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Math Help - Calculus: Analysis of Functions and Curve Sketching

  1. #1
    Newbie Mscscgrad2011's Avatar
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    Exclamation Calculus: Analysis of Functions and Curve Sketching

    Can anyone please help me complete the following given tasks for the functions :
    a. f (x) = 2x^3 - 3x^2 + 12x +9
    b. f (x) = x^2-1/x^2-16
    c. f(x) = x(x-1)^3
    d. f(x) = x-1/x^2-4

    1. Find the zeros of the function. Find the values at which the function is undefined.
    2. Find the domain of the function.
    3. Find the horizontal and vertical asymptotes.
    4. Find the x- and y-intercepts.
    5. Find the limits at infinity and negative infinity to determine the function’s end behavior.
    6. Find the first derivative and the second derivative.
    7. Find the critical numbers and the possible points of inflection.
    8. Use the x-intercepts and vertical asymptotes to divide the x-axis into open intervals. Complete the following table:
    Intervals

    Test Value

    Sign of f(x)
    + or -
    Conclusion
    Above or below x-axis

    9. Use the critical numbers to divide the x-axis to into open intervals. Complete the following table:
    Intervals

    Test Value

    Sign of f(x)
    + or -
    Conclusion
    Increasing or Decreasing


    10. Use the possible points of inflection to divide the x-axis to into open intervals. Complete the following table:
    Intervals

    Test Value

    Sign of f(x)
    + or -
    Conclusion
    Concave Up or Concave Down


    11. Determine the relative extrema and points of inflection based on analysis of your charts.
    12. Generate a graph of the function using your graphing utility.

    P.S. CAN U SHOW STEPS PLEASE. THANKS IN ADVANCE.
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Mscscgrad2011 View Post
    Can anyone please help me complete the following given tasks for the functions :
    a. f (x) = 2x^3 - 3x^2 + 12x +9
    b. f (x) = x^2-1/x^2-16
    c. f(x) = x(x-1)^3
    d. f(x) = x-1/x^2-4

    1. Find the zeros of the function. Find the values at which the function is undefined.
    2. Find the domain of the function.
    3. Find the horizontal and vertical asymptotes.
    4. Find the x- and y-intercepts.
    5. Find the limits at infinity and negative infinity to determine the functionís end behavior.
    6. Find the first derivative and the second derivative.
    7. Find the critical numbers and the possible points of inflection.
    8. Use the x-intercepts and vertical asymptotes to divide the x-axis into open intervals. Complete the following table:
    Intervals

    Test Value

    Sign of f(x)
    + or -
    Conclusion
    Above or below x-axis

    9. Use the critical numbers to divide the x-axis to into open intervals. Complete the following table:
    Intervals

    Test Value

    Sign of f(x)
    + or -
    Conclusion
    Increasing or Decreasing


    10. Use the possible points of inflection to divide the x-axis to into open intervals. Complete the following table:
    Intervals

    Test Value

    Sign of f(x)
    + or -
    Conclusion
    Concave Up or Concave Down


    11. Determine the relative extrema and points of inflection based on analysis of your charts.
    12. Generate a graph of the function using your graphing utility.

    P.S. CAN U SHOW STEPS PLEASE. THANKS IN ADVANCE.
    Wow that is a lot of problems, and you didn't show any work/attempt. Would you like us to write your term paper for you too?

    Please show at least a little work...
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  3. #3
    Newbie Mscscgrad2011's Avatar
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    Exclamation

    Sorry, I am struggling to start off (a): f(x) 2x^3 - 3x^2 + 12x + 9.
    I set 2x^3 - 3x^2 + 12x + 9 = 0 ( from there I am lose)

    for (b) f(x) = x^2 -1/x^2-16
    x^2 -1/X^2-16 = 0
    zero: x^2-1= 0
    x = +/- 1

    domain: x^2 - 16 = 0
    x = +/- 4
    (- infinity, 4) u (-4, 4) u (4, infinity)

    but c and d is confusing
    I just need help starting them off
    Thank you....
    Last edited by Mscscgrad2011; April 21st 2010 at 06:42 PM.
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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Mscscgrad2011 View Post
    Sorry, I am struggling to start off (a): f(x) 2x^3 - 3x^2 + 12x + 9.
    I set 2x^3 - 3x^2 + 12x + 9 = 0 ( from there I am lose)

    for (b) f(x) = x^2 -1/x^2-16
    x^2 -1/X^2-16 = 0
    zero: x^2-1= 0
    x = +/- 1

    domain: x^2 - 16 = 0
    x = +/- 4
    (- infinity, 4) u (-4, 4) u (4, infinity)

    but c and d is confusing
    I just need help starting them off
    Thank you....
    It's understandable that you'd have trouble finding the roots of f(x) defined in part (a). This is a cubic polynomial, and although the general cubic has a solution, it's not something taught in most school curricula. Using the rational root theorem reveals that there are no rational roots. Just use a graphing calculator. The root is approximately equal to -0.616119.

    The domain is all values for which f(x) is defined. If f(x) is defined for all real numbers, then the domain is the set of real numbers.

    The horizontal/vertical asymptotes question deals with limits as x approaches a certain value, or as x approaches positive or negative infinity.

    The f(x) defined in (b), is it supposed to be f (x) = x^2-1/(x^2-16)? The parentheses are very important. If so then your domain is fine except that you missed a negative sign.

    Of course you'll have to know how to do derivatives.

    Is that enough help to get you started?
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