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applications of integralsin this part were going to come across at some of the applications of integration it should be noted also that these
example for comparison test for improper integrals example find out if the following integral is convergent or divergentintinfin2 cos2 x x2
comparison test for improper integrals here now that weve seen how to actually calculate improper integrals we should to address one more topic about
following is some more common functions that are nice enough polynomials are nice enough for all xs if f x p x q x then fx will be nice enough
one-sided limits we do this along with one-sided limits as the name implies with one-sided limits we will just looking at one side of the point
velocity problem lets look briefly at the velocity problem several calculus books will treat it as its own problem in this problem we
rates of change or instantaneous rate of change now we need to look at is the rate of change problem it will turn out to be one of the most
rates of change and tangent lines in this section we will study two fairly important problems in the study of calculus there are two cause for
solve 2 ln radicx - ln 1 - x 2 solution firstly get the two logarithms combined in a single logarithm2 ln radicx - ln x - l 2ln radicx2 ln 1
solve 3 2 ln x 73 -4 solutionthis initial step in this problem is to get the logarithm by itself on one side of the equation along with a
solve following 4e13 x - 9e5-2 x 0 solutionhere the first step is to get one exponential on every side amp then well divide both sides by one of
solve following x - x e 5 x 2 0 solution the primary step is to factor an x out of both termsdo not divide an x from both termsnote as
exponential and logarithm equations in this section well learn solving equations along with exponential functions or logarithms in them well begin
change of base the final topic that we have to look at in this section is the change of base formula for logarithms the change of base formula
write following in terms of simpler logarithms a log3 9 x4 radicysolutionlog3 9 x4 radicy log shy3 9x4 - log y 12log shy3 9 log
properties1 the domain of the logarithm function is 0 infin in other terms we can just plug positive numbers into a logarithm we cant
logarithm functions in this section well discuss look at a function which is related to the exponential functions we will learn logarithms in this
definition of natural exponential function the natural exponential function is f x ex where e 271828182845905 hence since e gt 1 we also
natural exponential function there is a extremely important exponential function which arises naturally in several places this function is called as
general approach of exponential functions before getting to this function lets take a much more general approach to things lets begin with b 0 b ne
solve 5x tan 8x 3x solution firstly before we even begin solving we have to make one thing clear do not cancel an x from both sides whereas
solve 8 cos2 1 - x 13 cos1 - x - 5 0 solutionnow as specified prior to starting the instance this quadratic does not factor though that doesnt
solve 4 sin 2 t - 3 sin t 3 1 solutionbefore solving this equation lets solve clearly unrelated equation4x2 - 3x 1 rarr 4x2 - 3x -1 4x 1
standard trig equation now we need to move into a distinct type of trig equation all of the trig equations solved to this point were in some way more
solving trig equations with calculators part ii since this document is also being prepared for viewing on the web we split this section into two