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Math Help - 2 AP test questions I can't seem to solve

  1. #1
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    2 AP test questions I can't seem to solve

    Hi, the AP test is approaching and I'm struggling with these two Calculus questions and need some guidance. If you can provide a general (or detailed if possible =)) explanation on how you solved the problem, or the strategy you used for the problem, I would be extremely grateful. Solving even just one part will help me immensely! So here goes...

    1. Let f and g be the functions given by f(x)=e^x and g(x) = ln(x)
    a. Find the area of the region enclosed by the graphs of f and g between x = 1/2 and x=1.
    b. Find the volume of the solid generated when the region enclosed by the graphs of f and g between x= 1/2 and x=1 is revolved around the line y=4.
    c. Let h be the function given by h(x) = f(x) - g(x). Find the absolute minimum value of h(x) on the closed interval [1/2,1] and find the absolute maximum value of h(x) on the closed interval [1/2,1].



    2.
    t(minutes) 0 2 5 7 11 12
    r ' (t) (feet/min) 5.7 4.0 2.0 1.2 0.6 0.5 (Note: this is a table)


    The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above gives seleceted values of the rate of change, r ' (t), of the radius of the balloon over the time interval [0,12]. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V = 4/3(pi)(r^3).

    a. Find the rate of change of the volume of the balloon with respect to time when t=5. Indicate units of measure.
    b. Use a right reimann sum with the five subintervals indicated by the data in the table to approximate the integral(antidifferentiation) from 0 to 12 of r ' (t) dt.
    c. Using the correct units, explain the meaning of your solution from part b in terms of the radius of the balloon.


    So there it is...any help is appreciated, good luck, and thanks in advance :X
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  2. #2
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    Quote Originally Posted by nivek114 View Post
    Hi, the AP test is approaching and I'm struggling with these two Calculus questions and need some guidance. If you can provide a general (or detailed if possible =)) explanation on how you solved the problem, or the strategy you used for the problem, I would be extremely grateful. Solving even just one part will help me immensely! So here goes...

    1. Let f and g be the functions given by f(x)=e^x and g(x) = ln(x)
    a. Find the area of the region enclosed by the graphs of f and g between x = 1/2 and x=1.
    b. Find the volume of the solid generated when the region enclosed by the graphs of f and g between x= 1/2 and x=1 is revolved around the line y=4.
    c. Let h be the function given by h(x) = f(x) - g(x). Find the absolute minimum value of h(x) on the closed interval [1/2,1] and find the absolute maximum value of h(x) on the closed interval [1/2,1].



    2.
    t(minutes) 0 2 5 7 11 12
    r ' (t) (feet/min) 5.7 4.0 2.0 1.2 0.6 0.5 (Note: this is a table)


    The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above gives seleceted values of the rate of change, r ' (t), of the radius of the balloon over the time interval [0,12]. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V = 4/3(pi)(r^3).

    a. Find the rate of change of the volume of the balloon with respect to time when t=5. Indicate units of measure.
    b. Use a right reimann sum with the five subintervals indicated by the data in the table to approximate the integral(antidifferentiation) from 0 to 12 of r ' (t) dt.
    c. Using the correct units, explain the meaning of your solution from part b in terms of the radius of the balloon.


    So there it is...any help is appreciated, good luck, and thanks in advance :X
    For the first problem, I'm sorry to say, but you just really need to memorize the formulas for all of those. You are 100% GUARANTEED to see a problem like this on the AP test. I have taken both the AB and BC tests, and both teachers guaranteed me a problem much like that one on both tests.

    1a)For area between the curves, you just need to examine the curves. You basically wanna see which curve is the "upper" curve, or which one is on top, then the formula is very simple

    \int^a_b{f_{1}(x)-f_{2}(x)}

    f_{1} being the "upper curve" and f_{2} being the "lower curve"

    1b) For volume, it depends on what exactly you're doing (what is access or rotation, and what method you are most comfortable with)

    This situation would be most simple to use the "shells" method, where the formula is much the same as the one above, I will use the same notation of f_{1}(x) as upper function and f_2{x} as the lower function.

    The formula then becomes
    \pi\int^a_b{(f_{1}(x)-f_{2}(x))^2} You can visualize this integral as the "radius" of 2 circles. Since you need to rotate around y=4, you just have to add in a factor of 4 to this radius so it becomes
    \pi\int^a_b{(4-f_{1}(x)-f_{2}(x))^2}

    1c) For this last part, you are told h(x) so you can just use the rules you normally know for finding maxima and minima by simply taking the derivative. Keep in mind, however, that you must also test the end points of the interval.

    2a) This part is quite simple. Simply look at the chart, and see that r'(5)=2.0

    2b) This part is also quite simple. You must use the formula for a Right Hand Reimann Sum. The formula is quite simple and works out to be \frac{1}{2} \Delta t(y_{1}+2y_{2}+...+y_{n}) You need 5 subintervals, so your \Delta t will change, as you input values, but the formula remains the same. You just need to multiply each y_{n} by their respective \Delta t

    2c) This part is simply analysis. If you are given the rate of change of r, the to integrate said function over an interval would give you the total change in r that occurs over the interval, and the unit would, obviously, be in feet.

    Hope this helps! Good luck on the exam!
    Last edited by Kalter Tod; April 15th 2008 at 09:37 PM.
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  3. #3
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    2.
    t(minutes) 0 2 5 7 11 12
    r ' (t) (feet/min) 5.7 4.0 2.0 1.2 0.6 0.5 (Note: this is a table)


    The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above gives seleceted values of the rate of change, r ' (t), of the radius of the balloon over the time interval [0,12]. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V = 4/3(pi)(r^3).

    a. Find the rate of change of the volume of the balloon with respect to time when t=5. Indicate units of measure.
    b. Use a right reimann sum with the five subintervals indicated by the data in the table to approximate the integral(antidifferentiation) from 0 to 12 of r ' (t) dt.
    c. Using the correct units, explain the meaning of your solution from part b in terms of the radius of the balloon.



    Can't solve part a. It's asking for the rate of change of the "volume" with respect to time. Help!
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  4. #4
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    Quote Originally Posted by nivek114 View Post
    2.
    t(minutes) 0 2 5 7 11 12
    r ' (t) (feet/min) 5.7 4.0 2.0 1.2 0.6 0.5 (Note: this is a table)


    The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above gives seleceted values of the rate of change, r ' (t), of the radius of the balloon over the time interval [0,12]. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V = 4/3(pi)(r^3).

    a. Find the rate of change of the volume of the balloon with respect to time when t=5. Indicate units of measure.
    b. Use a right reimann sum with the five subintervals indicated by the data in the table to approximate the integral(antidifferentiation) from 0 to 12 of r ' (t) dt.
    c. Using the correct units, explain the meaning of your solution from part b in terms of the radius of the balloon.



    Can't solve part a. It's asking for the rate of change of the "volume" with respect to time. Help!
    Well, you are given the rate of change of the radius, so just use the radius to find \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}
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  5. #5
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    So would the answer be 4(pi)(30^2)(2), since the radius at time t = 5 is 30? This would be an extremely large number...
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  6. #6
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    Quote Originally Posted by nivek114 View Post
    So would the answer be 4(pi)(30^2)(2), since the radius at time t = 5 is 30? This would be an extremely large number...
    Yes, that should be the answer. your units are in ft^3/min though, so a high number isn't surprising
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