Angles and Proportions
Imagine a proportion in the form y = kx . The graph of this function will form an angle with the x axis of the graph. The rate of change of the slope of the function is one because the slope of kx is k , and the derivative of k with respect to itself (the rate of change of the slope as the slope varies) is one. Given this information, what is the rate of change of the angle formed by the graph and x axis in terms of k ?
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1 solution
We can see that the proportion forms a triangle with sides x and y and some interior angle we will call θ. Because this is a proportion, note that k = y / x . Because this is a proportion, dy / dx = y / x . Therefore, k = y / x . Now, back to the triangle. Because we have the opposite and adjacent and need to find the angle, we can use tan^-1 ( y / x ) to find θ. Remember, because k = y / x , θ = tan^-1 ( k ). Our final step is to find dθ/dk, the rate of change of the angle with respect to slope. The derivative of tan^-1 ( x ) w.r.t. x is 1/(1+x^2), and so dθ/dk=1/(1+k^2).