1 1. Find the function y=x? -2x？ Maximum and minimum values of +x+5 in the interval [- 1, 1].

Solution: Let y ′ = 3x? -4x+1= (3x-1) (x-1) = 0, then the stagnation point x? = 1/3，x？ = 1，x？ Is it the maximum point, x? Is a minimal point;

x？ ，x？ ∈ [- 1, 1], so the maximum value in the interval [- 1,1] = y (1/3) =1/27-2/9. Minimum value =y( 1)

= 1-2+ 1+5=5; The left end of the interval y (-1) =-1-2-1+5 = 3; Therefore, the maximum value on [- 1, 1] is139/27 = 5.148;

The minimum value is 3.

12。 Looking for uncertain extreme points:

( 1)。 ∫[2x/( 1+x？ )]dx=∫d( 1+x？ )/( 1+x？ )=ln( 1+x？ )+C

(2)。 ∫(xe^x+cosx)dx=∫x(e^x)dx+∫cosxdx=∫xd(e^x)+sinx=xe^x-∫(e^x)dx+sinx=xe^x-e^x+sinx+c

=(x- 1)e^x+sinx+c；

13。 It is known that y=xln[x+√( 1+x? )], find dy/dx.

Solution: dy/dx=ln[x+√( 1+x? )]+x[ 1+x/√( 1+x？ )]/[x+√( 1+x？ )]=ln[x+√( 1+x？ )]+x/√( 1+x？ )

14。 Find the limit x? ∞lim[(4x？ -3)? (3x-2)？ /(6x？ +7)? The highest degree of x in numerator and denominator is 10, so its limit is equal to.

Coefficient ratio of the highest term

Solution: The original formula =(4? ×3? )/6? =5 184/7776= 162/243=2/3

1 1。 Find the definite integral 0, 1 ∫ (2xsinx? +xe^x)dx

Solution: Original formula = 0, 1 [∫ sinx? d(x？ )+∫xd(e^x)]=[-cosx？ +(x- 1)e^x]∣0, 1=-cos 1-(-cos0- 1)=2-cos 1

12。 Find the function f(x)=4x? -Five times? Extreme value in all real number fields +6.

Solution: let dy/dx= 12x? -10x=2x(6x-5)=0, stagnation point x? =0，x？ =5/6; x？ Is it the maximum point, x? Is a minimal point;

Therefore, the maximum value f (x) = f (0) = 6; The minimum value f(x)=f(5/6)=4×(5/6)? -5×(5/6)? +6=523/ 108;

13。 Can't you find a fixed point ∫x? e^(x？ )dx

Solution: The original formula =( 1/2)∫x? d(e^x？ )=( 1/2)[x？ e^(x？ )-2∫xe^(x？ )dx]=( 1/2)[x？ e^(x？ )]-( 1/2)∫d(e^x？ )

=( 1/2)[x？ e^(x？ )]-( 1/2)(e^x？ )+C=( 1/2)(x？ - 1)e^(x？ )+C

14。 Find the curve y=x +3x? Concave-convex interval and inflection point coordinates -x- 1

Solution: Let y'=3x? +6x- 1=0, y"=6x+6=6(x+ 1), because y "< 0, the curve is concave downward in the interval (-∞,-1); Y "on [- 1, +∞] >: 0, so the curve is concave upward in the interval [- 1, +∞]. The coordinate of the inflection point is (-1, 2).