Question:
How do you find an angle of a parabola if you only know
the vertex and the ending point?

Replies:
This is a very easy problem for differential calculus. It can be solved
geometrically, but that is more involved. Without derivation, here is the
answer. The formula for a parabola is: y = kX^2. The slope of the tangent
is given by:
lim(d--->0) [Y(X + d) - Y(X)] / (X + d) - X =
lim(d--->0) [k*(X +d)^2 - k*X^2] / (X + d) - X
lim(d--->0) [k*(X^2 +2dX +d^2) - k*X^2 / (X + d) - X
lim(d--->0 [2dX + d^2] / d neglecting terms quadratic in d, i.e. d^2
this is:
lim(d--->0) [2dX] / d = 2 * X.
So the slope, i.e. the tangent at any point X is simply 2*X
Knowing that the tangent of the angle is 2*X, you can determine the angle
as the inverse tangent. This you can do on a calculator or look it up in a
trig table.

This same process is used to determine the slope (i.e. the tangent) of any
function. That is what differential calculus is all about.

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