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Math Help - A few Short Calculus Practice Problems- HELP!

  1. #1
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    A few Short Calculus Practice Problems- HELP!

    Can anyone please show me how to do these? I'm in short calculus (calc without trig).

    If <br /> <br />
\int_1^4 f(x)dx = 6 <br />
, what is the value of the integral  <br /> <br />
\int_1^4 f(5-x)dx ?
    A. 6 B. -6 C. 3 D. 0




    If y^2 - 2xy=16 then \frac{dy}{dx}=
    a. \frac{x}{y-x} b. \frac{2y}{y-x} c. \frac{y}{x-y} d. \frac{y}{y-x}




    The area enclosed by x(1-x) and the x axis is
    a. 1/6 b. 1 c. 5/6 d/ 2/3




    A particle travels in a straight line with a constant acceleration of 3 \frac{m}{s^2}. if the velocity of the particle is 10 m/s at time 2s, how far does the particle travel during the time interval when its velocity increases from 4 m/s to 10 m/s
    a. 20 b. 14 c. 7 d.6




    Let f(x)=ln\sqrt{x^2+1} then f'(1)=
    a. 1/3 b. 1/2 c. 3/2 d. 2/3




    At what value does the function f(x)= x^3*e^{-x} obtain its maximum value?
    a. 3 b. 0 c.e d. √3




    Compute \int_1^4 \sqrt{t}ln t dt
    4ln4_____4/3 ln4______16/3*ln4-28/9______22/3*ln4-25/9




    If y is defined at \frac{dy}{dx}+xy=x and y(0)=0, then y(1)=
    1-e___1-1/√e_____1-√e_______2-1/√e




    On which of the following intervals does x^5*3x^2+pi=0 have a solution?
    [1,2] [0,1] [-1,0] [-2,-1]




    Using the definition of the derivative and the fact that \frac{d}{dx}a^x = a^x*ln a, what is the value of the limit of (a^h -1)/h as h approaches 0
    ln a 1 DNE a^h*ln a




    Determine the equation of the tangent line to y=x-lnx at x=e




    For the equation F(v)= Av^2 + \frac{B}{v^2} where a and b are constants, the drag is minimized when v=5. Find the ratio of B/A



    Solve the following differential equations:
    \frac{dy}{dx}= \frac{1}{x+1} where y(0)=1
    \frac{dy}{dt}=y^2*xe^{-x}

    Integral from +inf to 1 of 1/√x dx

    Calculate the following integrals:
    \int \frac{ln{x}}{x^{2}}{dx} = <br />

     <br /> <br />
\int \frac{1}{\sqrt{x}} from 1 to +infinity =
    [/tex]



    Thanks for any help!
    Last edited by Sm10389; December 11th 2008 at 02:46 PM.
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  2. #2
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    For a "short" calculus course, this list of laundry covers a wide swath of concepts.

    I can provide some hints, but I don't think you'll get anyone to show you how to do all these "few" problems in detail. Good luck.

    Quote Originally Posted by Sm10389 View Post
    Can anyone please show me how to do these? I'm in short calculus (calc without trig).

    If the integral of f(x)dx=6 from 1 to 4, what is the value of the integral f(5-x)dx from 1 to 4?
    a. 6 b. -6 c. 3 d. 0

    use substitution ... let u = 5-x

    If y^2 - 2xy=16 then dy/dx=
    a. x/(y-x) b. 2y/(y-x) c. y/(x-y) d. y/(y-x)

    implicit differentiation ... do it.

    The area enclosed by x(1-x) and the x axis is
    a. 1/6 b. 1 c. 5/6 d/ 2/3

    A = definite integral of x(1-x) from 0 to 1

    A particle travels in a straight line with a constant acceleration of 3 m/s^2. if the velocity of the particle is 10 m/s at time 2s, how far does the particle travel during the time interval when its velocity increases from 4 m/s to 10 m/s
    a. 20 b. 14 c. 7 d.6

    integrate a = 3 and use the initial condition to find v(t) ... displacement will be the integral of v(t) from 0 to 2 sec

    Let f(x)=ln(√(x^2+1)) then f'(1)=
    a. 1/3 b. 1/2 c. 3/2 d. 2/3

    ln(√(x^2+1)) = (1/2)ln(x^2+1) ... derivative of ln(u) = u'/u

    At what value does the function f(x)= x^3e^-x obtain its maximum value
    a. 3 b. 0 c.e d. √3

    max will occur where f'(x) changes sign from positive to negative

    Compute the integral of √t*lnt dt on the interval 1 to 4.
    4ln4 4/3 ln4 16/3ln4-28/9 22/3ln4-25/9

    integration by parts ... u = ln(t) , dv = √t dt

    If y is defined at dy/dx+xy=x and y(0)=0, then y(1)=
    1-e 1-1/√e 1-√e 2-1/√e

    dy/dx = x(1-y) ... separate variables and integrate.

    On which of the following intervals does x^5-3x^2+pi=0 have a solution
    [1,2] [0,1] [-1,0] [-2,-1]

    use the Intermediate Value Theorem

    Using the definition of the derivative and the fact that d/dx*a^x = a^x*ln a, what is the value of the limit of (a^h -1)/h as h approaches 0
    ln a 1 DNE a^h*ln a

    f(x) = a^x ... f'(0) = [f(0+h) - f(0)]/h

    Determine the equation of the tangent line to y=x-lnx at x=e

    x = e , y = e-1 ... find the slope, y'(e), then use the point-slope form for a linear equation.

    For the equation F(v)= Av^2 + (B/v^2) where a and b are constants, the drag is minimized when v=5. Find the ratio of B/A

    F'(5) = 0 and F''(5) > 0

    Solve the following differential equations:
    dy/dx= 1/(x+1) where y(0)=1

    this one is very basic ... you do it.

    dy/dt=y^2*xe^-x

    I think you mean dy/dx ... separate variables and integrate.

    Calculate the following integrals:
    Integral of ln(x)/x^2 dx

    integration by parts ... u = ln(x) , dv = 1/x^2

    Integral from +inf to 1 of 1/√x dx

    you should already know that this integral diverges
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  3. #3
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    Thank you.
    I still have a few questions (for starters):

    #1.) I know substitution but don't know how to use it here.
    #2.) Didn't work out correctly.
    #3.) Figured it out, but don't know how you found the interval [0,1]
    #4.) Don't understand at all.
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  4. #4
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    More questions:
    #5: I got -1/2, which is not an option, correct answer is 3/2 but idk how.
    #6: Don't know how to find critical points with e's.
    #7: Used integration by parts, answer wasn't on here.
    #8: Wouldn't it be x(-1-y)?
    #9: Solved, thanks.
    #10: Still don't understand.
    #11: For the derivative I got 1-(1/x) what is the slope of that?
    #12: Still don't understand.
    #13: Still don't understand.
    #14: Solved, thanks.
    #15: Solved, thanks.
    #16: How would I know that by looking at it?

    Thanks.
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