MATH 1110 UNIFORM FINAL - FALL 1998

  1. Rewrite the following as a single fraction: . Show all work.




  2. Consider the two points P [ 13 , 6 ] and Q [ 5, e ]. Write an equation for the line determined by the points P and Q. Do not simplify.


  3. Solve the inequality . Write the solution in interval notation. Show all work.


  4. Given and .


    1. What is the domain of f ?


    2. Expand an expression for the composition




  5. In Figure 1 a plot of h(x) = x 2 is shown. The function f(x) = - ( x + 1 ) 2 + 2 is a transformation of h(x). On the coordinate system shown on the computer screen in Figure 1 do the following:
    1. sketch and label the graph of f(x),
    2. mark and label the vertex for f(x), and
    3. sketch and label the line of symmetry for f(x).




  6. Consider the invertible function . Compute its inverse . Show all work.


  7. Given .


    1. Identify any horizontal or slant asymptotes


    2. Identify the vertical asymptotes


    3. Find the x-intercept(s) of


    4. Find the y-intercept of


  8. Given :
    1. What is the degree of ?


    2. What is the leading coefficient of ?


    3. What is the leading term of ?
  9. The Rocky State Park deer population after t years is modeled by the function . A complete graph of this model is shown in Figure 1.
    1. What is the beginning population of deer?
    2. What is the maximum number of deer modeled?
    3. The model indicates that the deer population does not continue to grow unchecked after the first ten or so years.
      1. What conditions might cause the growth in the deer population to level off as the model predicts? Use the back of the sheet if necessary.
      2. Why is this model believable or unbelievable? Use the back of the sheet if necessary.



  10. On the coordinate axes shown in Figure 3 sketch a graph of the parabola passing through the point ( 2 , 1 ) with a vertex ( -1 , 3 ).

  11. In our text book we have the following definition: The value S of an annuity with n equal payments of R dollars at an interest rate i per compounding period is . Write an expression for the value S of an annuity for Kelly in which she makes weekly payments of $50.00 at an interest of 6% annually for a period of 10 years. Do not simplify.

  12. On a lab examination Jessie was asked to plot a complete graph of the polynomial P(x) = x 4 + 9x 3 - 34x 2 + 24x . Jessie adjusted the window so it looked like that shown in Figure 4 and printed it. Explain why Jessie's plot is not a complete graph of this polynomial.



  13. Solve for x: log ( x + 1 ) = log ( x ) + 1
  14. Solve for x: 2 ln x - 3 = 4
  15. Solve for x: e -5 x = 10
  16. Show that is a rational function.
  17. The total resistance R of two resisters connected in parallel with resistances of R 1 and R 2 is given by the expression . If the desired total resistance is to be R = 9 ohms and R 1 = 10 ohms, then what must be the value selected for R 2. Show all work.
  18. A cup of coffee has cooled from 96ºC to 67ºC after 10 minutes in a room at 20ºC.
    Newton's law of cooling states TEMPERATURE(t) = tm + ( t0 - tm ) ê - k t where
    TEMPERATURE(t) = temperature at time t
    tm = temperature of surrounding medium
    t0 = initial temperature of object
    k = heat exchange constant
    1. Use the given information to write an expression to compute the value of k.
    2. A decimal approximation for k is k = 0.048. Write an expression which can and be entered into a calculator or a computer to compute the time t when the coffee is 30ºC?
  19. The number of students infected with influenza (bovine type X) at Metronome Music College after t days is modeled by the function .
    1. What was the initial number of infected students?
    2. Solve this expression for the time t when 400 of the 5000 students in the student body are infected.
  20. Solve the following equation for x: .
    Hint: first multiply both sides of the equation by e x to create a quadratic expression in e x. Use completion of the square or the quadratic formula to solve for e x.
  21. Given the polynomial f(x) = x 5 - 2 and the polynomial h(x) = x 2 - 1. Show all work used to divide and find:
    1. the polynomial quotient q(x),
    2. the polynomial remainder r(x), and
    3. write the polynomial f(x) = q(x) h(x) + r(x).
  22. Consider all rectangles of area 100 square miles. If the length of the rectangle is x and the width of the rectangle is y, express the perimeter as a function of x, that is write an expression for P(x). Show all work.
  23. Find a partial fraction decomposition of . Show all work.
  24. This is a row reduced matrix for a system of linear equations .
    1. If the variables are named x, y, z, and w, then
      1. x = _____ ,
      2. y = _____ ,
      3. z = _____ , and
      4. w = _____ .
    2. What is a correct way of writing the solution of this system as a single expression?
  25. Write an augmented matrix for the system of equations
    8p - 3q + r = 5
    p + q + r + s + t = 3
    2p + 7t = -1
    s + 7t = -1
    q + r - 3s = 7    .


  26. Given a coefficient matrix , a variable matrix , and a constant matrix , what is a system of linear equations associated with these?