Introduction to Optimization

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Introduction to optimization : Optimization is one of the most common applications of calculus involves the determination of minimum and maximum values. In general we hear terms such as greatest profit, least cost, least time, greatest voltage, optimum size, least size, large as possible, maximum and minimum.

For optimization problems, the following results will be very useful.
1) For a square of side 'x', we have,
Area = $x^{2}$ and Perimeter = 4x.

2) For rectangle of sides x and y, we have,
Area = xy and Perimeter = 2(x + y)

3) For trapezoid, we have,
Area = 1/2( a + b)*h where a and b are two opposite parallel sides and h is the height.

4) For circle of radius r, we have,
Area = $\pi r^{2}$ , Circumference = r$\pi$r

5) For sphere of radius r, we have ,
Surface area = 4$\pi r^{2}$ , Volume = 4/3 $\pi r^{3}$

6) For a right circular cylinder of base radius r and height h, we have
Surface area = 2$\pi$ r(r + h), Curved surface area = 2$\pi$ rh, Volume = $\pi r^{2}$ h

7) For right circular cone of height h, slant height 'l' and radius of the base r, we have
Surface area = $\pi$ r (r + l) , Curved surface area = $\pi$ r l , Volume = 1/3 $\pi r^{2}$ h

8) For a rectangular prism of length x , y and z, we have
Surface area = 2(xy + yz + xz) and Volume = xyz

9) For a cube of edge length x, we have
Surface area = 6$x^{2}$ , Volume = $x^{3}$

10) Area of equilateral triangle = $\frac{\sqrt{3}(side)^{2}}{4}$

Remark : It should be noted that if k is a positive constant, then a function of the form kf(x), k + f(x),$[f(x)^{k}]$, $[f(x)^{1/k}]$, logf(x) will be maximum or minimum according as f(x) is maximum or minimum provided that f(x) > 0.

12th grade math


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