实验4微积分运算(四)积分运算等 本文档介绍求导数的另一种方法:直接执行菜单命令.以及其他的微积分运算 (1)求函数的导函数与在特定点处的导数值: 直接对函数表达式执行菜单命令求导 sin(x).e cos(x) 选中自变量x,执行菜单命令 symbols/ ariable/Differentiate得到: cos(x)·exp(x)+sin(x)·exp(x)-2·cos(x):sin(x) 一阶导数,重复操作得到: 2·cos(x)·exp(x)+2·sin(x)-2·cos(x) 二阶导数,重复操作得到: -2sn(x:exp(x)+2cosx;exp(x)+8smx:cox)三阶导数,重复操作得到: 4·sin(x)·exp(x)+8·cos(x)-8·sin(x) 四阶导数,重复操作得到: 定义函数,应用 Calculus运算板上的求导数按钮: f(x): sin(x).e + cos(x) f(x)→>cos(x)·exp(x)+sin(x)·exp(x)-2.cos(x)·sin(x) →-8·cos(x)·exp(x)+32.sin(x)--32.cos(x) -I d f(x)=3.027 f(x)=-29.427 f(x)→>2.cos dxd 3)2m()2 (2)求不定积分和定积分 f(x)dx→:cos(x):exp(x)+·sin(x)·exp(x)+÷·cos(x)·sin(x)+ fx)dk→1.csp(x)+1 13.641 6 cosx3dkx→ CoSIX sin( x)+
cos(x) x 3 óô ô õ d 1 3 cos(x) 2 × × sin(x) 2 3 ® + × sin(x) 0 p f(x) x ó ô õ d 1 2 × exp(p) 1 2 + × p 1 2 ® + = 13.641 0 1 2 3 3 6 9 f(y) p y f(x) x ó ô õ d -1 2 × cos(x) × exp(x) 1 2 + × sin(x) × exp(x) 1 2 + × cos(x) × sin(x) 1 2 ® + × x (2) 求不定积分和定积分: x x f(x) d d d d 2 cos 1 3 × p æ ç è ö ÷ ø × exp 1 3 × p æ ç è ö ÷ ø × 2 sin 1 3 × p æ ç è ö ÷ ø 2 × 2 cos 1 3 × p æ ç è ö ÷ ø 2 ® + - × 5 x f(x) d d 5 = -29.427 x f(x) d d x = 3.027 p 3 := n x f(x) d d n -8 × cos(x) × exp(x) 32 sin(x) 2 × 32 cos(x) 2 n := 6 ® + - × x f(x) d d f(x) sin(x) e ® cos(x) × exp(x) + sin(x) × exp(x) - 2 × cos(x) × sin(x) x × cos(x) 2 := + 定义函数, 应用Calculus运算板上的求导数按钮: -4 × sin(x) × exp(x) 8 cos(x) 四阶导数, 重复操作得到: 2 × 8 sin(x) 2 + - × 三阶导数, 重复操作得到: -2 × sin(x) × exp(x) + 2 × cos(x) × exp(x) + 8 × sin(x) × cos(x) 2 × cos(x) × exp(x) 2 sin(x) 二阶导数, 重复操作得到: 2 × 2 cos(x) 2 + - × cos(x) × exp(x) + sin(x) × exp(x) - 2 × cos(x) × sin(x) 一阶导数, 重复操作得到: sin(x) e 选中自变量x, 执行菜单命令symbols/Variable/Differentiate得到: x × cos(x) 2 + 直接对函数表达式执行菜单命令求导: (1) 求函数的导函数与在特定点处的导数值: 本文档介绍求导数的另一种方法: 直接执行菜单命令. 以及其他的微积分运算. 实验4 微积分运算(四） 积分运算等
x·exp(x)dx→>x·exp(x)-3·x·exp(x)+6x·exp(x)-6·exp(x) x+ 1)dx -p cos T p+a si eyp(a·)-sn(B·x)d→ex{-·πa a:=2b:=3 a" sin(x)+b cos(x)+1 (3)级数展开 si(x).e+co(x)2选中自变量x执行菜单命令 symbols/Variable/Expand to Series得到: +1·x+ f(t): sin(t)e+ cos(t) tst→1+t+1.3+1.4-1.5-1·使用符号运算板上的 Iseries按钮 (4)求函数的极限 k=1 In(sin(x)) y+x-13 x→0 im(1+3co(x)0)→exp(3)limx(mx+1)-ln(x)→0
x 0 lim x × (ln(x + 1) - ln(x)) + ® ® 0 p x (1 + 3 × cot(x)) sec(x) lim ® ® exp(3) x 0 x × sin(x) + 1 - 1 exp x 2 ( ) - 1 lim ® 1 2 ® x 0 1 + x - 1 3 1 + x - 1 lim ® 3 2 ® p 2 x ln(sin(x)) (p - 2 × x) 2 lim ® -1 8 ® n ¥ 2 n n! lim ® ® 0 n ¥ n 1 n k Õ (2 × k - 1) = 1 n k Õ (2 × k) = lim ® ® 1 1 ¥ k 1 k å 2 = 1 6 p 2 ® × n ¥ lim S(n) ® 1 6 p 2 S(n) ® × 1 n k 1 k å 2 = := (4) 求函数的极限 f(t) series,t, 7 1 + t 使用符号运算板上的series按钮 1 3 t 3 + × 1 3 t 4 × 1 30 t 5 - × 1 18 t 6 ® + - × f(t) sin(t) e t × cos(t) 2 := + 1 + 1 × x 1 3 x 3 + × 1 3 x 4 × 1 30 x 5 + - × O x 6 + ( ) sin(x) e 选中自变量x, 执行菜单命令symbols/Variable/Expand to Series得到: x × cos(x) 2 + (3) 级数展开 0 p 2 x 1 a 2 sin(x) 2 × b 2 cos(x) 2 + + 1 ó ô ô ô õ d 1 20 a := 2 b := 3 ® × 2 × p 0 p 2 exp(a × x) × sin(b × x) x ó ô ô õ d exp 1 2 × p × a æ ç è ö ÷ ø -b cos 1 2 × p × b æ ç è ö ÷ ø × a sin 1 2 × p × b æ ç è ö ÷ ø + × æ ç è ö ÷ ø a 2 b 2 ( + ) × b a 2 b 2 ( + ) ® + 0 1 x × ln(x + 1) x ó ô õ d 1 4 x ® 1 sin(x) m ó ô ô ô õ d x 1 sin(x) m ó ô ô ô õ ® d x x 3 × exp(x) óô ô õ d x 3 × exp(x) 3 x 2 ® - × × exp(x) + 6 × x × exp(x) - 6 × exp(x)
lim sin(o).e sn(x)-X·coS(X imnt".e-t→0 →0 (5)求弧长和旋转体体积 1+f(x)2dx=1415 0到π上曲线f(x)的弧长 x)-dx→-.π·exp(2 40=243.932
2 x ® t 0 sin(t) t 2 × sin 3t 3 ( ) lim ® 1 3 ® t 0 sin(t) e t × sin(3t) lim + ® 1 3 ® t ¥ 1 4 t - æ ç è ö ÷ ø t lim ® ® exp(-4) x 0 sin(x) - x × cos(x) sin(x) 3 lim ® 1 3 ® t ¥ t n e - t lim × ® ® 0 t 0 ln 3t 2 ( + 1) t × sin(t) lim ® ® 3 x 1 x 1 1-x lim ® ® exp(-1) x 1 (1 - x) tan p 2 × x æ ç è ö ÷ ø lim × ® 2 p ® x 0 ln 1 x æ ç è ö ÷ ø æ ç è ö ÷ ø x lim + ® ® 1 x ¥ 2 p × atan(x) æ ç è ö ÷ ø x lim ® exp -2 p æ ç è ö ÷ ø ® x 1 2 p × acos(x) æ ç è ö ÷ ø 1 x lim - ® ® 0 (5) 求弧长和旋转体体积 0 p 1 f(x) x 2 + ó ô õ d = 14.156 0 到 p 上曲线f(x)的弧长. 0 p p f(x) x 2 × ó ô õ d 1 8 × p × exp(2 × p) 2 5 + × p × exp(p) 3 8 p 2 + × 11 40 ® + × p = 243.932
∑ n:=10000,50000.100000 31 14149716394721 3.14157355512957 3.1415820433013
S(n) 1 n k 1 k å 2 = := n := 10000, 50000.. 100000 6 × S(n) 3.14149716394721 3.14157355512957 3.1415820433013 =
