英文版微积分

1、Review,一、Determine whether the series is convergent or divergent,Infinite Sequences and Series,二、Find the radius of convergence and interval of Convergence of the following series.,二、Find the sum of the following series.,二、Find the sum of the following series.,二、Find the sum of the following series.

2、,二、Find the sum of the following series and the Maclaurin Series for the sum.,A sequence,is defined recursively by the equation,Determine whether the series,is convergent or divergent. And find the sum of this series.,Example,Example,(19) 求幂级数,的收敛域及和函数,.,(9) 若级数,收敛,则级数 ( ) (A),收敛 (B),收敛 (C),收敛 (D),收

3、敛,(19) (本题满分9分) 设级数,的和函数为S(x). 求： (I) S(x)所满足的一阶微分方程； (II) S(x)的表达式.,求幂级数,的和函数f(x)及其极值.,True or false,1.,2.,3.,If ,then is convergent.,If is divergent ,then is divergent.,If for all ,and is divergent,then is divergent.,4.If and converges, then,True or false,5.If then is convergent.,6.If and converge

4、s, then,is convergent.,Choose the best answer for each of the following,1.If,then,A.,B.,C.,D.Diverges,2.We have the following four statement:,(1).If is convergent,then,is convergent.,(2).If is convergent,then,is convergent.,(3).If ,then,is divergent.,(4).If is convergent,then,are both convergent.,Th

5、en the correct statement is,A.(1) (2) B.(2)(3) C.(3)(4) D.(4)(1),1.If diverges when,then,and converges at,2. The sum of is,Example,1.Prove that,2. Suppose that,Show that the convergence of,implies the convergence of,.Is the converse true， prove it or give a counter-example.,1.If calculate and,2.If c

6、alculate and,If is the angle between the nonzero vectors and then,Vectors and Geometry of Space,Direction Angles and Direction Cosins,The direction angles of a nonzero vector are the angles and , that makes with the positive,and z-axes.,o,and are called the directio cosin of,We have,The vector is a

7、unit vector in the direction of,Example Find the scalar projection and the vector projection of onto,Solution,The scalar projection is,The vector projection is,The length of the cross product is equal to the area of the parallelogram determined by and,Properties of the cross product,prlelgrm,The vol

8、ume of the parallelepiped determined by the vector and is the magnitude of their scalar triple product:,Partial Derivatives,1.If calculate and,2.If calculate and,Chain rule,Show that when Laplaces equation,Example,is written in spherical coordinates, it becomes,Implicit Differentiation,Example,1.Fin

9、d and if is defined implicitly as a function of and by the equation,Implicit Differentiation,Example,2.Find and if and are both defined implicitly as function of by the equation system.,Implicit Differentiation,Example,3.If is defined implicitly as a function of and by the equation,Directional Deriv

10、atives and the Gradient Vector,Example,Given that,is defined by, find,Where, U=(1,1).,Example Given f(x, y) = x3 +y3 3xy, find the extreme values of function f. Solution f has continuous second partial derivatives so the critical points of f are those at which fx and fy are 0. Since fx(x, y) = 3x2 3

11、y and fy(x, y) = 3y2 3x, Solving for x, y from 3x2 3y = 0, 3y2 3x = 0, we obtain the critical points of f, which are (1, 1) and (0, 0).,For (1, 1) we have A = 6, B = -3, C = 6, and D=27 0. Since A0, we know that f(1, 1) = -1 is a local minimum. For (0, 0) we have A = 0, B = -3, C = 0, and D= -9 0, h

12、ence f has no extreme value at (0, 0).,Example,solution,Step1 we first find the critical points.,since,Evaluate,Let,We get the only critical point ,and the value of,is,Step2 We look at the values of on the boundary of D,it is the circle,On the circle, we have and,So its minimum value is and its maxi

13、mum value is,Thus on the boundary,the minimum value of is 22 and the maximum is 94.,since,when,We have,Step3 We compare these values in step 1 and step 2,We conclude that the absolute maximum value of on D is and the absolute minimum value is,A manufacturer can produce three distinct products in qua

14、ntities,respectively, and thereby derive a profit,. Find the values of,that maximize profit if production is subject to the constraint,.,Example,Find the extreme values of subject to the constrait,Solution:,Using Lagrange multipliers,We look for values of and such that,This gives the equations,we ge

15、t,is the maximzing level of output,1.If,then,=_.,2.If,then,=_.,3.If then,=_.,4.If has a local extreme value at,=_.,then,If, then,has at point,(A),(B),(C),(D),Choose the best answer for each of the following,Integrals,1.The Substitution Rule,Example,Integrals,2. Integration by Parts,Integrals,3. Trigonometric substitution,Integrals,4.Rationalizing Substitutions,5. Improper Integrals,1.