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珀斯代寫 ETF2700/ETF9700 Mathematics for Business WEEK 0

    珀斯代寫 ETF2700/ETF9700  Mathematics for Business WEEK 05 Tutorial Exercises 
    (Week 04 lecture content)
     
    QUESTION 1 
    TB Progress Exercises 8.1: p433, #4, #12, 18.
    Integrate each of the following:
    4.                  12.                   18. 
     
    QUESTION 2
    TB Progress Exercises 8.2: p439, #7, #17, #26.
    Integrate each of the following:
    7.    dx           17.        dQ            26.   dx
     
    QUESTION 3
    TB Progress Exercises 8.3: p445, #18, #25.
    18.  Evaluate  correct to 4 decimal places.
     
    25. 
    (a)   Sketch f(Q), the function to be integrated, over the interval of integration.
    (b)   Calculate the net area enclosed over the interval of integration correct to 5 decimal places.
    (c)   Calculate the areas above and below the axis separately correct to 5 decimal places.
    珀斯代寫 ETF2700/ETF9700  Mathematics for Business WEEK 05 Tutorial Exercises 
     
    QUESTION 4
    TB Progress Exercises 8.5: p457, #5, #11.
    Determine the general solution for the following differential equation:
    5.   
     
    Find the particular solution for the following differential equation:
    11.  , given P = 10 when Q = 0.
     
    QUESTION 5
    TB Progress Exercises 8.6 p461, #7.
    7. A firm expects its annual income, (Y), and expenditure to change according to the equations
    Y(t) = 250 + 0.5t  and  E(t) = t2 – 17t + 287.5 (t is time in years).
    (a)   Sketch the graphs of income and expenditure on the same diagram over the interval 0 ≤ t ≤ 20
    and confirm the break-even points algebraically.
    (b)   Use integration to calculate the net savings (total income – total expenditure) over the
    intervals   (i)    0 ≤ t ≤ 2.5            (ii)   2.5 ≤ t ≤ 15
     

     
    QUESTION 6
    TB Progress Exercises 8.7: p465, #3, #5.
    Find the general solution and particular solutions of the following differential equations.
    3.    , given N = 812 at t = 2.           5.    , given y = 50 at t = 0.
     
    QUESTION 7
    TB Progress Exercises 8.8: p469, #4, #7, #8. 
    4.         A capital investment (I) depreciates continually at the rate of 5% annually, 
    (a)   If the investment is valued at $12,000 initially (i.e., at t = 0), calculate the value of the
    investment after 5.5 years.
    (b)   How many years will it take for the value of the investment to fall to $5,000?
     
    7.         A hospital is considering the installation of two alternative heating systems. The rate of increase in costs for each system is given by the equations

    (a)   Derive the expressions for the total cost of each system at any time t given that cost = $50,000
    at t = 0.
    (b)   Graph both cost functions from t = 0 to t = 10 years.
    Which system is more cost-effective in the long run?  Give reasons.
     
    8.    The price elasticity of demand is given by ε = .
    (a)   Write down the differential equation in terms of P and Q.
    (b)   Find the equation of the demand function if P = 20 and Q = 0.
     
     
     
     
     
    Matlab Tips
    ·         Names in Matlab are case-sensitive.
    ·         Matlab uses the same conventions as Excel for calculations and algebraic expressions,
    eg, x2 « x^2;  4´3 « 4*3;  e1.68 « exp(1.68);  log101000 « log10(1000);  loge23 « log(23).
    ·         syms x    declares x to be a symbolic variable, eg, for integration.
    ·         int    to integrate an algebraic expression.
    eg,      syms x
        int(x^2)         à   x^3/3
    int(exp(2*x))      à   exp(2*x)/2
    ·         int    to calculate a definite  integral.
    eg,      integrate f(x) = 3x; x = 0 to 5.
    syms x
        f = 3*x
    int(f, x, 0, 5)       à 75/2
    ·         clear       removes all variables from current workspace, releasing them from system memory.
    ·         clc         clears all input and output from the Command Window display, giving a clean screen.
     

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