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Arc length Formula
L = ∫ ds
Where
ds √ (1+ (dy/dx)2 ) dx if y = f(x), a < x < b
ds √ (1+ (dx/dy)2 ) dy if x = h(y), c < y < d
Note that there is no limits were put on the integral as the limits will depend on the ds that we are using. By using the first ds will need x limits of integration and by using the second ds will need y limits of integration.
Idea of the arc length formula as a single integral with dissimilar ways to define ds will be suitable when we run across arc lengths in future sections. As well, this ds notation will be a nice notation for the later section as well.
With reference to Fig. 1(a) show that the magnification of an object is given by M=SID/SOD. With reference to Fig. 1(b) show that the size of the penumbra (blur) f is given by f
reduction
Extended product rule : As a last topic let's note that the product rule can be extended to more than two functions, for instance. ( f g h )′ = f ′ gh + f g ′ h+ f g h′ ( f
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Determine y′′ for x 2 + y 4 = 10 Solution: We know that to get the second derivative we required the first derivative and to get that w
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