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. . ( )
. - () Oxy - -, . x y . , . - : A1x+B1y+C1=0;A2x+B2y+C2=0 , .. 1/2 1/2, - - . . - -: 1) - -, - -: y-y1=k(x-x1). 2) - k- , - , M1(x1, y1), , Oy . -:- , - A(3;-2), y+2=k(x-3). 3) - , - : - :k=y2-y1/x2-x1. y-y1=y2-y1/x2-x1 * (x-x1). 4) - - - x/a +y/b=1. 5) - : ,( ) - (+By+C=0) - Oxy. +By+C=0 - - .
| 33. . 1- . . - - , -, . . -. - - , - y=f(x) , . Dy=f(u) du - - -. 34. . ( ). F(x) f(x) , , : F(x)=f(x) F(x)+C y=f(x) y=f(x) ⌠f(x)dx=F(x) +C 1.(f(x)dx)=f(x) - (⌠f(x)dx)=(F(x)+C)=F(x)+C=f(x) 2.d(⌠f(x)dx)=f(x)dx - 3.⌠dF=F+C - 4.⌠()=⌠() 5.⌠(()+-())= ⌠()+-⌠() 38. . . . . Φ(x)= x⌠f(x)dx = x⌠f(t)dt a a Φ(x ) =f(x) - Φ(x )=lim ∆φ/∆x= lim φ(x+∆x)- φ(x)/ ∆x= lim x+∆x ∫ f(t)dt-x⌠ f(t)dt/ ∆x = lim x⌠ f(t)dt+ + x+∆x ⌠f(t)dt- x⌠ f(t)dt/∆x = lim (x+∆x-x)*f(ξ)/ ∆x = lim f(ξ)= f(x) a a .. Φ(x)- f(x). .. x⌠ f(t)dt = F(x)+C a -. a b [a;b]. b⌠f(x)dx = F(b)-F(a) a 1. x=a a⌠f(t)dt=F(a)+C a F(a)+C→C= -F(a) 2.x=b b⌠f(t)dt=F(b)-F(a) a 39. . ( ). . +∞∫ f(x) dx - f(x) [a;+∞] - - (t) t, +∞. +∞∫-∞ e-x2/2 dx -. 26. ( ). - ( .-). - - , . .-:- - x1 x2 . x2>x1, x1,x2 - ., f(x2)>f(x1). f(x2)-f(x1)=f(a)(x2-x1), 1< a <x2 => f(a)>0. f(x2)-f(x1)>0 f(x2)>f(x1). - ( .-). - - , . | 35. . - ∫ f(x) dx = ∫ f(φ(t)) φ (t) dt; - . - b∫a f(x) dx = b∫a f(φ(t)) φ dt. 36. ( ). . U=U(x) V=V(x) .. d(uv) = (uv)dx=uvdx+uvdx= du*v+u*dv, , ⌠d(uv)= ⌠vdu+⌠udv uv=⌠vdu+⌠udv , . . ⌠lnx*x8dx = {u=lnx;dv= x8dx; du = 1/8dx; v= ⌠ x8dx= x9/9}=lnx* x9/9-⌠ x9/9-1/xdv=lnx* x9/9-1/9⌠ x8dx=lnx* x9/9-1/9* x9/9+C . b⌠udv=(uv-⌠vdu)b│ a a u=u(x), v=v(x) b⌠udv=uvb│-⌠ b vdu a a a 37. . . . max i , . y=f(x) [a,b], b∫a f(x) dx, - y=f(x) - [a,b]. - .: 1. . 2. - -. 3. , . 4. . 5. . - y=f(x) [a,b], ( a<b), ξ [a,b], b∫a f(x) dx = f(ξ)(b-a). 50. y= (1+x)m . . y= (1+x)m, m f(x) = (1+x)m f(x) = m+(1+x)m-1 f(x) = m(m-1)(1+x)m-2 f(x) = m(m-1)(m-2)(1+x)m-2 f(n)(x) = m(m-1).(m-n+1)(1+x)m-n x=0 f(0) = 1 f(0) = m f(0)= m(m-1) f(0)= m(m-1)(m-2) f(n) (0) = m(m-1).(m-n+1) (1+x)m = 1+mx+m(m-1)/2!x2 + m(m-1)(m-2)/3!x3 +..+ m(m-1)(m-n+1)/n!xn - (-1;1) | 43. 1- . . - - , y=g(y/x). - - - - -. - y=f(x,y) - k ( x y), α f(αx, αy)=αk f(x,y) -: f(x,y)=x2 xy. f(αx, αy)=(αx)2 (αx)(αy)=α2(x2 xy)= α2 f(x,y), - - 2. - - - , y+f(x)y=g(x). , - g(x) , - - -, -. 44. . . (). . , . u1+u2+.un= ∑ un=S Sn ( ) S=lim Sn 1.1-+1-1+1-1+1. S1=1; S2=0; S3=1 - , , , n→∞, - . -. -, Un n→∞, 0 lim Un=0 n→∞, lim Un=lim (Sn-Sn-1)= limSn-lim Sn-1= S-S=0 n→∞, -. lim Un≠0 - -. . ∑4n+5/3n+7 n=1 lim 4n+5/3n+7= lim 4n/3n≠0 - n→∞, 47. . . . , , . a1-a2+a3-a4+an an>0 -, 2 1. ., . an≥an+1 2. lim an=0 n→∞ . - 1-½2+⅓2 +(-1)n-1/n2 .. 1>½2>⅓2 lim 1/n2=0 -. , - , . , -, , -. - , . , , .., - . 1-1/2+1/3-1/4. -. | ||
| 28. ( ). . . 0 - - y=f(x) , 0 - y=f(x), , - . . , .. (, 0) (f (x) >0), (0, b) (f (x) < 0). f(x) (, 0) (0, b). f(x0) > f(x) (, 0), f(x) < f(x0) (0, b), .. f(x0)≥f(x) (, b), , 0 y=f(x). . . f(x) 0, f(x0) , 0 f(x); f(x0) , x0 . . f(x0) =0, f (x0) >0. , f (x) = (f(x0)) >0 0, .. f(x) (a, b), 0. f(x0) =0, , (, 0) f (x) >0, .. f (x) 0 , .. 0 . | 40. . .
1. - y=f(x) [a,b]. . . S y=f(x) [a,b] ., .. S = b∫a f(x)dx. 2. - y=f(x) [a,b]. S = b∫a (-f(x)) dx, .. S = - b∫a f(x)dx. 3. - . , S=S1+S2+S3, .. . .: S = c∫a f(x)dx - d∫c f(x)dx + b∫d f(x)dx. 4.-. - y=f1(x) y=f2(x) , f2(x) > f1(x). S , y=f2(x) y=f1(x), : S = b∫a (f2(x) f1(x)) dx. -: - -, . y=x2-2, y=x.(.11.18).-: : y=x2-2 y=x => (-1;-1) (2;2). - [-1,2] x > x2-2. f2(x)=x, f1(x)=x2-2. S=2∫-1 (x-(x2-2)) dx = x2/2 2|-1 x3/3 2|-1 +2x 2|-1 =1/2(4-(-1)2) 1/3(23-(-1)3) +2(2-(-1)) = 4,5 (.2).
| 48. . . =x (). . - .: - f(x) - .,.. f(x)=∞∑n=0cnxn. [;b], - (-R;R), - f(x) - , -, . : ∫b f(x)dx = ∫bc0dx + ∫bc1xdx + + ∫bcnxndx + , - . : f(x) = c1 + 2c2x + 3c3x2 +... + ncnxn-1 +... - R. () = (0) + f(0) + ((f(0))/2!)2 + ((f(0))/3!)x3 +.. + ((f(n)(0))/n!) xn +.. , f(x) f (x)=S n (x) + r n (), Sn(x)- n- ; rn(x) - n- . - =. 1. = () = f() = f() =.. = f(n)(x) = ex f(0) = f(0) =f(0) =.. = (n)(0) = e0=1. - = 1 + + 2/2! + 3/3! + + n/n! + - (-∞;∞). | 42. 1- ( , ) . . - - - , , .. , . y=f(x,y). -. - - - f(x,y) f/y Oxy. : 1) (x0,y0) - y=y(x) -, - y0=y(x0); 2) y=y1(x) y=y2(x) - x=x0, .. y1(x0)=y2(x0), , . -= - () = . =1 = =() 0=(0) =0-0 =() ==0-0=0-0 =0 |
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