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quadric surfacesearlier we have looked at lines and planes in three dimensions or r3 and when these are used fairly heavily at times in a calculus
scalar equation of planea little more helpful form of the equations is as follows begin with the first form of the vector equation and write a vector
write down the equation of the line which passes through the points 2 -1 3 and 1 4 -3 write all three forms of the equation of the linesolutionto
definition1 given any x1 amp x2 from an interval i with x1 lt x2 if f x1 lt f x2 then f x is increasing on i2
the shape of a graph part i in the earlier section we saw how to employ the derivative to finds out the absolute minimum amp maximum values of a
equations of lines in this part we need to take a view at the equation of a line in r3 as we saw in the earlier section the equation y mxb does
find out the absolute extrema for the given function and interval g t 2t 3 3t 2 -12t 4 on -4 2solution all we actually need to do here is
finding absolute extrema of fx on ab0 confirm that the function is continuous on the interval ab1 determine all critical points of fx
finding absolute extrema now its time to see our first major application of derivatives specified a continuous function fx on an interval ab we
fermats theorem if f x contain a relative extrema at x c amp f prime c exists then x c is a critical point of f x actually it will be a
three dimensional spacesin this section we will start taking a much more detailed look at 3-d space or r3 this is a major topic for mathematics
provide the vector for each of the followinga the vector from 2 -7 0 - 1 - 3 - 5 b the vector from 1-3-5 - 2 - 7 0c the position vector for - 90
position vectorthere is one presentation of a vector that is unique in some way the presentation of the v a1a2a3 that begins at the point a
extrema note as well that while we say an open interval around x c we mean that we can discover some interval a b not involving the endpoints
definition1 we say that fx consist an absolute or global maximum at x c if f x le f c for every x in the domain we are working on2 we
minimum and maximum values several applications in this chapter will revolve around minimum amp maximum values of a function whereas we can all
critical point of exponential functions and trig functionslets see some examples that dont just involve powers of xexample find out all the
critical point definition we say that x c is a critical point of function fx if f c exists amp if either of the given are truef prime c
two cars begin 500 miles apart car a is into the west of car b and begin driving to the east that means towards car b at 35 mph amp at the
vectors this is a quite short section we will be taking a concise look at vectors and a few of their properties we will require some of this
estimating the value of a seriesone more application of series is not actually an application of infinite series its much more an application of
fourier series - partial differential equationsone more application of series arises in the study of partial differential equations one of the
series solutions to differential equationshere now that we know how to illustrate function as power series we can now talk about at least some
application of rate changebrief set of examples concentrating on the rate of change application of derivatives is given in this
important formulasd ab dx 0 important formulasd ab dx 0