7.求函數 f(x,y,z) = x²y + y²z + z²x - 23 的偏導函數, 並求曲面 x²y + y²z + z²x = 23 在點 (1,2,3) 處的切平面的方程式。(13 分)

(13) A logistic population model with relative growth rate 0.1 per year and carrying capacity 50 thousand can be expressed by the differential equation =0.1P(1 - ), with P in thousands and t in years. Given that the initial population is 9 thousand. Find the population size after 20 years. (If you memorized the formula, then you need to derive it for this problem.)