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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
series - convergencedivergencein the earlier section we spent some time getting familiar with series and we briefly explained convergence and
inverse sine lets begin with inverse sine following is the definition of the inverse siney sin -1 x harr
derivatives of inverse trig functions now we will look at the derivatives of the inverse trig functions to derive the derivatives of inverse trig
obligatory applicationinterpretation problem next we need to do our obligatory applicationinterpretation problem so we dont forget about themexample
differentiate following functionsa r w 4w - 5 log9 wb f x 3ex 10x3 ln xsolution a it will be the only example which
logarithm functions now lets briefly get the derivatives for logarithms in this case we will have to start with the following fact regarding
theorem if an is bounded and monotonic then an is convergent be cautious to not misuse this theorem it does not state that if a sequence is
determine if the following sequences are monotonic andor boundeda -n2infinn0b -1n1infinn1c 2n2infinn5solution-n2infinn0this sequence is a
monotonic upper bound and lower boundgiven any sequence an we have the following terminology1 we call or denote the sequence increasing if an
derivatives of exponential and logarithm functions the next set of functions which we desire to take a look at are exponential amp logarithm
differentiate following functions g x 3sec x -10 cot x solution there
sequenceslet us start off this section along with a discussion of just what a sequence is a sequence is nothing much more than a list of numbers
sequences and seriesin this section we will be taking a look at sequences and infinite series in fact this section will deal approximately
arc length and surface area revisited we wont be working any instances in this part this section is here exclusively for the aim of summarizing
surface area with polar coordinates we will be searching for at surface area in polar coordinates in this part note though that all were going to
extended product rule as a last topic lets note that the product rule can be extended to more than two functions for instance f g h prime f
example of quotient rule lets now see example on quotient rule in this unlike the product rule examples some of these functions will require the
quotient rule if the two functions fx amp gx are differentiable that means the derivative exist then the quotient is differentiable
product ruleif the two functions fx amp gx are differentiable ie the derivative exist then the product is differentiable
product and quotient rule firstly lets see why we have to be careful with products amp quotients assume that we have the two functions f x
derivative with polar coordinatesdydx drdtheta sin theta r cos theta drdtheta costheta - r sinthetanote rather than trying to keep in mind this
tangents with polar coordinateshere we now require to discuss some calculus topics in terms of polar coordinateswe will begin with finding tangent