By Bob Miller
Bob Miller's humor-laced, step by step studying assistance make even the main tough math difficulties regimen. in accordance with greater than 28 years of educating and pupil suggestions, his easy-to-grasp techniques supply scholars much-needed self assurance.
<pPYes, moment semester calculus can truly be enjoyable. With this cartoon-filled identify, differentiation and integration healthy simply into students' lives.
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Extra info for Bob Miller's Calc for the Cluless: Calc II
It is not!!!! Example 8— Integral converges. Area is 3. Diverges!!!! Amazing!!!!! Here is one more example I like a lot because it is slightly nasty, slightly tricky, and verrry instructive. " Break up using partial fractions! Law 6 of logs!!! Now Nice example!!! Chapter 9— Parameters, Polar Coordinates Arc Length, Surface of Revolution If the cartesian coordinate system were superior to all others, this chapter would not be necessary. However, it is not. Parameters are variables that are introduced to make life easier.
If we have sin x or cos x in the denominator, and if they are added or subtracted to each other (with one being multiplied by a number or being added or subtracted by itself from a number), then the substitution is u = tan (x/2)!!!!!!!! I don't know who discovered it or why, but it works. Let us derive all the parts. Otherwise you would never believe it. Let tan (x/2) = u = u/1. Draw the triangle for x/2. We get sin (x/2) = u/(1 + u2 )1/2 and cos (x/2)= 1/(1 + u 2)1/2. Draw triangle for angle x.
This is one of his favorite tricks. Thanks, David. Example 11— Find the area of r = 4 cos 6θ. We know we can slide the curve y = f(x) + a units to the right by replacing x by x - a. In the same way, we can rotate r = f(θ) through a counterclockwise angle +α by replacing θ by θ -α. Thus, by rotating our curve by 15º = π/12 radians, r = 4 cos 6(θ - π/12) = 4 cos (6θ - π/2) = 4 sin 6θ, which is exactly the curve in Example 10!!!!!! Rotating a curve doesn't affect its area. So... the answer must be the same as in Example 10!!