大一高数复习资料(Ahighnumberofreviewmaterials).doc

大一高数复习资料(A high number of review materials)A large number of review information. Txt7 is a Durian warmth of spring; warm is written on the face of the shadow; warming is bound to respond with warmth is strict in demands・The 8 respect is a spring, a spring, a give a person warm Shuxin pill, a dose of inspiring arm of Higher Mathematics (undergraduate class type)The first chapter of function and limitThe first section functionThe basic function, (high function knowledge) (* * *)In the neighborhood (go to the heart of the neighborhood) (*)The second section of the sequence limitProof, limit (*)[example] known sequence types, proof[example] that language1. by simplifying it,LThat is, for 2. , at that time, there is always inequality,The third day limit functionProof of function limit, when (b)[questions] example prove the known function.[example] that language1.by simplifying it,LThat is, for 2. , at that time, there is always inequality, LProof of function limit, when (b)[questions] example prove the known function.[example] that language1. by simplifying it,LThat is, for 2., at that time, there is always inequality, The fourth section infinity and infinitesimalInfinity and infinitesimal, the essence of (*)The function is infinitely smallInfinite functionIn the infinitesimal and infinite theorems and inferences (・) (theorem three) is assumed to be bounded, as is infinitesimal,(theorem four) in a change of variables in the process, if it is infinite, is infinitely small; conversely, if it is infinitesimal, and is infinite[example]: (or questions)Any of the 1. dreams than to heart function in the formula in the neighborhood is bounded;(二dreams, bounded on the R function;)The 2. function is the infinite small:(that is the infinitesimal function;)By theorem 3.The fifth day limit algorithmFour arithmetic operations, the limit (* *)(theorem a) and law(theorem two) and lawLimit operation of polynomial, business typeSet:Is there(in particular, when (not stereotypes), usually the numerator and denominator divisor is about to go to solve the discontinuous points can limit, can also be used for Robida rule)[example] evaluation questionsFor example: [] because, so we can get the solution, so the original typeThe function to break pointIf the use of Robida law solution (see Chapter third section second): Solution:Through continuous function, theorem (Solving Limit Function) (* *)(theorem five) if the function is a continuous function defined on the domain, so,[example] evaluation questions:[for example]The sixth day limit existence criteria and two important limitsIn the clamping criterion (P53) (* * *)The first important limit:Dreams, R(in particular,)In the bounded monotone convergence criteria (P57) (* * *)Second important limits:(typically, one)[example] evaluation questions:[for example]The seventh section of infinitesimal order (comparison ofinfinitesimal) 2003 (equivalent infinitesima1.)OneTwo(for multiplication and division, and no)[example] evaluation questions:[for example]The eighth section of continuous functionThe definition, the continuity of the function (・)In the discontinuous point classification (P67) (*)(in particular, to break point in the corresponding fraction in to commonfactor)[questions] a sample function, how to choose the number, make continuous function on?[for example]1.dreams2.by definition of continuous functionThe Ninth Section Properties of continuous functions on closed intervalThe zero point theorem, (*)[example]: proof type equation has at least one root between and among[example] proof1.(an auxiliary function) continuous function in closed interval;2.dreams (ISO, endpoint)3.star by the zero point theorem, in the open interval at least one point, which is (,)4.this equation shows that the equation has at least one root in the open intervalThe second chapter of derivative and differentialThe first section of the derivative conceptThe definition and derivative, in Higher Mathematics (P83) (* *) [questions] examples of known function, in place to guide, for, [for example]1.dreams,2.by function can be defined byL[]for example questions in tangent and normal equation at(or: tangent and normal equation of image point)[for example]1,2.tangent equation:Normal equation:The second section and function (difference), and taking the product differentiation ruleFunction, and (difference), product derivation rule and quotient (* * *)1.linear combination (Ding Liyi):In particular, when there are2. product derivation rule function (theorem two):Taking the 3. function derivative rule (Ding Lisan):The third section derivation rule of compound function and inverse functionIn the derivation rule of inverse function (*)[example] for function derivative questions[example] by solving problem for direct function in the domain can be monotonous, is scheduled to guide, and dry;Derivation rule of compound function, (* * *)[example] questions, for[for example]The fourth section high order derivative,(or) (*)[example] derivative type for function[for exampleThe fifth section derivative of implicit function and parameter equation functionIn the derivation of implicit function (equation of derivative) (* * *)[example] : find the questions given by the equation of the curve in tangent equation and equation of normal point[example] from both sides of the derivative solutionThat Jane wasLL tangent equation:Normal equation:In the parameter equation type function derivation[questions] set parameter equation for example・[for example] 1. 2.The sixth section changes and related problems of rate of change rate (not required)The seventh section differential functionThe basic elementary function, differential equation and differential algorithm (* * *)The third chapter is the application of mean value theorem and derivativeThe first section theoremLemma (lemma, Fermat) (*)Rolle, (* * *) theorem[example] it is assumed that the function in question is continuous in the guide, try to prove:,The establishment of[example] proof1.(an auxiliary function) OrderObviously, continuous function in closed interval, in the open interval can guide;2.dreamsThat is3.star by Rolle theorem of knowledgeThat makes upThe Lagrange theorem (*),[example]: the proof of inequality questions,[example] proof1.(an auxiliary function) to function on, apparently continuous function in closed interval, in the open interval can be guided, and;2.by Lagrange mean value theorem, the equation was established,And dreams, r,Jane, that proved: at that time,[example]: the proof of inequality questions,[example] proof1.(am aux订iary function) to the function, continuous function in closed interval, in the open interval can be guided, and;2.by Lagrange mean value theorem, the equation was established, Jane, and dreams,So, 1,The permit was,The second section Robida ruleThe basic steps, using the rules of Robida limit operation (* *)1.or Equivalent Infinitesimal Replacement (to simplify operation)2.judgment limit amorphous type and whether or not to meet the three conditions using Robida lawA.belongs to the two basic stereotypes () and meet the conditions of operation:(then 1, 2 steps repeatedly until the results)B.You do not belong to the two basic stereotypes (into basic stereotypes)The type (except for the ride, construction of fractional)[example] evaluation questions: [for example](typically, one)The type (through the construction of fractional denominator, observation)[example] evaluation questions:[for example]The type (log limit method)[example] evaluation questions:[for example]The type (log limit method)[example] evaluation questions:[for example]The type (log limit method)[example] evaluation questions:[for example]The basic ideas, using the rules of Robida limit operation (* *) The way to get fractional (usually accompanied by Equivalent Infinitesimal Replacement)The reciprocal fraction obtained (the product form into a fractional form)The logarithmic gain product (by logarithmic index in advance)The third section Taylor theorem (not required)Monotonicity and convexity curve of section fourth functions・Continuous function, monotonicity (monotone interval) (* * *)[questions] sample trial to determine the monotone interval function[for example]1.dreams function in its definition domain is continuous and differentiableL2.order solution:3.(three line)maximum valueMinimum value4.star function monotonically increasing interval;Monotone interval[questions] example demonstrated that the,[example] proof1.(construction of auxiliary function),() (2.).L3.card: at that time,[questions] example demonstrated that the,[example] proof1. (construction of auxiliary function),() (2.).L3.card: at that time,Continuous, convex function (* * *) [example] questions try to discuss the monotonicity, concavity dnd inflection point, extreme value [example] proofOne2.order solution:3.(four line)4. a monotone increasing function interval, monotonic interval,; The minimum value in the function to take, for,The maximum value when take, for;The function in the interval, concave, convex in the interval:The function of the inflection point coordinatesThe fifth section extreme value function and the maximum and minimum valuesThe extreme value of function, and the value of (* * *)The design function of the domain, if a neighborhood, so on, are suitable for inequality,We say that the function has a maximum value at the point;orderIs the maximum function in closed interval values:■9The design function of the domain, if a neighborhood, so on, are suitable for inequality,We say that the function has a minimum value at the point; orderIs the minimum function in closed interval to meet:■9[]for example the value type function in the[for example]1.dreams function in its definition domain is continuous and differentiableL2.order,Solution:3.(three line)Minimum value maximum value4.dreamsLThe sixth section describes the function diagram (not required) The seventh section curvature (not required)The eighth section of the approximate solution of equation (not required)The fourth chapter of indefinite integralThe first section conceptions and properties of indefinite integralsThe concept, function and indefinite integral (* *)The concept of the original function:Assume that in any given interval, derivative differentiable function, i. e. when the independent variable, or set up, a primary function is called theThe original function theorem: (* *)If the function is continuous on interval, in there must be a differentiable function which, also is to say: there must be a continuous function of original function (derivative must be continuous)The concept of indefinite integral (* *)In the interval of definition, function with arbitrary constant primary function is called indefinite integral in the definition of interval,That is:(called the integral numbet, called the integrand, called integral representation, called integral variables)In the basic table (* * *)In the indefinite linear integral property (partial integral formula) (* * *)The second section element integral methodFirst, change element method (Minato differential) (in * * *) (the reverse application)[questions] for example [for example][questions] for example[for example]Second, change element method (to (positive application)For a radical (1):So, then,The original type ofFor the root and the square (form) Order: (a),Thus, the original type of;The square root of the difference A.: (to),Thus, the original type of;B.: (to),Thus, the original type of: []for example questions (a type)[for example][]for example types (triangle element) [for example]Section third subsection integral methodIn subsection integral method (* *)The design function, with continuousderivative, the integral formula can beexpressed as:The subsection integral function, thesort order: "anti, power, three"The basic steps, using the subsectionintegral calculation of indefinite integral:To comply with subsection integral function of the integrand sort sort order;The nearest together: (differential)The use of integral formula:The expansion of a judgment, form) (* *)in form ():A. if the indefinite integral is easy to solve, is to directly calculate the answer (to use basic table for element method with integral of rational function can easily solve the result);If the B. is still quite complex, indefinite integral cannot be solved by a method, then repeat the branch, until the emergence of indefinite integral can be solved easily; if the cycle repeated process, then the simultaneous equations, but the final note added constant[questions] for example[for example][questions] for example[for example]LThe indefinite integral of rational functions in section fourth(b), rational functionSet:For rational functions, the number of times when less than when the rational function is true when the number is greater than the numbet fraction; when the fractional rational function is falseIn the rational function (fraction) of indefinite integral solution of (*)It will be the denominator of rational function split into product of two have no common factor of Polynomials: one can be expressed as a polynomial factorization; another polynomial can be expressed as two times the quality factor (,);That is:In general, the parameter:The parameterIt is a rational function of the spin off and type:amongThe parameter by the method of undetermined coefficient (comparison method) to calculateIt was split after the breakdown of integral can be solved[](for example test method)[for example]The use of integral section fifth (not required)The fifth chapter integral and its applicationThe first concept of definite integralsThe definition of definile integral, (b)(called the integrand, called the integrand expressions, called integral variables, called the lower limit of the integral, called integral upper limit, called integral interval)Nature, integral (* * *)123The (linear properties)(the additivity of integral interval)If such functions satisfy in the integral interval, then;(d a)If the function, function satisfies in the integral interval is;(corollary two)In the mean value theorem of integral (not required)The second section calculus basic formulaIn Newton Leibniz formula (* * *) (theorem three) if the function is a function of the original continuous function in the interval, thenVariable limit integral, derivative formula (* * *) (on the guide - under the guide)[questions] for example[for example]The third section for definite element method and integral integral methodIn the fixed element method for integral (* * *)1(first element method)[questions] for example[for example]2(second method)Set the function, function satisfies:A., the;B. in the interval or continuous.Then: [questions] for example[for example]The (integral method)Even, odd times (* *)We have the following conclusion:If it is.If it is.The application of the fourth section integral in geometry of (temporarily not required)The application of section fifth of the definite integral in the physical world (not required)The sixth section (not required) abnormal integralSuch as: proof of indefinite integral formula・Many students in class can not prove, then at the end of the semester, I give such a proof method to explain the problem:So, indefinite integral formula is easily proved, hope that we carefully try to figure out, understand.Finally, the limited to the level limit, data errors and omissions can hardly be avoided, I hope the students to actively learn from each other so that improved.99• •99• •99• •99• •Review of advanced mathematics data of first pages (9 pages)。