函數(shù)的極限-read

1、實驗3 導數(shù)及偏導數(shù)運算,實驗目的：,1. 進一步理解導數(shù)概念及幾何意義；,2. 學習Matlab的求導命令與求導法。,學習 Matlab 命令導數(shù)概念求一元函數(shù)的導數(shù)求多元函數(shù)的偏導數(shù)求高階導數(shù)或高階偏導數(shù)求隱函數(shù)所確定函數(shù)的導數(shù)與偏導數(shù),實驗內(nèi)容：,1. 學習Matlab命令,建立符號變量命令 sym 和 syms 調(diào)用格式：,x=sym(‘x’),建立符號變量 x；,syms x y z,建立多個符號變量 x

2、,y,z；,Matlab 求導命令 diff 調(diào)用格式：,diff(f(x))，,求 的一階導數(shù) ；,diff(f(x),n)，,diff(f(x，y), x)，,求 對 x 的一階偏導數(shù) ；,diff(函數(shù)f(x，y),變量名 x,n)，,求 對 x 的 n 階偏導數(shù) ；,jacobian([f(x,y,z),g(x,y

3、,z),h(x,y,z)],[x,y,z]),matlab 求雅可比矩陣命令 jacobian，調(diào)用格式：,2. 導數(shù)的概念,導數(shù)為函數(shù)的變化率，其幾何意義是曲線在一點處的切線斜率。,1). 點導數(shù)是一個極限值,例1 .,解：,syms h; limit((exp(0+h)-exp(0))/h,h,0),ans=1,2). 導數(shù)的幾何意義是曲線的切線斜率,畫出 在x=0處(P(0,1))的切線及若干條割線，

4、觀察割線的變化趨勢.,例2,解:在曲線 上另取一點 ，則PM的方程是:,即,取h=3,2,1,0.1,0.01，分別作出幾條割線.,h=[3,2,1,0.1,0.01];a=(exp(h)-1)./h;x=-1:0.1:3;plot(x,exp(x),'r');hold onfor i=1:5;plot(h(i),exp(h(i)),'r.')

5、plot(x,a(i)*x+1)endaxis square,作出y=exp(x)在x=0處的切線y=1+x,plot(x,x+1,’r’),從圖上看，隨著M與P越來越接近，割線PM越來越接近曲線的割線.,3. 求一元函數(shù)的導數(shù),例3 .,1) y=f(x)的一階導數(shù),解：,輸入指令,syms x;dy_dx=diff(sin(x)/x),得結果: dy_dx=cos(x)/x-sin(x)/x^2.,pretty(dy_

6、dx) cos(x) sin(x) ------ - ------ x 2 x,,,,,,在 matlab中，函數(shù) lnx 用 log(x)表示， log10(x) 表示 lgx。,例4

7、,解：,輸入指令,syms x;dy_dx=diff(log(sin(x))),得結果: dy_dx=cos(x)/sin(x).,,,,,,例5,解：,輸入指令,syms x;dy_dx=diff((x^2+2*x)^20),得結果: dy_dx=20*(x^2+2*x)^19*(2*x+2).,,,,,,例6,解：,輸入指令,syms a x;a=diff([sqrt(x^2-2*x+5),cos(x^2)+2*cos(2

8、*x),4^(sin(x)),log(log(x))]),Matlab 函數(shù)可以對矩陣或向量操作。,a = [ 1/2/(x^2-2*x+5)^(1/2)*(2*x-2), -2*sin(x^2)*x-4*sin(2*x), 4^sin(x)*cos(x)*log(4), 1/x/log(x)],解：,輸入命令,2) 參數(shù)方程確定的函數(shù)的導數(shù),例7,dy_dx =

9、 sin(t)/(1-cos(t)),syms a t;dx_dt=diff(a*(t-sin(t)));dy_dt=diff(a*(1-cos(t))); dy_dx=dy_dt/dx_dt.,syms x y z;du_dx=diff((x^2+y^2+z^2)^(1/2),x)du_dy=diff((x^2+y^2+z^2)^(1/2),y) du_dz=diff((x^2+y^2+z^2)^(1/2),z)a=jac

10、obian((x^2+y^2+z^2)^(1/2),[x y,z]),,解：輸入命令,4. 求多元函數(shù)的偏導數(shù),例8,du_dx=1/(x^2+y^2+z^2)^(1/2)*x du_dy =1/(x^2+y^2+z^2)^(1/2)*y du_dz = 1/(x^2+y^2+z^2)^(1/2)*z,解：,輸入命令,syms x y;diff(atan(y/x),y),ans = -y/x^2/(1+y^2/x^2),sy

11、ms x y;diff(atan(y/x),x),ans = 1/x/(1+y^2/x^2),syms x y;Jacobian([atan(y/x),x^y],[x ,y]),ans = [ -y/x^2/(1+y^2/x^2), 1/x/(1+y^2/x^2)][ x^y*y/x, x^y*log(x)],5. 求高階導數(shù)或高階偏導數(shù),例10,syms x ;diff(x^2

12、*exp(2*x),x,20),解：輸入命令,ans = 99614720*exp(2*x)+20971520*x*exp(2*x)+1048576*x^2*exp(2*x),例11,syms x y ;dz_dx=diff(x^6-3*y^4+2*x^2*y^2,x,2)dz_dy=diff(x^6-3*y^4+2*x^2*y^2,y,2)dz_dxdy=diff(diff(x^6-3*y^4+2*x^2*y^2,x),y),

13、解：輸入命令,dz_dx = 30*x^4+4*y^2 dz_dy = -36*y^2+4*x^2 dz_dxdy =8*x*y,6. 求隱函數(shù)所確定函數(shù)的導數(shù)或偏導數(shù),例12,syms x y ;df_dx=diff(log(x)+exp(-y/x)-exp(1),x)df_dy=diff(log(x)+exp(-y/x)-exp(1),y)dy_dx=-df_dx/df_dy,解：,df_dx = 1/x+y/x^2*e

14、xp(-y/x)df_dy = -1/x*exp(-y/x)dy_dx = -(-1/x-y/x^2*exp(-y/x))*x/exp(-y/x),例13,syms x y z;a=jacobian(sin(x*y)+cos(y*z)+tan(x*z),[x,y,z])dz_dx=-a(1)/a(3)dz_dy=-a(2)/a(3),解：,a = [ cos(x*y)*y+(1+tan(x*z)^2)*z,

15、cos(x*y)*x-sin(y*z)*z, -sin(y*z)*y+(1+tan(x*z)^2)*x]dz_dx =(-cos(x*y)*y-(1+tan(x*z)^2)*z)/(-sin(y*z)*y+(1+tan(x*z)^2)*x)dz_dy =(-cos(x*y)*x+sin(y*z)*z)/(-sin(y*z)*y+(1+tan(x*z)^2)*x),,輸入命令：,syms x y;f=(x^2-2*x)*exp(