厦门大学数学科学学院：《高等代数》课程教学资源（应用与实验）MATLAB Lecture 7 - Calculus

MATLAB Lecture 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr MATLAB Lecture 7-Calculus 微积分 Ref: Symbolic Math Toolbox-Using the Symbolic Math Toolbox Calculus ● Vocabulary ulus微积分 function函数 composite/compound function复合函数 inverse function反函数 limit极限 derivative导数 erentiation微分 differential quotient微商,导数 definite integral不定积分 definite integral定积分 Taylor series泰勒级数 Taylor expansion泰勒展开式 Taylor formula泰勒公式 item/term项 summation求和 accumulate累加 singleton单元素 dimension维数 ndex/subscript/suffix下标 theN- th order difference n阶微分 ● Some functions compose inverse limit diff int symsum taylor *gradient *sum *prod ● Calculus This section explains how to use the Symbolic Math Toolbox to perform many common mathematical operations Y Function composition compose(fg)returns f(g(y)) where f=f(x)and g=gly). Here x is the symbolic variable of f as defined by findsym and y is the symbolic variable of g as defined by findsym. compose(f,g, z) returns f(g(z) where f= f(x),g=g(y), and x and y are the symbolic ariables of f and g as defined by findsym compose(f,g, x, z)returns f(g(z)and makes x the independent variable for f. That is, if f cos(x /t), then compose(f, g, x, z) returns cos(g(z)t) whereas compose (fg, t, z)returns cos(x/g(z) compose(fg, x,y, z)returns f(g(z))and makes x the independent variable for f and y the independent variable for g. For f=cos(x/t) and g= sin(y/u), compose (fg,,y, z)returns s(sin(z/u)/t) whereas compose(f, g, x, u, z)returns cos(sin(y/z)/t) > syms yatu, >>f=1/(1+x2): g=sin(y); h=xt: p=exp(-y/u); > compose(f, g) 1+si > compose(f g, t):%

MATLAB Lecture 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec7­1 MATLAB Lecture 7 – Calculus 微积分 Ref: Symbolic  Math Toolbox→Using  the  Symbolic  Math Toolbox  →Calculus l Vocabulary: calculus  微积分 function  函数 composite/compound function  复合函数 inverse function  反函数 limit  极限 derivative 导数 differentiation  微分 differential quotient  微商,  导数 indefinite integral  不定积分 definite integral  定积分 Taylor series  泰勒级数 Taylor expansion  泰勒展开式 Taylor formula  泰勒公式 item/term  项 summation  求和 accumulate 累加 singleton  单元素 dimension  维数 index/subscript/suffix  下标 the N­th order difference N 阶微分 l Some functions compose finverse limit  diff int symsum taylor *gradient  *sum  *prod  l Calculus This section explains  how to use the Symbolic Math Toolbox  to perform many common  mathematical operations.  ² Function composition compose (f,g) returns f(g(y)) where f = f(x) and g = g(y). Here x is the symbolic variable of f as defined by findsym and y is the symbolic variable of g as defined by findsym.  compose (f,g,z) returns  f(g(z)) where f =  f(x),  g =  g(y),  and  x  and y are the symbolic variables of f and g as defined by findsym.  compose (f,g,x,z) returns f(g(z)) and makes x the independent variable for f. That is, if f =  cos(x/t), then compose (f,g,x,z) returns cos(g(z)/t) whereas compose (f,g,t,z) returns cos(x/g(z)).  compose (f,g,x,y,z) returns  f(g(z)) and makes  x  the independent  variable for f and y the  independent  variable for g.  For f =  cos(x/t)  and g  =  sin(y/u),  compose (f,g,x,y,z) returns  cos(sin(z/u)/t) whereas compose (f,g,x,u,z) returns cos(sin(y/z)/t).  Examples:  >> syms x y z t u;  >> f = 1/(1 + x^2); g = sin(y); h = x^t; p = exp(­y/u);  >> compose(f,g);  % 2 1  1+ sin y >> compose(f,g,t);  % 2  1 1+ sin t

ATLAB Lecture 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr > compose(h, g, x, z) > compose(h g, t, z): %x > compose(h,p, x, y, z) %oe > compose(h, p, t, u, z); %x NOTICE: the order of the codes is important. >>clear; syms x y z: z-exp(y): y=sin(x); compose(z, y),z ans exp(sin(x)) exple) > clear; syms x y z, y=sin(x); z=exp(y ) compose(z, y),z %NOT RECOMMEND exp(sin(sin(x))) exp(sin(x)) > clear; syms uv y z; y=sin(u): z=exp(v); compose(z, y) %RECOMMEND to use different variables as independent variable for different functions ☆ Functional inverse g= inverse(f)returns the functional inverse of f. f is a scalar sym representing a function of exactly one symbolic variable, say 'x'. Then g is a scalar sym that satisfies g(f(x))=x g= inverse(f, v) uses the symbolic variable v, where v is a sym, as the independent variable Then g is a scalar sym that satisfies g(fv))=v. Use this form when f contains more than one symbolic variable > inverse(1/tan(x)): %returns atan( >> inverse(fy): %oreturns-x2+y > inverse(f) Warning: inverse(x 2+y) is not unique >In sym. inverse at 43 (-y+x)^(1/2) Limit of an expression limit(F, x, a)takes the limit of the symbolic expression F asx -a limit(F, a)uses findsym(F)as the independent variable limit(F)uses a=0 as the limit point limit( E,x, a,'right)or limit(F x, a,'left) specify the direction of a one-sided limit

MATLAB Lecture 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec7­2  >> compose(h,g,x,z);  %sint z >> compose(h,g,t,z);  % sin z  x >> compose(h,p,x,y,z);  % ( / ) t z u e - >> compose(h,p,t,u,z);  % y / z  e x - NOTICE: the order of the codes is important. >> clear; syms x y z; z=exp(y); y=sin(x); compose(z, y), z ans =  exp(sin(x)) z =  exp(y) >> clear; syms x y z; y=sin(x); z=exp(y); compose(z, y), z %NOT RECOMMEND  ans =  exp(sin(sin(x))) z =  exp(sin(x)) >> clear; syms u v y z; y=sin(u); z=exp(v); compose(z, y) %RECOMMEND to use…  different variables as independent variable for different functions  ² Functional inverse g = finverse(f) returns the functional inverse of f. f is a scalar sym representing a function of exactly one symbolic variable, say 'x'. Then g is a scalar sym that satisfies g(f(x)) = x.  g = finverse(f,v) uses the symbolic variable v, where v is a sym, as the independent variable.  Then g is a scalar sym that satisfies  g(f(v)) =  v. Use this form when f contains  more than one symbolic variable.  Examples: >> finverse(1/tan(x));  %returns atan(1/x).  >> f = x^2+y;  >> finverse(f,y);  %returns ­x^2+y.  >> finverse(f) Warning: finverse(x^2+y) is not unique.  > In sym.finverse at 43  ans =  (­y+x)^(1/2)  ² Limit of an expression limit(F,x,a) takes the limit of the symbolic expression F as x  → a.  limit(F,a) uses findsym(F) as the independent variable.  limit(F) uses a = 0 as the limit point.  limit(F,x,a,'right') or limit(F,x,a,'left') specify the direction of a one­sided limit

ATLAB Lecture 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr > syms x a th >>limit(sin(x)/x) %evaluate lim sIn x limit(x-2)/(x/2-4).2) %evaluate lim--f-2 >> limit(1+2*t/x)(3*x),x,inf) %evaluate lim[ 1+ exp(6 t) >> limit(1/x, x, 0,'right) %evaluate lim x→0 >>limit( 1/x, x, 0, left) %evaluate lim >> limit(sin( x+h)-sin(x))h,h, 0) % evaluate lim sin(x+h)sin x h cos(x) >>V=[(1+ax)^x,exp(-x) >>limit(v, x, inf, "left) ans I exp(a) 0] Y Difference and approximate derivative diff (X), for a vector X, is [X(2)-X(1)X(3)-X(2) X(n)-X(n-D)I diff (X), for a matrix X, is the matrix of row differences, X(2: n,: )-X(I: n-1, )1 diff (X, N)is the N-th order difference along the first non-singleton dimension(denote it by dim). If N > size(X, dim), diff takes successive differences along the next non-singleton dimension diff(x, N, DIM) is the nth difference function along dimension dim. Ifn > size(X, dIM) diff returns an empty array

MATLAB Lecture 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec7­3  Examples:  >> syms x a t h;  >> limit(sin(x)/x)  %evaluate  0  sin lim x x  Æ x ans =  1  >> limit((x­2)/(x^2­4),2) %evaluate  2  2  2 lim  x 4 x  Æ x - - ans =  1/4  >> limit((1+2*t/x)^(3*x),x,inf) %evaluate  3  2 lim 1 x x t Æ• x Ê ˆ + Á ˜ Ë ¯ ans =  exp(6*t)  >> limit(1/x,x,0,'right')  %evaluate  0  1 lim x x Æ + ans =  Inf >> limit(1/x,x,0,'left')  %evaluate  0  1 lim x x Æ - ans =  ­Inf >> limit((sin(x+h)­sin(x))/h,h,0) %evaluate  0  sin( ) sin lim h  x h x  Æ h + - ans =  cos(x) >> v = [(1 + a/x)^x, exp(­x)];  >> limit(v,x,inf,'left')  %evaluate  lim 1 , lim x x x x a e  x - Æ• Æ• Ê ˆ + Á ˜ Ë ¯ ans =  [ exp(a),  0] ² Difference and approximate derivative diff (X), for a vector X, is [X(2)­X(1) X(3)­X(2) ...  X(n)­X(n­1)].  diff (X), for a matrix X, is the matrix of row differences, [X(2:n,:) ­ X(1:n­1,:)].  diff (X,N) is the N­th order difference along the first non­singleton dimension (denote it by  dim).  If N  >= size(X,dim),  diff takes  successive differences  along the next  non­singleton  dimension.  diff (X,N,DIM) is the Nth difference function along dimension DIM. If N >= size(X,DIM),  diff returns an empty array

ATLAB Lecture 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr h=.001:x=0h sin(d)-sin(i-h) > diff(sin(x. 2))/h; i=hh: pi, there are approximation pl >>diff(1: 10). 2) %evaluate(i+D)'-i4, i 3 51719 >>X=375: 257; diff(X) %the same as diff(X, 1, 1) ans diff(X, 1, 2 >> diff(X, 2, 2) %the 2nd order difference along the dimension 2 diff(X,3, 2) Empty matrix: 2-by-0 > syms x y, y=atan((x+l(x-1)) (1/x-1)(x+1)(x-1)^2)/(1+(x+1)^2/(x-1)^2) (-2/x-1)^2+2°(x+1)(x-1)^3)(1+(x+1)^2/x-1)^2)-(1(x-1)-(x+1)(x-1)^2 (1+(x+1)y^2/(x-1)^2)2*(2*(X+1)x-1)^2-2*(x+1)2/(x-1)3) >> syms x y >>z=x^4+y4-cos(2*x+3y) cos(2x+3 >>diff(z, x) 4*x^3+2*sin(2*x+3*y)

MATLAB Lecture 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec7­4  Examples:  >> h = .001; x = 0:h:pi; >> diff(sin(x.^2))/h;  % 2 2 sin( ) sin( ) , : : i i h  i h h pi  h - - = , there are approximation …  to 2*cos(x.^2).*x, x=0:h:pi >> diff((1:10).^2) % evaluate  2 2  (i +1) -i ,i =1, 2,...,9 ans =  3  5  7  9  11  13  15  17  19  >>X = [3 7 5; 2 5 7]; diff(X) %the same as diff(X,1,1) ans =  ­1  ­2  2  >> diff(X,1,2) ans =  4  ­2  3  2  >> diff(X,2,2) %the 2nd order difference along the dimension 2  ans =  ­6  ­1  >> diff(X,3,2) ans =  Empty matrix: 2­by­0  >> syms x y; y=atan((x+1)/(x­1));  >> yx=diff(y,x) yx =  (1/(x­1)­(x+1)/(x­1)^2)/(1+(x+1)^2/(x­1)^2) >> yxx=diff(y,x,2) yxx =  (­2/(x­1)^2+2*(x+1)/(x­1)^3)/(1+(x+1)^2/(x­1)^2)­(1/(x­1)­(x+1)/(x­1)^2)/  (1+(x+1)^2/(x­1)^2)^2*(2*(x+1)/(x­1)^2­2*(x+1)^2/(x­1)^3) >> syms x y  >> z=x^4+y^4­cos(2*x+3*y);  >> zx=diff(z,x) %compute 4 4 (x y cos(2x 3y )) x ¶ + - + ¶ zx =  4*x^3+2*sin(2*x+3*y) >> zy=diff(z,y) %compute z y ¶ ¶ zy =  4*y^3+3*sin(2*x+3*y)

MATLAB Lecture 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr >>zxxdiff(zx, x)% equivalent to zxx=diff(z, x, 2) ZXX- 12*x^2+4*cos(2*x+3y) ◆ Integrate t(s)is the indefinite integral of s with respect to its symbolic variable as defined by findsym. S is a sym(matrix or scalar). If s is a constant, the integral is with respect to'x int(S, v) is the indefinite integral of S with respect to v. v is a scalar sym. int(S, a, b)is the definite integral of S with respect to its symbolic variable from a to b a and b are each double or symbolic scalars int(s, v, a, b)is the definite integral of s with respect to v from a to b > syms x xI alpha t >>A=[cos(x*t), sin(x*t): -sin(x*t), cos(x*t); /(1+x2))%compute dx without constant item C atan(x) > int(sin(alpha*u), alpha)%compute sin(au )da without constant item C 1/u cos(alpha u) > int(x1*log(1+x1),0, 1) %compute L x In(1+x,dx 1/4 t(4*x°tx,2,sin(t) 2°t°(Sint)2-4) > int(exp(O),exp(alpha D) %compute [e'dt, e"dt exp(t), 1/alpha*exp(alpha t) > int(A, t compute[ cos xtd, sin xtd;[-sin xtdt, cos xdr [I/x*sin(x* t),-cos(x*t)x y Taylor series expansion taylor(f) is the fifth order Maclaurin polynomial approximation to f

MATLAB Lecture 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec7­5  >> zxx=diff(zx,x) %compute  2 2 z x ¶ ¶ , equivalent to zxx=diff(z,x,2) zxx =  12*x^2+4*cos(2*x+3*y) ² Integrate int(S) is the indefinite integral  of S with respect  to its  symbolic  variable as  defined by  findsym. S is a sym (matrix or scalar). If S is a constant, the integral is with respect to 'x'.  int(S,v) is the indefinite integral of S with respect to v. v is a scalar sym.  int(S,a,b) is the definite integral of S with respect to its symbolic variable from a to b. a and b  are each double or symbolic scalars.  int(S,v,a,b) is the definite integral of S with respect to v from a to b.  Examples:  >> syms x x1 alpha u t; >> A = [cos(x*t), sin(x*t);­sin(x*t), cos(x*t)];  >> int(1/(1+x^2)) %compute  2  1 1 dx  + x Ú without constant item C ans =  atan(x) >> int(sin(alpha*u),alpha) %compute sin(au)da Ú without constant item C ans =  ­1/u*cos(alpha*u) >> int(x1*log(1+x1),0,1) %compute 1  1 1 1  0  x ln(1+ x )dx Ú ans =  1/4  >> int(4*x*t,x,2,sin(t)) ans =  2*t*(sin(t)^2­4) >> int([exp(t),exp(alpha*t)]) %compute [ , ] t t  e dt e dt a Ú Ú ans =  [ exp(t), 1/alpha*exp(alpha*t)] >> int(A,t)  %compute [ cos xtdt, sin xtdt; -sin xtdt, cos xtdt] Ú Ú Ú Ú ans =  [ 1/x*sin(x*t),  ­cos(x*t)/x]  [ cos(x*t)/x, 1/x*sin(x*t)] ² Taylor series expansion taylor(f) is the fifth order Maclaurin polynomial approximation to f

ATLAB Lecture 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr Three additional parameters can be specified, in almost any order taylor(f, n)is the(n-1)-st order Maclaurin polynomial taylor(f, a) is the Taylor polynomial approximation about point a taylor(f x)uses the independent variable x instead of findsym(f) > taylor(exp(-x)) %evaluate the first 6 items of Taylor series expansion at O 1-X+1/2*x^2-1/6*x^3+1/24*x^4-1/120*x^5 > taylor(log(x), 6, 1) %evaluate the first 6 items of Taylor series expansion at 1 l-x+1/2*x^2-1/6*x^3+1/24*x4-1/120*x > taylor(sin(x), 6, pi/2) %evaluate the first 6 items of Taylor series expansion at pi/ 1-1/2*(x-1/2*pi)^2+1/24*(X-1/2pi)4 >>taylor(sin(x)t, t) %evaluate the first 6 items of Taylor series expansion responding ans > taylor(sin(x)t, x)%evaluate the first 6 items of Taylor series expansion responding to x x*t-1/6*x^3+1/120°t*x^5 令 Sum of elements S= sum(X) is the sum of the elements of the vector X If X is a matrix. s is a row vector with the sum over each column If X is floating point, that is double or single, S is accumulated natively, that is in the same class as x and s has the same class as x If X is not floating point, S is accumulated in double and S has class double S= sum(X, DIM) sums along the dimension dim >>X=[012,345] >> sum(X) %evaluate a row vector with the sum over each column ans 357 > sum(X, 1) %evaluate a row vector with the sum over each column > sum(, 2) %evaluate a column vector with the sum over each row ans- Lec7-6

MATLAB Lecture 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec7­6  Three additional parameters can be specified, in almost any order.  taylor(f,n) is the (n­1)­st order Maclaurin polynomial.  taylor (f,a) is the Taylor polynomial approximation about point a.  taylor (f,x) uses the independent variable x instead of findsym(f).  Examples:  >> taylor(exp(­x)) %evaluate the first 6 items of Taylor series expansion at 0  ans =  1­x+1/2*x^2­1/6*x^3+1/24*x^4­1/120*x^5  >> taylor(log(x),6,1) %evaluate the first 6 items of Taylor series expansion at 1  ans =  1­x+1/2*x^2­1/6*x^3+1/24*x^4­1/120*x^5  >> taylor(sin(x),6,pi/2) %evaluate the first 6 items of Taylor series expansion at pi/2  ans =  1­1/2*(x­1/2*pi)^2+1/24*(x­1/2*pi)^4  >> taylor(sin(x)*t,t) %evaluate the first 6 items of Taylor series expansion responding …  to t  ans =  sin(x)*t  >> taylor(sin(x)*t,x)  %evaluate the first 6 items of Taylor series expansion responding …  to x  ans =  x*t­1/6*t*x^3+1/120*t*x^5  ² Sum of elements S = sum(X) is the sum of the elements of the vector X.  If X is a matrix, S is a row vector with the sum over each column.  If X is floating point, that is double or single, S is accumulated natively, that is in the same class as X, and S has the same class as X.  If X is not floating point, S is accumulated in double and S has class double.  S = sum(X,DIM) sums along the dimension DIM.  Examples:  >> X = [0 1 2; 3 4 5];  >> sum(X) %evaluate a row vector with the sum over each column  ans =  3  5  7  >> sum(X,1) %evaluate a row vector with the sum over each column  ans =  3  5  7  >> sum(X,2) %evaluate a column vector with the sum over each row  ans =  3

MATLAB Lecture 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr ◆ Symbolic summation symsum(S)is the indefinite summation of s with respect to the symbolic variable determined symsum(S, v)is the indefinite summation with respect to v symsum(S, a, b)and symsum(s, v, a, b) are the definite summation from a to b > syms k n > alsimple( symsum(k)) %evaluate 12+k*(k-1) >>a2-=simple(symsum(k,0, n-1)%evaluate >ik l/2*n°(n-1) >>a3=simple(symsum(k,0, n) %evaluate >ik n°(n+1) le(symsum(k 2, 0, n)) %evaluate 1/6°n°(n+1)°(2*n+1) >>symsum(k,0, 10) %evaluate 510 385 > symsum(k/2, 11, 10) %evaluate >mk2)2%aae∑1 Psi( > symsum(1/k/2, 1, Inf) %evaluate 2 1/6°p^2 Lec7-7

MATLAB Lecture 7  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec7­7  12  ² Symbolic summation symsum(S) is the indefinite summation of S with respect to the symbolic variable determined  by findsym.  symsum(S,v) is the indefinite summation with respect to v.  symsum(S,a,b) and symsum(S,v,a,b) are the definite summation from a to b.  Examples:  >> syms k n  >> a1=simple(symsum(k)) %evaluate  1  0 k i  i -  = a1 =  1/2*k*(k­1) >> a2=simple(symsum(k,0,n­1)) %evaluate  1  0 n  k k -  = a2 =  1/2*n*(n­1) >> a3=simple(symsum(k,0,n)) %evaluate  0 n  k k  = a3 =  1/2*n*(n+1) >> a4=simple(symsum(k^2,0,n)) %evaluate  2  0 n  k k  = a4 =  1/6*n*(n+1)*(2*n+1) >> symsum(k^2,0,10) %evaluate  10  2  k 0 k  = ans =  385  >> symsum(k^2,11,10) %evaluate  10  2  k 11 k  = ans =  0  >> symsum(1/k^2) %evaluate  1  2  k 1 i  i  = ans =  ­Psi(1,k) >> symsum(1/k^2,1,Inf) %evaluate  1  2  1 k k •  = ans =  1/6*pi^2