Solution: Let y'= 2x- 1/x = 1 and get 2x? -x-1= (2x+1) (x-1) = 0, then x? =- 1/2 (truncation); x？ = 1; Correspondingly, y? = 1.
That is, the coordinates of the tangent point are (1, 1).
2. Find the curve y=-x? +x？ A graphic area surrounded by +2x and x axes.
Solution: make y=-x? +x？ +2x=-x(x？ -x-2)=x(x-2)(x+ 1)=0，x？ =- 1; x？ =0; x？ =2.
When x, the domain of the curve is (-∞,+∞).
It is infinite and cannot be calculated. Only the areas within [- 1, 0] and [0,2] can be calculated. The area in [- 1, 0] is negative, so it should be absolute.
S=│(- 1，0)∫(-x？ +x？ +2x)dx│+(0，2)∫(-x？ +x？ +2x)dx
=│[-x？ /4+x？ /3+x？ ](- 1，0)│+[-x？ /4+x？ /3+x？ ](0,2)
=│-(- 1/4- 1/3+ 1)│+[-4+8/3+4]
=│-5/ 12│+8/3=5/ 12+8/3=37/ 12.