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Math Help - I need help on a few things

  1. #1
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    I need help on a few things

    1. The limit, as x tends to 2, of the function y=\frac{x^2 + bx - 28}{x - 2} exists, and is equal to 16. What is the value of the whole number coefficient b?

    2. The first derivative of y = (6x^3 + 11)^9 is of the form ax^2(6x^3 + 11)^n , where a is a positive number. What is the value of the whole number coefficient a?

    3. For the ellipse 4x^2 + 9y^2 = 724, the slope of the tangent at the point (10, -6) is b. Correct to the nearest hundredth, the value of b is ______________.

    4. The number I is defined as \int_{1}^{2} 4~sin~x~cos~xdx. The value of I, correct to the nearest hundredth, is ______________.

    5. A spring has a motion that can be modelled by the differential equation \frac{d^2 y}{dt^2} = -5y. One solution of this equation is y = 4~cos~kt, where k is a positive number to be determined. What is the value of k, correct to two decimal places?

    6. An LRT train travels for 120 s between two stations. It accelerates for 30 s, maintains a constant velocity for 70 s, and brakes to a stop in 20 s. The velocity function v(t), with velocity measured in m/s, is defined as follows:
    v(t) = 20sin[\frac{\pi t}{60}]~~~~~~~~0 \geq t \geq 30
    v(t) = 20 ~~~~~~~30 \geq t \geq 100
    v(t) = -0.05(t - 100)^2 + 20~~~~~~~~100 \geq t \geq 120

    a. Sketch a graph of the velocity as a function of time, assuming the velocity function to be continuous at t = 30 and t = 100. (Be sure to label your axes.)

    b. Determine the total distance travelled by the train in the 120-second time interval.

    4. Let the function f be defined as f(x) = \frac{tan~x - sin~x}{x^3}.
    a. Use a suitable numerical value of x to estimate the limit, if it exists, of f(x) as x approaches zero.
    b. Use trigonometric identities and limit theorems to confirm the estimate made in part a above.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by wren17 View Post
    1. The limit, as x tends to 2, of the function y=\frac{x^2 + bx - 28}{x - 2} exists, and is equal to 16. What is the value of the whole number coefficient b?

    2. The first derivative of y = (6x^3 + 11)^9 is of the form ax^2(6x^3 + 11)^n , where a is a positive number. What is the value of the whole number coefficient a?

    3. For the ellipse 4x^2 + 9y^2 = 724, the slope of the tangent at the point (10, -6) is b. Correct to the nearest hundredth, the value of b is ______________.

    4. The number I is defined as \int_{1}^{2} 4~sin~x~cos~xdx. The value of I, correct to the nearest hundredth, is ______________.

    5. A spring has a motion that can be modelled by the differential equation \frac{d^2 y}{dt^2} = -5y. One solution of this equation is y = 4~cos~kt, where k is a positive number to be determined. What is the value of k, correct to two decimal places?

    6. An LRT train travels for 120 s between two stations. It accelerates for 30 s, maintains a constant velocity for 70 s, and brakes to a stop in 20 s. The velocity function v(t), with velocity measured in m/s, is defined as follows:
    v(t) = 20sin[\frac{\pi t}{60}]~~~~~~~~0 \geq t \geq 30
    v(t) = 20 ~~~~~~~30 \geq t \geq 100
    v(t) = -0.05(t - 100)^2 + 20~~~~~~~~100 \geq t \geq 120

    a. Sketch a graph of the velocity as a function of time, assuming the velocity function to be continuous at t = 30 and t = 100. (Be sure to label your axes.)

    b. Determine the total distance travelled by the train in the 120-second time interval.

    4. Let the function f be defined as f(x) = \frac{tan~x - sin~x}{x^3}.
    a. Use a suitable numerical value of x to estimate the limit, if it exists, of f(x) as x approaches zero.
    b. Use trigonometric identities and limit theorems to confirm the estimate made in part a above.
    1)For the first one, there are two choices, the limit does not exist, or it is indeterminate (i.e. the top has a factor of x-2). And since the limit exists the latter must be true. Your choices are divide, or use L'hopital's. Either way once you have simplified this is just a simple matter of subbing and and solving for b

    2)Use chain rule

    y'=9(6x^3+11)^8\cdot(6x^3+11)'

    Then visually inspect and equate coefficients

    3) Your best bet here is to just multiply through by 36, and then use implicit differentiation

    4) I=4\int_1^{2}\sin(x)\cos(x)dx

    You can decimate this one by noting that

    \sin(2x)=2\sin(x)\cos(x)\Rightarrow\sin(x)\cos(x)=  \frac{1}{2}\sin(2x)]


    5) This is an SDE (Seperable Differential Equation), seperate variables and integrate. If you have any problems report back

    6) Here, you must just sketch it, and remember that

    \text{Total Distance}=\int_a^{b}|v(t)|dt

    where || denotes absolute value

    7) I am assuming that this is just the limit as x goes to zero?

    For the approximation put in x=.00001 and x=-.00001

    For the actual limit, I assume you do not know Maclaurin series, so I would suggest L'hopital's. Or a clever usage of algaebraic manipulation.


    If you have any problems just post back

    Mathstud
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  3. #3
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    Thanks for the help, but is it possible to get the actual solutions to these, these were just some examples that i didn't get the solutions for and i still have more similar questions like these that i need to do i just wanted to have these to compare to.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by wren17 View Post
    Thanks for the help, but is it possible to get the actual solutions to these, these were just some examples that i didn't get the solutions for and i still have more similar questions like these that i need to do i just wanted to have these to compare to.
    I will help you get the final solutions, if you post your work unto now.
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  5. #5
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    Well i haven't done any work for these yet, i was working on other problems which i had more knowledge on. These were the ones i wasn't quiet as familiar with. Thats why i posted them here because i don't feel ill be able to complete these ones accurately and i still have more similar to these to do.
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by wren17 View Post
    Well i haven't done any work for these yet, i was working on other problems which i had more knowledge on. These were the ones i wasn't quiet as familiar with. Thats why i posted them here because i don't feel ill be able to complete these ones accurately and i still have more similar to these to do.
    Well, I gave you a start for each. So why not work on them a little and when, if you do, get caught come back and we will work on them together?
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  7. #7
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    Ok ill try that, thanks for the help, i doubt ill be able to finsh them today though since im going out of town in a few hours.
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  8. #8
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by wren17 View Post
    Ok ill try that, thanks for the help, i doubt ill be able to finsh them today though since im going out of town in a few hours.
    Well, whenver you get a chance to post them, I am sure someone on here will be glad to help.
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