Try graphing this on the TI-89. r 2sin 2.15 0 16 To find the slope of a polar curve: dy dy d dx dx d d r sin d d r cos d r sin r cos
r cos r sin We use the product rule here. To find the slope of a polar curve: dy dy d dx dx d d r sin d d r cos d r sin r cos r cos r sin
dy r sin r cos dx r cos r sin Example: r 1 cos r sin sin sin 1 cos cos Slope sin cos 1 cos sin 2 2 sin cos cos sin cos sin sin cos 2 2
sin cos cos 2sin cos sin cos 2 cos sin 2 sin Area Inside a Polar Graph: The length of an arc (in a circle) is given by r . when is given in radians. For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula: A 1 bh 2 r r d 1 1 2
dA rd r r d 2 2 1 2 dA r d 2 We can use this to find the area inside a polar graph. 1 dA r 2 d 2 1 2 A r d 2 Example: Find the area enclosed by: r 2 1 cos
1 2 0 2 r d 2 1 2 4 1 cos d 0 2 2 2 2 1 2 cos cos d 2 0 2 0 1 cos 2 2 4 cos 2 d 2
2 0 1 cos 2 2 4 cos 2 d 2 2 3 4 cos cos 2 d 0 1 3 4sin sin 2 2 2 0
6 0 6 Notes: To find the area between curves, subtract: 1 2 2 A R r d 2 Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross. When finding area, negative values of r cancel out: r 2sin 2 Area of one leaf times 4: 2 1 2 A 4 2sin 2 d
2 0 A 2 Area of four leaves: 2 1 2 A 2sin 2 d 2 0 A 2 To find the length of a curve: Remember: ds dx 2 dy 2 For polar graphs: x r cos y r sin
If we find derivatives and plug them into the formula, we (eventually) get: 2 dr ds r d d 2 So: Length 2 dr r d d 2
2 dr r d d 2 Length There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: 2 dr S 2 y r
Chapter 9: Chemical bonding - The Lewis model. Why chemical bonds form. Energy is a dominating factor in determining which bonds form and which bonds do not form. Coulomb's law recap. ... Ionic crystal lattice. Lattice energy. Ionic compounds have...
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= mk against HA: at least one pair of means are different If H0 is accepted we know that all means are equal (not significantly different) If H0 is rejected we conclude that at least one pair of means is...
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