根據(jù)導數(shù)的幾何意義,函數(shù)y ＝ f(x)在點x0處的導數(shù)f'(x0)在幾何上表示曲線y = f(x)在點M(x0,f(x0))處的切線的斜率,即
f'(x0) = tanα, 其中α是切線的傾角.
由直線的點斜式方程,知曲線y = f(x)在點M(x0,y0)處的切線方程是
y - y0 = f'(x0)(x-x0)
過點M(x0,y0)且與切線垂直的直線叫做曲線y=f(x)在點M處的法線.如果f'(x0)≠0 ,法線的斜率為 -1/f'(x0),從而法線方程為
y - y0 = -1/f'(x0)(x - x0)
對于本題,
y = x^2+2x+3 的導數(shù)為
y' = 2x +2
則在點M0(2,11)處的導數(shù)為 f'(2) = 2*2 +2 = 6
即過點M0(2,11)的切線斜率為k = 6
則切線方程為
y-11 = 6(x-2)
即 y = 6x - 1
易知法線斜率為 k' = -1/6 = -1/6
所以在點M0(2.11)處的法線方程為
y -11 = -1/6(x-2)
即 x + 6y -68 = 0