Applications of Integration

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eutas1

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Please refer to attachment A - at first, I had no clue how to even attempt (d), but after taking a quick peak at the worked solution, I can see that since you know the values for the opposite and adjacent lines (refer to attachment B), you would be able to use tan because tan is opposite over adjacent (O/A). However, I'm not sure where to go from here with this information - examining the worked solution did not help me either since I don't understand their working, or why they even used the derivative function rather than the y function from (a)...

Thank you!
 

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eutas1 said:
Please refer to attachment A - at first, I had no clue how to even attempt (d), but after taking a quick peak at the worked solution, I can see that since you know the values for the opposite and adjacent lines (refer to attachment B), you would be able to use tan because tan is opposite over adjacent (O/A). However, I'm not sure where to go from here with this information - examining the worked solution did not help me either since I don't understand their working, or why they even used the derivative function rather than the y function from (a)...Thank you!
Click to expand...
How is dy/dx related to the angle (Θ) the rod makes with the x-axis (for a given x specifically at point A)
 
Also, I think a better way of thinking about this question is to transform it onto a cartesian plane and rewrite the question.

Let 'O' be the origin and 'A' a point 2 units along the positive x - axis.
Then it is given that:
f'(x) = dy/dx = 0.001(3x2 - x3 - 2)

Assuming you have done all the previous parts, the question is...
What is the angle the tangent at A makes with the horizontal (x - axis)?

Not sure if this helps but I feel like questions like these are easier to imagine on a cartesian plane.
 
Subhotosh Khan said:
Geometrically,

What does derivative of a function (at a point) represent?
Click to expand...

Muddyakka said:
dy/dx is can be thought of as "a small change in y over a small change in x" or otherwise "rise over run"

This has a geometric significance in (d)
Click to expand...

Oh! Geometrically, the derivative of a function at a point represents the slope - in the case of this question, it would be the line that forms the angle from the horizontal (green line in attachment 'LL') ???

So you would calculate the derivative at point A where x = 2, then set that equal to tan(theta), then calculate what theta is. I see.

However, when I do this, I do not get 0.115 degrees as my answer (please refer to attachment 'R')... Obviously I have done something very wrong because why would the answer be the same as the derivative...
 

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