I have similar ones … either math or .. French !!! Weird
]]>Suppose hit a baseball and it pops up, goes for some distance, and then hits the (flat) ground. Suppose we are told that the path of the ball is a parabola so that for some constants a and b, at distance x from you across the ground the height of the ball isf(x) = xb – ax^2.For the slope of the ball at x, for a relatively small positive number h, we can approximate the slope by(f(x + h) – f(x))/ hThen we can let h go to 0 and see what limit we get; that will be the slope.For the term bx, we get(b(x + h) – bx) / h= bh / h = b.For the term -ax^2, we get(-a(x + h)^2 + ax^2) / h= (-ax^2 -2axh -ah^2 + ax^2)/h= -2ax – ahso that as h goes to 0 the limit is just-2ax.Denote the slope by f'(x). Then adding the two results, we have that the slope isf'(x) = b – 2axThe top of the parabola will be where the slope is 0, and that isx = b/(2a)Then the height isf(b/(2a)) = b^2/(2a) – a(b/(2a))^2= b^2/(2a) – b^2/(4a)= b^2/(4a)The slope f'(x) is called the ‘first derivative’.What’s not to like?
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