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Thread: slope problem - calculus

  1. #1 slope problem - calculus 
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    Hi there!

    Been learning calculus, find the slope of the function at point :



    Have I done something wrong? I worked out the problem by taking small values for and the result appeared to converge on .

    Is that the way I should have done it?

    Thanks!


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  3. #2  
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    Rgba,
    What you're doing is not finding the slope of the function, but the derivative, which, while a slope, is not that of the function (but of a linear line that relates to the function).

    If you'd like to find the derivative of this function, than it's easier to use the so-called "power rule" rather than the difference quotient. Since simplifies to , and the power rule states that a function has a derivative , than the derivative of the function is .


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  4. #3 Re: slope problem - calculus 
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    Quote Originally Posted by rgba
    Hi there!

    Been learning calculus, find the slope of the function at point :



    Have I done something wrong? I worked out the problem by taking small values for and the result appeared to converge on .

    Is that the way I should have done it?

    Thanks!
    Firstly, how do you get that lim h->0 ((sqrt(a+h)-sqrt(a))/h)=lim h->0 ((a+h-a)/h*h). Secondly, lim h->0 ((a+h-a)/h*h)=lim h->0 (1/h) = infinity, not zero.
    373 13231-mbm-13231 373
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  5. #4  
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    Thanks for the replies!

    Quote Originally Posted by Ellatha
    Rgba,
    What you're doing is not finding the slope of the function, but the derivative, which, while a slope, is not that of the function (but of a linear line that relates to the function).

    If you'd like to find the derivative of this function, than it's easier to use the so-called "power rule" rather than the difference quotient. Since simplifies to , and the power rule states that a function has a derivative , than the derivative of the function is .
    Cool, so not slope of the actual function, but the slope of the rate of change. Haven't done the power laws yet, but now that I can't work this one out properly, I'd like to see how to work it out.

    thyristor:

    What I did was;

    [LaTeX ERROR: Compile failed]

    I guess I'm not doing it right?
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  6. #5  
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    Rgba,
    Slope is rate of change. The "average" rate of change is the slope of the function, while the "instantaneous" rate of change is the derivative; the slope of the line tangent to some point of the function.

    Consider the following, hypothetical scenario and problem:

    "A piece of metal falls from a rocket that's three hundred feet in the air. The height of the metal is proportional to three times the square of the time, in seconds, of which the piece of metal has fallen. What is the average speed of the piece of metal at impact with the ground and what is its instantaneous speed?"

    To find its "average" speed we must write a function to express the height of the ball. The function will be , with respect to time , that is . To find its average rate of change (speed) we must find the time at , which can be solved through the following quadratic equation:






    Now, to find the average speed at impact we will divde the output of by the input, . Therefore, the average speed of the piece of metal is 30 feet per second (the negative is due to the fact that the distance traveled is negative).

    In order to find the instantaneous rate of change (the instantaneous speed) we must differentiate the function, which is a polynomial, which means we must differentiate every term of the polynomial.

    The derivative of the function is . Now, plugging in h'(10) we get the instantaneous velocity, which is , or 60 feet per second.

    I hope this has helped you to some extant to understand what the derivative is and one of its many applications.
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  7. #6  
    Ots
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    What I did was;

    [LaTeX ERROR: Compile failed]

    I guess I'm not doing it right?
    Instead of squaring you should have multiplied numerator and denominator by [(a+h)^1/2 + (a)^1/2]/[(a+h)^1/2 + (a)^1/2)], which of course is equal to 1, but more importantly it eliminates the radicals in the numerator and then allows cancelation of the h-term top and bottom so you can take the limit without the denominator going to zero.
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  8. #7  
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    Hi guys, thanks for the replies!

    Quote Originally Posted by Ellatha
    Rgba,
    Slope is rate of change. The "average" rate of change is the slope of the function, while the "instantaneous" rate of change is the derivative; the slope of the line tangent to some point of the function.

    Consider the following, hypothetical scenario and problem:

    "A piece of metal falls from a rocket that's three hundred feet in the air. The height of the metal is proportional to three times the square of the time, in seconds, of which the piece of metal has fallen. What is the average speed of the piece of metal at impact with the ground and what is its instantaneous speed?"

    To find its "average" speed we must write a function to express the height of the ball. The function will be , with respect to time , that is . To find its average rate of change (speed) we must find the time at , which can be solved through the following quadratic equation:






    Now, to find the average speed at impact we will divde the output of by the input, . Therefore, the average speed of the piece of metal is 30 feet per second (the negative is due to the fact that the distance traveled is negative).

    In order to find the instantaneous rate of change (the instantaneous speed) we must differentiate the function, which is a polynomial, which means we must differentiate every term of the polynomial.

    The derivative of the function is . Now, plugging in h'(10) we get the instantaneous velocity, which is , or 60 feet per second.

    I hope this has helped you to some extant to understand what the derivative is and one of its many applications.
    Ah yeah, ok, slope is rate of change. Average rate of change is the slope between two points on a function. The derivative, the instantaneous rate of change, is the limit of the function as point approaches point , which is the tangent to the function at point .

    On the rocket/ball example; average speed is simply the distance traveled in a certain time, in this case 300 feet in 10 seconds, so the average speed is 30 feet per second. This is the slope of the function of the balls travel over time. This slope goes from to and is negative as the ball travels downwards (negative direction).

    The instantaneous speed (the derivative) is the limiting value of the rate of change of the balls speed as we approach 10. This means we take ; we try to bring as close as possible to , which gives us the tangent to (brings closer and closer to ).

    Got that right?

    Hey Ots, thanks! Have worked it out in full;





















    Did I get it right?

    Thanks!
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  9. #8  
    Ots
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    Oh yeah. Ain't that cool? It's better than sex!
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  10. #9  
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    Quote Originally Posted by Ots
    Ain't that cool? It's better than sex!
    Depends who you're having sex with? If Russel Crowe then definitely, otherwise if Kate Beckinsale then bugger that
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  11. #10  
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    Rgba,
    You are correct.
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  12. #11  
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    on both counts.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  13. #12  
    Ots
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    I don't know. I think Russel would need bigger hooters.
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  14. #13  
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    Mod note C'mon, kids, behave. Let's have no more playground smut
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  15. #14  
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    Maybe the 'smut' should be cut off, don'tcha think? Guitarist?
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  16. #15  
    Ots
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    Oh yeah, it's *sigh* really bad.
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