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Applied Mathematics for Business and Economics

87 Pages · 2010 · 1.79 MB · English

  • Applied Mathematics for Business and Economics













    Applied Mathematics


    for


    Business and


    Economics









    Norton University


    Year 2010



    Lecture Note













    Applied Mathematics for



    Business and Economics Contents



    Page


    Chapter 1 Functions


    1 Definition of a Function (of one variable) .........................................................1


    1.1 Definition ..................................................................................................1


    1.2 Domain of a Function ...............................................................................1


    1.3 Composition of Functions .........................................................................2



    2 The Graph of a Function ....................................................................................3



    3 Linear Functions ................................................................................................5


    3.1 The Slope of a Line ...................................................................................5


    3.2 Horizontal and Vertical Lines ...................................................................6


    3.3 The Slope-Intercept Form .........................................................................6


    3.4 The Point-Slope Form ...............................................................................6



    4 Functional Models .............................................................................................8


    4.1 A Profit Function ......................................................................................8


    4.2 Functions Involving Multiple Formulas ...................................................8


    4.3 Break-Even Analysis ................................................................................9


    4.4 Market Equilibrium .................................................................................11


    Chapter Exercises ...................................................................................................12



    Chapter 2 Differentiation: Basic Concepts


    1 The Derivative


    Definition .........................................................................................................19



    2 Techniques of Differentiation ..........................................................................20


    2.1 The Power Rule.......................................................................................20


    2.2 The Derivative of a constant ...................................................................21


    2.3 The Constant Multiple Rule ....................................................................21


    2.4 The Sum Rule .........................................................................................21


    2.5 The Product Rule ....................................................................................21


    2.6 The Derivative of a Quotient ..................................................................21



    3 The Derivative as a Rate of change .................................................................22


    3.1 Average and Instantaneous Rate of Change ...........................................22


    3.2 Percentage Rate of Change .....................................................................23



    4 Approximation by Differentials; Marginal Analysis .......................................23


    4.1 Approximation of Percentage change .....................................................24


    4.2 Marginal Analysis in Economics ............................................................25


    4.3 Differentials ............................................................................................27



    5 The Chain Rule ................................................................................................27



    6 Higher-Order Derivatives ................................................................................29


    6.1 The Second Derivative ............................................................................29


    6.2 The nth Derivative ...................................................................................30 7 Concavity and the Second Derivative Test ......................................................30



    8 Applications to Business and Economics ........................................................34


    8.1 Elasticity of Demand ..................................................................................... 34


    8.2 Levels of Elasicity of Demand ................................................................36


    8.3 Elasticity and the Total Revenue ............................................................36


    Chapter Exercises ...................................................................................................38



    Chapter 3 Functions of Two Variables


    1 Functions of Two Variables .............................................................................49



    2 Partial Derivatives ............................................................................................50


    2.1 Computation of Partial Derivatives ........................................................50


    2.2 Second-Order Partial Derivatives ...........................................................52



    3 The Chain Rule; Approximation by the Total Differential ..............................53


    3.1 Chain Rule for Partial Derivatives ..........................................................53


    3.2 The Total differential ..............................................................................55


    3.3 Approximation of Percentage Change ....................................................56



    4 Relative Maxima and Minima .........................................................................56



    5 Lagrange Multipliers ........................................................................................59


    5.1 Contrained Optimization Problems.........................................................59


    5.2 The Lagrange Multiplier .........................................................................61



    Chapter Exercises ...................................................................................................62



    Chapter 4 Linear Programming (LP)


    1 System of Linear Inequalities in Two Variables ..............................................72


    1.1 Graphing a Linear Inequality in Two Variables .....................................72


    1.2 Solving Systems of Linear Inequalities ..................................................73



    2 Geometric Linear Programming ......................................................................74



    Chapter Exercises ....................................................................................................... 77



    Bibliography ............................................................................................................. 81




































    This page is intentionally left blank. Lecture Note Function



    Chapter 1


    Functions




    1 Definition of a Function


    1.1 Definition


    Let D and R be two sets of real numbers. A function f is a rule that matches each


    number x in D with exactly one and only one number y or f (x) inR. D is called the


    domain of f and R is called the range of f . The letter x is sometimes referred to as


    independent variable and y dependent variable.



    Examples 1:



    Let f(xF)in=dx3 −2x2 +3x+100. f (2).



    Solution:


    f (2)=23 −2×22 +3×2+100=106



    Examples 2


    A real estate broker charges a commission of 6% on Sales valued up to $300,000. For


    sales valued at more than $ 300,000, the commission is $ 6,000 plus 4% of the sales


    price.


    a. Represent the commission earned as a function R.


    b. Find R (200,000).


    c. Find R (500,000).


    Solution


    ⎧0.06x for 0≤ x≤300,000


    a. R(x)=⎨


    ⎩0.04x+6000 for x>300,000


    b. Use R(x)=0.06xsince 200,000<300,000


    R(200,000)=0.06×200,000=$12,000


    c. Use R(x)=0.04x+6000 since 500,000>300,000


    R(500,000)=0.04×500,000+6000=$26,000


    1.2 Domain of a Function


    The set of values of the independent variables for which a function can be evaluated is


    called the domain of the function.


    D ={x∈(cid:92)/∃y∈(cid:92),y = f (x)}


    Example 3


    Find the domain of each of the following functions:


    1


    a. f (x)= , b. g(x)= x−2


    x−3


    Solution


    a. Since division by any real number except zero is possible, the only value of x


    1


    for which f (x)= cannot be evaluated isx=3, the value that makes the


    x−3


    denominator of f equal to zero, or D =(cid:92)−{3}.


    1


    Lecture Note Function



    b. Since negative numbers do not have real square roots, the only values of x for


    which g(x)= x−2can be evaluated are those for whichx−2 is nonnegative,


    that is, for whichx−2≥0 or x≥2 or D =[2,+∞).



    1.3 Composition of Functions


    The composite function g⎡h(x)⎤is the function formed from the two functionsg(u)


    ⎣ ⎦


    andh(x)by substituting h(x)for u in the formula forg(u).



    Example 4


    Find the composite function g⎡h(x)⎤if g(u)=u2 +3u+1andh(x)= x+1.


    ⎣ ⎦



    Solution


    Replace u by x+1 in the formula for g to get.


    g⎡h(x)⎤ =(x+1)2 +3(x+1)+1= x2 +5x+5


    ⎣ ⎦



    Example 5


    An environmental study of a certain community suggests that the average daily level


    of carbon monoxide in the air will be C(p)=0.5p+1parts per million when the


    population is p thousand. It is estimated that t years from now the population of the


    community will beP(t)=10+0.1t2thousand.


    a. Express the level of carbon monoxide in the air as a function of time.


    b. When will the carbon monoxide level reach 6.8 parts per million?



    Solution


    a. Since the level of carbon monoxide is related to the variable p by the equation.


    C(p)=0.5p+1


    and the variable p is related to the variable t by the equation.


    P(t)=10+0.1t2


    It follows that the composite function


    C⎡P(t)⎤ =C(10+0.1t2)=0.5(10+0.1t2)+1=6+0.05t2


    ⎣ ⎦


    expresses the level of carbon monoxide in the air as a function of the variable t.



    b. Set C⎡P(t)⎤equal to 6.8 and solve for t to get


    ⎣ ⎦


    6+0.05t2 =6.8


    0.05t2 =0.8



    t2 =16


    t =4


    That is, 4 years from now the level of carbon monoxide will be 6.8 parts per million.






    2


    Lecture Note Function



    2 The Graph of a Function


    The graph of a function f consists of all points (x,y) where x is in the domain of f and


    y = f (x).


    How to Sketch the Graph of a Function f by Plotting Points



    1 Choose a representative collection of numbers x from the domain of f and


    construct a table of function values y = f (x)for those numbers.


    2 Plot the corresponding points(x,y)


    3 Connect the plotted points with a smooth curve.



    Example 1


    Graph the functiony = x2. Begin by constructing the table.


    x −2 −1 0 1 2


    y = x2 4 1 0 1 4



    y




    4




    3





    -2 -1 1 2


    x




    Example 2 Graph the function


    ⎧2x, if 0≤ x<1



    ⎪2


    f (x)=⎨ , if 1≤ x<4


    x



    ⎪3, if x≥4



    Solution


    When making a table of values for this function, remember to use the formula that is


    appropriate for each particular value of x. Using the formula f (x)= 2xwhen0≤ x<1


    , the formula f (x)=2 xwhen1≤ x<4and the formula f (x)=3whenx≥4, you can


    compile the following table:



    x 0 1/2 1 2 3 4 5 6


    f (x) 0 1 2 1 2/3 3 3 3



    Now plot the corresponding point(x, f (x))and draw the graph as in Figure.





    3


    Lecture Note Function




    y










    3



    2



    1


    ½ 1 2 3 4 5 6 x



    Comment


    The graph of y = f (x)=ax2 +bx+c is a parabola as long asa≠0. All


    parabolas have a U shape, and y = f (x)=ax2 +bx+copens either up


    (ifa>0) or down (ifa<0). The “Peak” or “Valley” of the parabola is called


    b


    its vertex, and in either case, the x coordinate of the vertex isx=− .


    2a



    Note that to get a reasonable sketch of the parabolay =ax2 +bx+c, you need


    only determine.


    1 The location of the vertex


    2 Whether the parabola opens up (a>0) or down (a<0)


    3 Any intercepts.



    Example 3


    For the equation y = x2 −6x+4


    a. Find the Vertex.


    b. Find the minimum value for y.


    c. Find the x-intercepts.


    d. Sketch the graph.


    Solution


    −6


    a. We havea =1,b=−6, and c =4. The vertex occurs atx =− =3


    2×1


    Substituting x = 3 givesy =32 −6×3+4=−5. The vertex is(3,−5).


    b. Since a=1>0and the parabola opens upward, y =−5is the minimum


    value for y.


    c. The x-intercept are found by setting x2 −6x+4=0and solving for x


    6± 36−16


    x= =3± 5


    2


    d. The graph opens upward becausea=1>0.The vertex is(3,−5)


    The axis of symmetry isx=3.


    The x-intercepts arex =3± 5.





    4



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