From 5d7fe27f4b29c564a1710c443732f1501b97222c Mon Sep 17 00:00:00 2001 From: jade Date: Mon, 3 Oct 2022 00:40:17 +0800 Subject: [PATCH] =?UTF-8?q?=E5=8A=A0=E5=85=A5=E6=96=B0=E8=AF=BE=E4=BB=B6?= =?UTF-8?q?=EF=BC=9A=E5=BE=AE=E7=A7=AF=E5=88=86=E7=AC=AC=E4=BA=8C=E7=AB=A0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../微积分/Chapter_2.tm | 1426 +++++++++++++++++ 1 file changed, 1426 insertions(+) create mode 100644 北京师范大学-香港浸会大学联合国际学院/微积分/Chapter_2.tm diff --git a/北京师范大学-香港浸会大学联合国际学院/微积分/Chapter_2.tm b/北京师范大学-香港浸会大学联合国际学院/微积分/Chapter_2.tm new file mode 100644 index 0000000..faefb87 --- /dev/null +++ b/北京师范大学-香港浸会大学联合国际学院/微积分/Chapter_2.tm @@ -0,0 +1,1426 @@ + + +> + +<\body> + <\hide-preamble> + >>>>> + + + + + + + + + + \; + + |gr-frame|>|gr-geometry||gr-grid||1>|gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||1>|gr-edit-grid-old||1>|gr-fill-color|red|gr-color|blue|||>||>|>|>>||)>|>>||)>|>>||>>||>>|-f|b-a>*+f|>>| + \; + >>| + <\with|color|> + In fact, + + <\equation*> + m=lima> -f|b-a> + + + >>| + tangent line: + + <\equation*> + y=m*x+f + + + \ for some slope m + >>>> + + + + <\unfolded-plain> + Let > be the position of an object at time + . The average velocity from to is + + <\equation*> + -S|b-a> + + + The instantaneous velocity is the limit of the average velocity in + shorter and shorter time intervals, i.e. + + <\equation*> + lima> -S|b-a> + + + <\remark> + Limit is always a dynamic process! + + <|unfolded-plain> + \; + + + + + + + <\definition> + For a given function >, and a point + >, we say + + <\equation*> + a> + f=L>>, + + + read as: the limit of >, as approaches + , equals , if > gets arbitrarily + close to as long as is sufficiently close to + . + + + <\example> + Let =x>, then\ + + <\equation*> + lim3> f=9 + + + + <\wide-block> + |||| + <\example> + Find the equation of the tangent line to the graph of + =x> at . + + <\answer*> + Draw a secant line from > to + |)>>. Its slope is + + <\equation*> + m=-1|x-1>=-1|x-1>. + + + Then the slope of the tangent line is + + <\equation*> + lim1> m=lim1> + -1|x-1>=lim1> x+1=2. + + + So the equation of the tangent line is + + <\equation*> + y=2*+1=2*x-1. + + + + >>> + + + <\question> + \; + + <\equation*> + lim0> =? + + + Hint: draw a circle. + + |gr-frame|>|gr-geometry||gr-grid||2>|gr-grid-old||2>|gr-edit-grid-aspect|||>|gr-edit-grid||2>|gr-edit-grid-old||2>|gr-snap||gr-arrow-end|\|gr-color|blue|||>||>>||>>|>>|>>||>>|||>>|>>|||>>|>|0> + =1|>>>> + + + <\example> + \; + + <\equation*> + lim0> sin =? + + + The limit does not exist!\ + + + <\example> + \; + + <\equation*> + lim0> x*sin =0 + + + + + In the definition of a> + f=L>, the function value > is + irrelevant. In fact, can even be undefined on . + > + + <\example> + Let + + <\equation*> + f=|0,>>|,>|0,>>||>>>>g=|0,>>|,>|0.>>>>> + + + Find 0> f> and + 0> g>. + + <\answer*> + <\equation*> + lim0> f=1\f + + + <\equation*> + lim0> g=1,g + + + + + + + + <\definition> + For a given function >, and a point , + \ we say + + <\equation*> + >> f=L>>, + + + read as: the of >, as + approaches , equals , if + > gets arbitrarily close to as long as + a> is sufficiently close to . + + Definition of the is similar.\ + + + <\example> + Find the left and right hand limits of + + <\equation*> + f=|0,>>||0.>>>>> + + + <\answer*> + <\equation*> + lim0> + f=-1,lim0> + f=3. + + + + + \; + + Relation between limit and one-sided limits: + + <\itemize> + If the limit exists, the both left and right hand limit exsits + and equal to the limit value itself + + If both left and right hand limit exists and equal to the same + value, then the limit exists and equal to the same value. + + If both left and right hand limit exists but they are not equal, + then the limit does not exist.\ + + + \; + + <\example> + \ + + <\wide-tabular> + || + |png>|0.5par|||> + |<\cell> + <\itemize> + 2> + g=3>> + + 2> + g=1>> + + 2> + g>> not exist! + + > is undefined! + + 5> + g=2>> + + 5> + g=2>> + + 5> + g=2>> + + \1.2> + + 3> + g=1.5=g>> + + >>> + + + + + + <\definition> + We write + + <\equation*> + lima> f=\ + + + if > gets arbitrarily large when is + sufficiently close to . + + Similarly, we can define + + <\equation*> + lima> f=-\ + + + + <\note*> + In these cases we will not say the limit exists.\ + + + <\example*> + <\equation*> + lim0> >=\ + + + + can be defined similarly + + <\equation*> + lima> f=\-\ + + + <\equation*> + lima> f=\-\ + + + <\example*> + <\equation*> + lim0> + =\,lim0> + =-\. + + + + <\example*> + \; + + <\equation*> + lim0> ln + x=-\,lim|2>|)>> + tan x=\,lim|2>|)>> + tan x=-\ + + + + <\definition> + If the left/right hand limit of > is + > or > as approaches , + then the line is called a vertical asymptote of .\ + + + \; + + + + <\wide-block> + |||| + (for limit or one-sided limits) + + <\enumerate> + Sum Law: a> + +g|]>=lima> + f+lima> g>> + provided the limits on the right-hand-side exist.\ + + Difference Law: a> + -g|]>=lima> + f-lima> g>> + provided the limits on the right-hand-side exist.\ + + Constant Multiple Law: a> + c*f=c*lima>> f> + provided the limit on the right-hand-side exists, where is + a constant. \ + + Product Law: a> + *g|]>=lima> + f*lima> g>> + provided the limits on the right-hand-side exist.\ + + Quotient Law: \ a> + |g>=a> + f>|a> + g>>>> provided the limits on the right-hand-side + exist and a> + g\0>.\ + + >>> + + + \; + + Generalize to any of functions: e.g.\ + + <\equation*> + lima> 2*f+5*g=2*lima> + f+5*lima> g. + + + Power Law: a> + f=a> + f|]>>.\ + + Root Law: a> + f=a> + f|]>> ( must be odd if + a> f\0>).\ + + Two simple limits: + + <\equation*> + lima> c=c,lima> + x=a. + + + \; + + <\wide-block> + |||| + for polynomials and rational + functions: + + <\equation*> + lima> P=P,lima> + |Q>=|Q>,Q\0. + + + where are polynomials.\ + >>> + + + \; + + <\wide-block> + |||| + <\example> + \; + + <\equation*> + lim2> -4x+5|)>=2-4\2+5=5. + + + <\equation*> + lim1> -2x|x+1>==-. + + + <\equation*> + lim1> -2x|x-1>=lim1> + |+x+1|)>> + does not exist> + + + In fact + + <\equation*> + lim1> + |+x+1|)>>=\,lim1> + |+x+1|)>>=-\ + + + Another example: + + <\equation*> + lim1> -x|x-1>=lim1>|+x+1|)>>=lim1>+x+1|)>>=. + + + >>> + + + \; + + <\wide-block> + |||| + If =g> when a>, then + a> f=lima> + g>, provided the limits exist. + >>> + + + \; + + <\theorem> + If \g> when is + near (not including itself), then + + <\equation*> + lima> f\lima> + g + + + provided both limits exist.\ + + + \; + + <\with|theorem-text| (The Squeeze Theorem)>> + <\theorem*> + If \g\h> + when is near (not including itself), and + + <\equation*> + lima> f=lima> + h=L, + + + then + + <\equation*> + lima> g=L. + + + Note: the theorem also holds for one-sided limits.\ + + + + <\wide-centered> + |png>|0.3par|||> + + + <\example*> + <\equation*> + lim0> x*sin =? + + + For simplicity, first consider the case 0>. Then + + <\equation*> + -x\ x*sin \x, + + + and + + <\equation*> + lim0>-x=lim0> + x=0. + + + By the squeeze theorem, we know + + <\equation*> + lim0> x*sin =0. + + + Similarly, we have + + <\equation*> + lim0> x*sin =0. + + + So + + <\equation*> + lim0> x*sin =0. + + + + \; + + + + + + <\definition> + A function is said to be at , if + + <\equation*> + lima> f=f. + + + If is defined near and is not continuous at + , we say is at .\ + + + \; + + <\wide-block> + |||| + <\example> + At which numbers is continuous? + + |png>|0.4par|||> + + |<\cell> + <\itemize> + is discontinuous at since + > is undefined + + is discontinuous at since + 3> f> does not exist.\ + + is discontinuous at since + 5> + f\f>.\ + + is continuous at other points.\ + + >>> + + + \; + + <\note> + A polynomial > is continuous for every + >.\ + + + <\example> + Where are each of the following functions discontinuous? + + <\enumerate-alpha> + =-x-2|x-2>> + + <\itemize> + If 2>, then is continuous at + + If , then is discontinuous at , + since > is undefined. However,\ + + <\equation*> + lim2> f=lim2> + -x-2|x-2>=lim2> + *|x-2>=lim2> + =3. + + + If we define + + <\equation*> + =-x-2|x-2>,>|2>>||>>>> + + + Then is continuous everywhere (including ). Thus + this kind of discontinuity is called .\ + + + =|||||>>|>|0>>||>|>>>>|\>>,\ + + <\equation*> + lim0> f=\ + + + So the limit does not exist, and can not be continuous at + . This type of discontinuity is called .\ + + =|||||-x-2|x-2>>|>|2>>||>|>>>>|\>> + + Similar to part (a), is discontinuous at since\ + + <\equation*> + lim2> f=3\f + + + This is still a removable discontinuity.\ + + =|x|\>>, + the greatest integer smaller than or equal to . For example + + <\equation*> + |1.3|\>=1,|1|\>=1,|-2.7|\>=-3. + + + is continuous everywhere except when + \>: the set of all integers. The reason is + + <\equation*> + limn> + f=n-1,limn> + f=n\limn> + f*does not exist> + + + where \>. So there is a jump at . We + call it a .\ + + + \; + + There are other cases of discontinuity. For example + + <\equation*> + f=sin + + + > has no left or right hand limit as + 0>.\ + + + <\definition> + A function > is said to be at if + + <\equation*> + lima> f=f. + + + Similarly we can define .\ + + + \; + + For example, the function =|x|\>> + is continuous from the right at \>.\ + + \; + + <\definition> + A function > is said to be if it's continuous at every point + I>. Here can be all types of intervals, such as + + <\equation*> + ,,,|)>,,a|)>*etc.> + + + + \; + + <\theorem> + If and are both continuous at . Then + + <\equation*> + f+g,f-g,f*g,c*f + + + are continuous at , where is a constant. Moreover,\ + + <\equation*> + + + + is continuous at provided \0>. + + + \; + + <\theorem> + A polynomial > is continuous for all . A + rational function /Q> is + continuous in its domain.\ + + + \; + + \; + + + + From the geometrical definition of , we know + + <\equation*> + lim0> sin x=0=sin 0 + + + So is continuous at . Similarly we can show + is continuous at .\ + + How about any ? + + <\equation*> + sin =*+*\sin + ax\0 + + + So is continuous at every . Similarly we can show + is continuous at every . From the quotient rule, \ + + <\equation*> + tan x= + + + is continuous in its domain, i.e. every except + *\> for \>.\ + + <\question> + What type of discontinuity does have at + \ *\> for \>? + + + Similarly, we deduce that all the trigonometric and inverse trigonometric + functions are continuous in their domains.\ + + \; + + Other functions: root functions, exponential functions, logarithmic + functions are continuous at every number in the domains.\ + + <\theorem> + If is continuous at and + a> g=b>, then + a> f|)>=f>, + i.e.\ + + <\equation*> + lima> f|)>=fa> + g|)>. + + + + \; + + <\theorem> + If is continuous at and is continuous at + >, then g> is continuous at + .\ + + + \; + + <\example> + The function + + <\equation*> + f=sin |)> + + + is continuous at every number.\ + + + <\example> + The function + + <\equation*> + f=+sin x|> + + + is continuous everywhere except where , i.e. + > for \>.\ + + + <\render-theorem|> + Suppose that is continuous on the closed interval + > and let be any number between + > and >, where + \f>. Then there exists a + number in > such that + =m>. + + + \; + + \; + + Example. Show that the equation -x-1=> has a + solution for x\2>. + + Let + + <\equation*> + f=x-x-1- + + + We want to find such that =0>. Now + + <\equation*> + f\0,f\0 + + + So is a number between > and + >. Moreover, is continuous in + >. \ By the intermediate value theorem, there + exists an > such that + =0>.\ + + We can now pick the midpoint of and , that is + >, then + + <\equation*> + f|)>=--1-\0. + + + Using the Intermediate value theorem again in + ,2|]>>, we know there exists an + ,2|)>> such that + =0>.\ + + \; + + Repeating this procedure, we can get more and more accurate + approximations of the root of >. This is called the + . + + + + motivation: \> + -1|x+1>> + + <\equation*> + lim\> -1|x+1>=lim\>>|1+>>==1 + + + <\definition> + Suppose > is defined for a> for + some . Then we define + + <\equation*> + lim\> f=L + + + if > can be close to + as long as is large. Similarly + we can define + + <\equation*> + lim-\> f=L + + + + \; + + <\definition> + If + + <\equation*> + lim\> + f=Lorlim-\> + f=L, + + + we call a of . + + + \; + + <\example> + Let =>. Then + \> f=0>. So + is horizontal asymptote of . Moreover, + 0> f=\>, + so is a vertical asymptote of . + + + <\example> + \ + + <\equation*> + lim\> + -x-2|5*x+4*x+1>=lim\> + ->|5*++>>=. + + + + <\example> + \ + + <\equation*> + lim\> + +1>|3*x*-5>=lim\> + >>|3**->=|3> + + + <\equation*> + lim-\> + +1>|3*x*-5>=lim-\> + +1>|3**->=lim-\> + >>|3**->=-|3> + + + So we have two horizontal asymptote /3> and + /3>. Moreover + + <\equation*> + lim> + +1>|3*x*-5>=\,lim> + +1>|3*x*-5>=-\. + + + So is a vertical asymptote. + + + + \> sin =?> + |<\answer*> + <\equation*> + lim\> sin + =lim0> sin t=sin 0=0. + + > + + + \> x*|)>=?> + |<\answer*> + <\equation*> + lim\> x*|)>=lim\> *|>=lim0> =1 + + > + + <\definition> + Suppose is defined at every a> for some + . Then we define + + <\equation*> + lim\> f=\ + + + if > can be arbitrarily large as long as + is sufficiently large. Similarly we can define + + <\equation*> + lim\> + f=-\,lim-\> + f=\,lim-\> + f=-\ + + + + <\example> + \ + + <\equation*> + lim\> + x=\,lim-\> + x=\,lim-\> + x=-\. + + + + <\note> + If > is a polynomial, then + + <\equation*> + lim\> + P=\\,lim-\> + P=\\. + + + + <\example> + A business manager determines that the total cost of producing + units of a particular commodity may be modeled by the function + + <\equation*> + C=7.5*x+120,000 + + + The average cost is =|x>>. + Find +\> A> and + interpret your result. + + + \ + + <\equation*> + lim\> =7.5 + + + + + + + tangent line, velocity, derivative, rates of change + + \; + + The slope of the secant line for a function > + from to is + + <\equation*> + m=-f|b-a> + + + If a>, then the slope approaches the slope of the + tangent line if it exists at . + + <\definition> + If is continuous at , then we define + + <\equation*> + f=lima> + -f|x-a> + + + to be the of at . If has + a derivative at , we say is at + . + + + \; + + \; + + Other motivations: velocity.\ + + Suppose > is the displacement of an object at time + . Then the from to + is + + <\equation*> + -f|b-a>. + + + The at is + + <\equation*> + v=lima> -f|x-a>=f. + + + Similarly, the acceleration is the derivative of velocity: + =f>. + + \; + + Generally speaking, > is the + of with respect to at . + + \; + + <\example> + Find where the function => is + differentiable and find its derivative. + + <\wide-centered> + |gr-frame|>|gr-geometry||gr-grid||gr-edit-grid-aspect|||>|gr-edit-grid||gr-grid-old||1>|gr-edit-grid-old||1>|gr-color|blue|gr-auto-crop|true|||>>|||>>||>>||>>|>|>>|>>|>>> + + + <\enumerate> + If 0>, + + <\equation*> + lima> -f|x-a>=lima> + -|x-a>=lima> + =1\f=1. + + + If 0>, + + <\equation*> + lima> -f|x-a>=lima> + -|x-a>=lima> + =-1\f=-1. + + + If ,\ + + <\equation*> + lim0> -f|x>=lim0> + |x> \f + + + + + + Other examples where the function is not differentiable. + + <\wide-centered> + |gr-frame|>|gr-geometry||gr-grid||gr-edit-grid-aspect|||>|gr-edit-grid||gr-dash-style|11100|gr-color|red|gr-auto-crop|true|magnify|0.75|gr-grid-old||1>|gr-edit-grid-old||1>|||>|||>||>|||||>||>>|>|>>|>>>> + + + \; + + \; + + <\question> + Show that if > is defined and + + <\equation*> + lima> -f|x-a> + , + + + then is continuous at . + + + <\summarized-plain> + + + definition, graph, other notations + <|summarized-plain> + \; + + + <\math> + <\equation*> + f=lima> + -f|x-a>=lim0> + -f|h> + + + + Now for a variable , we obtain a function + + <\equation*> + f=lim0> + -f|h> + + + called the of , or simply the + of . + + <\example> + Let =x>, find + >. + + \; + + + + <\equation*> + lim0> -f|h>=lim0> + -x|h>=lim0> + |h>=lim0> + =2*x + + + So + + <\equation*> + f=2*x,x\. + + + + <\exercise> + What is > if + =x,n\\>. + + + <\example> + Let =>, find + >. + + \; + + + + <\equation*> + lim0> -f|h>=lim0> + -|h>. + + + Consider three cases + + <\enumerate> + 0> + + <\equation*> + lim0> -|h>=lim0> + =1. + + + 0> + + <\equation*> + lim0> -|h>=lim0> + +x|h>=-1. + + + + + <\equation*> + lim0> -|h>=lim0> + |h> . + + + + So + + <\equation*> + f=|0,>>||0,>>|>|>>>> + + + + Given the graph of , we can sketch the graph of + >. + + \; + + <\example> + \; + + \; + + <\wide-tabular> + | + |gr-frame|>|gr-geometry||gr-grid||gr-edit-grid-aspect|||>|gr-edit-grid||gr-grid-aspect-props|||>|gr-grid-aspect||>|gr-edit-grid-old||1>|gr-color|red|gr-auto-crop|true|gr-grid-old||1>||>|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>|>|>|>||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>|>|>>>|>|>>>>>>> + >>> + + + + <\render-theorem> + Other notations for derivative + <|render-theorem> + For a function >, the derivative function is + + <\equation*> + f,y,,D + f,D f + + + The derivative at is + + <\equation*> + f,||\|>,D + f,D f. + + + + \; + + \; + + \; + + \; + + \; + + \; + + \; + > + + +<\initial> + <\collection> + > + + + + + + > + + + + + > + + + + + > + + + + + + + + + + + + + + + + + + +<\references> + <\collection> + > + > + > + > + > + > + > + > + > + > + > + > + > + > + + + +<\auxiliary> + <\collection> + <\associate|toc> + |math-font-series||font-size||Chapter + 2: Limits and Derivatives> |.>>>>|> + + + |math-font-series||1The + Tangent and Velocity Problems> |.>>>>|> + + + |1.1The tangent problem (how to + find the tangent line) |.>>>>|> + > + + |1.2The velocity problem (how to + find instantaneous velocity) |.>>>>|> + > + + |math-font-series||2The + Limit of a Function> |.>>>>|> + + + |2.1Intuitive definition of a + limit |.>>>>|> + > + + |2.2One-sided limits + |.>>>>|> + > + + |2.3Infinite limits + |.>>>>|> + > + + |math-font-series||3Calculating + Limits Using the Limit Laws> |.>>>>|> + + + |math-font-series||4(skip)> + |.>>>>|> + + + |math-font-series||5Continuity> + |.>>>>|> + + + |math-font-series||6Limits + at Infinity; Horizontal Asymptotes> + |.>>>>|> + + + |math-font-series||7Derivatives + and Rates of Change> |.>>>>|> + + + |math-font-series||8The + Derivative as a Function> |.>>>>|> + + + + \ No newline at end of file