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proof of various limit propertiesin this section we are going to prove several of the fundamental facts and properties about limits which we saw
substitution rule for definite integralsnow we need to go back and revisit the substitution rule as it also applies to definite integrals at some
properties1 intbaf x dx -intba f x dx we can interchange the limits on any definite integral all that we have to do is tack a minus
substitution rulemostly integrals are fairly simple and most of the substitutions are quite simple the problems arise in correctly getting the
determine or find out the direction cosines and direction angles for a 2 1 -4solution we will require the magnitude of the vectora radic 4116
evaluate following integrals 1 - 1 w cos w - ln w dwsolutionin
determine the projection of b 2 1 -1 onto a 1 0 -2there is a requirement of a dot product and the magnitude of aararr bull brarr 4
determine the function f x f prime x 4x3 - 9 2 sin x 7ex f 0 15solutionthe first step is to
evaluate following integrals a int 3ex 5 cos x -10 sec2 x dx b 23 y 2 1 6 csc y cot y 9 y dysolutiona int 3ex 5 cos x -10
evaluate following indefinite integrals a int 5t 3 -10t -6 4 dt b int dysolution a int 5t 3 -10t -6 4 dttheres not whole lot to do here
properties of the indefinite integral1 int k f x dx k int f x dx where k refer for any number thus we can factor multiplicative
integration variable the next topic which we have to discuss here is the integration variable utilized in the integral in fact there isnt actually a
indefinite integrals in the past two chapters weve been given a function f x and asking what the derivative of this function was beginning
now lets move onto the revenue amp profit functions demand function or the price function firstly lets assume that the price which some item can be
properties of dot product - proofproof of if vrarr bull vrarr 0 then vrarr 0rarrthis is a pretty simple proof let us start with vrarr v1 v2
marginal cost amp cost function the cost to produce an additional item is called the marginal cost and as weve illustrated in the above example
properties of dot producturarr bull vrarr wrarr urarr bull vrarr urarr bull wrarr cvrarr bull wrarr vrarr
the production costs per week for generating x widgets is given byc x 500 350 x - 009 x2 0 le x le 1000answer
an apartment complex contains 250 apartments to rent if they rent x apartments then their monthly profit is specified by in
proof for properties of dot productproof of urarr bull vrarr wrarr urarr bull vrarr urarr bull wrarrwell begin with the three vectors urarr u1 u2
business applicationsin this section lets take a look at some applications of derivatives in the business world for the most of the part these
use newtons method to find out an approximation to the solution to cos x x which lies in the interval 02 determine the approximation to six
newtons method if xn is an approximation a solution of f x 0 and if given by f prime xn ne 0 the next approximation is given
compute the dot product for each of the subsequent equation a vrarr 5irarr - 8jrarr wrarr irarr 2jrarr b ararr 0 3 -7 brarr 2
dot product- vector the other topic for discussion is that of the dot product let us jump right into the definition of dot product there is given