Team Geometry

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22

[Ana Grom]. Four methods for constructing a strait line

1st: By a series of points: i.e. assuming [volms] for x or y

2nd: Making x or y = 0, and finding the other

3rd: By means of the Tangent which the line makes with x

4th: Solving with respect to x, and there by means of the Tangent, which the line makes with axis of y

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Problems in Calculus (Page 98)

diagram of triangle with inscribed rectangle

(4th Example) = Let x = one part u = x^m(a-x)ⁿ [Let] a-x = other

du/dx = mx^m-1(a-x)ⁿ-nx^m(a-x)^n-1 = x^m-1(a-x)^n-1(m(d-x)-nx) = ma-mx+nx + m(a-x)-nx = 0

= Ans

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5th Ex... Let a = Altitude of triangle [Let] b = Base " " " x = Altitude of rectangle " y = Base " " ∴ xy = Area of [rectangle]

b:a::y:a-x, ∴ ay = ba-bx & xy = u = = b/a(ax-x²)^a Neglecting the constant b/a, we have du/dx = a-2x = 0, x = a/2 = Ans.

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6th Ex. diagram of a cylinder Let x = Altitude, y = Radius of Base, Solidity (s) = πy²x,

Surface of cyclinder (n) = 2πxy + πy² = + πy² = 2s/y + πy² =

πy³ - s = 0 =

πy²x = πy³, divide by πy², and we have x = y = Ans.

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7 Ex. diagram of a circle with inscribed rectangle Let a = Radius of circle, x = Alt of rectangle, y = Base of [Do]

n = xy = area of rectangle

4R² - 2y² = 0 2R² - y² = 0

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