MATH 158 Lecture Notes - Lecture 1: Telescoping Series, Divergent Series, Geometric Series
13 Apr 2016
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Area between curves int |f(x)-g(x)| dx (s8. 6): find where f(x)=g(x) to nd where f(x) > g(x) and where g(x) > f(x) De nite integrals, fundamental theorem of calculus: int_a^b f(x) dx = f(b)-f(a) and. Basics: x^n, n not 1, 1/x, a^x, sin, cos, tan, sec^2, sec tan, 1/(1+x^2), 1/sqrt(1-x^2) Rules: sum, scaling, polynomials, reversing the integral. E. g. : xe^x, x sin(x), ln x, x sqrt(x+1), ln(x)/x^2, Improper integrals (s9. 5) (integrating over an in nite interval) Higher derivatives f_x, f_xx, f_xy = f_yx. De nitions: critical point, saddle point, relative maximum, relative minimum. Compute f_xx, f_yy, f_xy and d = f_xx f_yy - (f_xy)^2 at each critical point. D must be positive for an extremum, and the sign of f_xx tells if it is a max or min. Make tables for x, y, x^2, xy, and add them up. Write out the normal equations (x^2-sum) m + (x-sum) b = (xy-sum), (x-sum) m + n.
