MATLAB Exercise 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr MATLAB Exercise 7-Calculus 1. Try to express y to be a compose function of x 1)yl=√1+u2,u=ex 2)y2=√1+u2,u=lnv,v 3)y=v1+u, u=Inv, v=sinw,w=e 2. Try to generate the converse function of y. 1)y=√1+ln2sin 2)y=x+Insin u, where u is as the independent variable 3)y=x+Insin u, where x is as the independent variable 3. Letx= Try to evaluate the value of sin x(express the result as 1/22 (1/2))and arcsin(sin x)(express the result as pi/4 4. Compute the following limits, and simplify them tan nx-sin mx 1)li 3)lim lim 5.1)C the limit lim (x+h)2-x and simplify it h 2)Compute (x")by diff 6. Compute the following derivatives x3y-5 g(x,y)= 2xsin 3 7. Let A=La,a2,,,]is a vector with A=[1,5,8,-2,6,3]J,h generate a new vector 1)B=[a Ex7-1
MATLAB Exercise 7 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Ex71 MATLAB Exercise 7 – Calculus 1. Try to express y to be a compose function of x. 1) 2 1 1 , x y u u e - = + = ; 2) 2 2 1 , ln , x y u u v v e - = + = = ; 3) 2 1 , ln , sin , x y u u v v w w e - = + = = = . 2. Try to generate the converse function of y. 1) 2 y = 1+ ln sin x 2) y = x + ln sin u , where u is as the independent variable 3) y = x + ln sin u , where x is as the independent variable 3. Let 4 x p = . Try to evaluate the value of sin x (express the result as 1/2*2^(1/2) ) and arc sin(sin x) (express the result as pi/4 ). 4. Compute the following limits, and simplify them. 1) 0 tan sin lim x nx mx Æ x - 2) lim x y x y e e Æ x y - - 3) 3 lim x 2 100 x Æ+• x + 4) 3 lim x sin x Æ-• x 5) tan 2 4 lim (tan ) x x x p + Æ 6) lim tan x 2 x p Æ - 5. 1) Compute the limit 0 ( ) lim n n h x h x Æ h + - , and simplify it. 2) Compute ( ) n x ¢ by diff. 6. Compute the following derivatives 1) x 0 dg dx = for 3 2 5 ( ) 2 7 x g x x - = + 2) 2 x 1 d g dxdy = for 3 2 5 ( , ) 2 7 x y y g x y x - = + 3) x 1, y 2 dg dy = = for 3 2 5 ( , ) 2 7 x y y g x y x - = + 4) (5) f for f = sin x sin 2x sin 3x 7. Let 1 2 [ , ,..., ] A n = a a a is a vector with n elements, say A = [1,5,8,-2,6,3], how can we generate a new vector 1) 1 2 2 3 1 [ , ,..., ] B n n a a a a a a = - - - - ?
MATLAB Exercise 7 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr 2)C=[a1-2a2+a3,a2-2a3+a4…,an-2-2an1+an] Help Select Matlab Help in the toolbar, then select Index and input diff to see its different 8. Calculate the following calculus )∫d 1 2) x x+1 (x2y+1) l) Help Select Matlab Help in the toolbar, then select Index and input int to know the usage of this function, for example: int(f,x, -inf, inf) 9. Let f=x +1, compare it with the results of int( diff(f)and diff( int(f), respectively 10. Compute the following summations ∑k2 k ∑ 11. Evaluate Taylor series expansion to to the first 15 iter 2)f(x)=e-at point-I to the first 9 items 3)f(x)=e- the first 5 items of Taylor series expansion responding to x 12. *Compare the result.(cosx+2x)dr with sinl a+b),,(a+b) 2(b-a)when b equals to a+10, a+5z, a+T, a+1/2T, a+1/64, a+1/2567,respectively.What conclusion you may reach? 13.*Examine integral mean-value theorem, that is for any f(eCLa, b, there is a e(a, b) such that/(xkr=f(5)(b-a). For example, try to find out the 5E(0,1),such (x+1)(2+1)2 Ex7-2
MATLAB Exercise 7 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Ex72 2) 1 2 3 2 3 4 2 1 [ 2 , 2 ,..., 2 ] C n n n a a a a a a a a a = - + - + - - - + . Help Select Matlab Help in the toolbar, then select Index and input diff to see its different usage. 8. Calculate the following calculus 1) 1 1 dx x + Ú 2) 1 0 1 1 dx x + Ú 3) 0 1 1 t dx x + Ú 4) 2 sin ( 1) y dx x y +• -• + Ú 5) 2 sin ( 1) y dxdy x y +• +• -• -• + Ú Ú . Help Select Matlab Help in the toolbar, then select Index and input int to know the usage of this function , for example: int(f,x,inf,inf) 9. Let 2 f = x +1, compare it with the results of int(diff(f)) and diff(int(f)), respectively. 10. Compute the following summations 1) 3 1 n k k  = 2) 1 2 1 1 k k • = -  3) 2 2 1 1 k k • = -  4) 2 1 k k k x •  = 11. Evaluate Taylor series expansions of 1) 2 ( ) x f x = e at point 0 to the first 15 items; 2) 2 ( ) x f x = e at point 1 to the first 9 items; 3) 2 ( ) xy f x = e the first 5 items of Taylor series expansion responding to x. 12. *Compare the result (cos 2 ) b a x + x dx Ú with ( ) sin 2 ( ) 2 2 a b a b b a Ê Ê + ˆ + ˆ Á ˜ + - Á ˜ Ë Ë ¯ ¯ when b equals to a +10p , a + 5p , a +p , a+1/2p , a +1 64p , a +1 256p , respectively. What conclusion you may reach? 13. *Examine integral meanvalue theorem, that is for any f (x)ŒC[a,b], there is ax Œ(a,b) , such that ( ) ( )( ) b a f x dx = f x b - a Ú . For example, try to find out the x Œ(0,1) , such that 1 2 2 0 1 1 ( 1) ( 1) dx x x = + + Ú
