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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
cardioids and limaconsthese can be split up into the following three cases1 cardioids r a a cos theta and r a a sin thetathese encompass a graph
determine equation of the tangent line to f x 4x - 8 radicx at x 16 solution we already know that the equation of a tangent line is specified
circles - common polar coordinate graphs let us come across at the equations of circles in polar coordinates1 r a this equation is saying that there
some interpretations of the derivativeexample is f x 2 x3 300 4 increasing decreasing or not changing at x -2 solution we already
differentiate following functions h t 2t5 t2- 5 t2we can
differentiate following functionsa f x 15x100 - 3x12 5x - 46b h x x pi - x radic2 solutiona f x 15x100 - 3x12 5x - 46in
lines- common polar coordinate graphsa few lines have quite simple equations in polar coordinates1 theta betawe are able to see that this is a
basic computation formulas next lets take a quick look at some basic computation formulas that will let us to actually compute some
differentiation formulas we will begin this section with some basic properties and formulas we will give the properties amp formulas in this
velocity recall that it can be thought of as special case of the rate of change interpretation if the situation of an object is specified by ft
polar to cartesian conversion formulasx r cos thetay r sin thetaconverting from cartesian is more or less easy lets first notice the
interpretations of the derivative before moving on to the section where we study how to calculate derivatives by ignoring the limits we were
surface area with parametric equationsin this final section of looking at calculus applications with parametric equations we will take a look at
arc length for parametric equationsl intbetaalpha radic dxdt2 dydt2 dtnote that we could have utilized the second formula for ds above is we had
find out the area under the parametric curve given by the following parametric equations x 6 theta - sin thetay 6 1 - cos theta0 le theta le