the graph of the following rational functions: (a) <ÐBÑ œ B B B B B B (b) <ÐBÑ œ c) <ÐBÑ œ 28 B B B B '

9 B %B & %B (d) <ÐBÑ œ (e) <ÐBÑ œ Hole at ˆ!ß B B B B (f) <ÐBÑ œ B B B ) Let B be the amount invested in the 8% account. Then!!! B represents the amount invested at 8%. Since MœT<>, the interest from the 8% account after year is Þ!)BÞ The interest from the )% account after year is Þ) a!!! BbÞ If we add these two interest amounts it should total 00. Hence we solve: Þ!)B Þ) a!!! Bb œ!!! Þ!)B '!!Þ)B œ!!!!þb œ '! B œ ß '!! Thus,600 was invested at 8% and 00 was invested at 8%. 8.) Let B be the number of tickets produced and : be the price per ticket Þ First we create the demand equation. Note that ÐBß :Ñ is a point on this linear graph. We know that when B œ (!!!, : œ!þ If : œ *, then B œ!!!!. Thus the slope of the demand equation is

10 ?:? B!!! and the demand equation is given by!!! œ : œ B * or B œ &(!!!!!!:Þ The revenue is V œb:þ So the revenue function is VÐ:Ñ œ :Ð&(!!!!!!:Ñ œ &(!!!:!!!: This is a quadratic function with + œ!!! and, œ &(!!!, so the maximum value occurs when, &(!!! Bœ + œ Ð!!!Ñ œ*þ& This shows that the revenue is maximized when B œ *Þ&Þ So the price should be se at )(a)We want to express area EœBCas a function of CÞThe constraint in this problem is the perimeter. Note that no fencing in needed for one of the sides. Thus the perimeter is B C œ %!! or B œ %!! CÞ Subsituting this into the objective function yields EÐCÑ œ CÐ%!! CÑ œ %!!C C Þ (b) Since +œ, function will yield a maximum. To find it, use the vertex formula:, %!! C œ + œ Ð Ñ œ '!! If C œ '!!ß then B œ %!! Ð'!!Ñ œ!!þ Thus the area is maximized with a rectangle 200' 600'..) Z ÐBÑ œ Ð6/8>2ÑÐA3.>2ÑÐ2/32>Ñ Z ÐBÑ œ BÐ' BÑÐ) BÑ 2.) (a) Let : be the price per bushel and Bbe the quantity demanded. The price demand equation is of the form : œ B,Þ Note that when : œ Þ&!ß demand B œ *Þ) and when : œ Þ!, demand B œ (Þ)Þ There the slope of the price demand equation is?: Þ! Þ&! œ œ œ!þ%þusing point slope form the equation is? B (Þ) *Þ) : Þ&! œ!þ% ab *Þ) b : œ!þ%b 'Þ% For the price supply function, let : be the price per bushel and B the quantity supplied. Note that when : œ Þ&!ß supply B œ )Þ& and when : œ Þ!, supply B œ!þ&þ There the slope of the price demand equation is?: Þ! Þ& œ œ œ!þ%þ Using point slope form the equation is? B!Þ& )Þ& : Þ&! œ!þ% ab )Þ& b

11 : œ!þ%b!þ* (b) Recall that the equilibrium point is when the price supply and price demand equations intersect. Therefore!Þ%B!Þ* œ!þ%b 'Þ%!Þ)B œ (Þ B œ *Þ& : œ!þ% a*þ& b!þ* œ Þ(' The equilibrium point is 9.5 million bushels at 2.6 per bushel.