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differentiate y x xsolution weve illustrated two functions similar to this at this pointd xn dx nxn
interpretation of the second derivative now that weve discover some higher order derivatives we have to probably talk regarding an interpretation of
determine yprimeprime determine yprimeprime
determine the second derivative for following
interval of convergenceafter that secondly the interval of all xs involving the endpoints if need be for which the power series converges is termed
example determines the first four derivatives for
radius of convergencewe will be capable to illustrate that there is a number r so that the power series will converge for x - a lt r and will diverge
higher order derivatives lets begin this section with the given function
power series we have spent quite a bit of time talking about series now and along with just only a couple of exceptions weve spent most of that time
strategy for series now that we have got all of our tests out of the way its time to think regarding to the organizing all of them into a general set
root test- sequences and seriesthis is the final test for series convergence that were going to be searching for at like with the ratio test this
proof for absolute convergencevery first notice that an is either an or it is - an depending upon its sign the meaning of this is that we can
absolute convergence while we first talked about series convergence we in brief mentioned a stronger type of convergence but did not do anything with
related rates in this section we will discussed for application of implicit differentiation for these related rates problems usually its best to
suppose that x x t and y y t and differentiate the following equation with respect to
interpretations of derivativesexample find out the equation of the tangent line
simple derivativesexample differentiate following 5x3 - 7 x 15 f x 5 y x 5solution here with the first function were being asked
alternating series test - sequences and seriesthe final two tests that we looked at for series convergence has needed that all the terms in the
determine yprime for xy 1 solution there are in fact two solution methods for this problemsolution 1 it is the simple way of doing the problem
implicit differentiation to this instance weve done quite a few derivatives however they have all been derivatives of function of the form y f x
comparison test assume that we have two types of series suman and sumbn with an bn ge 0 for all n and an le bn for all n thena if sumbn is
differentiate following functionsa f x 2 x5 cosh xb h t sinh t t 1solutiona f prime x 10x4 cosh x 2x5 sinh xb hprime t t 1 cosh t -
a radiograph is made of an object with a width of 3 mm using an x-ray tube with a 2 mm focal spot at a source-to-film distance of 100 cm the object
telescoping series its now time to look at the telescoping series in this section we are going to look at a series that is termed a
series - special series in this part we are going to take a concise look at three special series in fact special may not be the correct term