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Math 254 TA: Gangyong Lee § 14 The deﬁnition of the limit, tangent plane and the partial deriva- tive 1) In the function y = f (x) if the value of the right limit and the value of the left limit are the same then there exists the limit whose value is the same as the above value. That is, if lim− f (x) = lim+ f (x) then lim f (x) = lim− f (x) = lim+ f (x) x→a x→a x→a x→a x→a We call lim f (x) = A x→a if for any , there exists δ > 0 such that |x − a| < δ ⇒ |f (x) − A| < . 2) In the function z = f (x, y) we call lim f (x, y) = A (x,y)→(a,b) if for any , there exists δ > 0 such that (x − a)2 + (y − b)2 < δ ⇒ |f (x, y) − A| < . 3) tangent line i) the general linear equation; ax + by + c = 0 ii) the equation of the line which goes through the point (x0 , y0 ); a(x − x0 ) + b(y − y0 ) = 0 iii) the equation of the tangent line of y = f (x) at the point (x0 , y0 ); y − y0 = f (x0 )(x − x0 ) 1 4) tangent plane i) the general plane equation; ax + by + cz + d = 0 ii) the equation of the plane which goes through the point (x0 , y0 , z0 ); a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 iii) the equation of the tangent plane of z = f (x, y) at the point (x0 , y0 , z0 ); z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) 5) The diﬀerence between the derivative and the partial derivative. i) If we represent z as the function only w.r.t. t then we use dz . dt ii) If we represent z as the function with respect to more than 1 variable then we use ∂z . ∂t y Example #5 In the problem w = xe z and x = t2 , y = 1−t, z = 1+2t 1−t since w = t2 e 1+2t we have to ﬁnd dz instead of ∂z . dt ∂t Example #7 In the problem z = x2 + xy + y 2 and x = s + t, y = st since z = (s + t)2 + (s + t)st + (st)2 we have to ﬁnd ∂z and ∂z instead ∂s ∂t of dz and dz , respectively. ds dt 2