Multivariate Calculus - Bruce E. Shapiro

98 LECTURE 13. TANGENT PLANES
Therefore
f (4) (x) = − 15
16 (x + 1)−7/2 ⇒ f (4) (0) = − 15
16
P 3 (x) = f(0) + xf ′ (0) + x2
2 f ′′ (0) + x3
3! f (3) (0)
= 1 + 1 2 x + 1 (
− 1 )
x 2 + 1 ( 3
x
2 4 6 8)
3
= 1 + 1 2 x − 1 8 x2 + 1
16 x3
and
R 3 (x) = f (4) (c)
x 4 = −15(c + 1)−7/2 x 4 15x 4
= −
4!
384
384(c + 1) 7/2
for some c between 0 and x. Since smaller denominators make larger numbers,
the remainder is maximized for the smallest possible value of c, which occurs when
c = 0. So the error at x is bounded by
|R 3 (x)| ≤ 15x4
384
For example, the error at x = 1 is no more than 15/384.
Looking more closely at the Taylor series
f(x) = f(a) + (x − a)f ′ (a) + 1 2! (x − a)2 f ′′ (x 0 ) + · · ·
we observe that the first two terms
f(x) = f(a) + (x − a)f ′ (a) + · · ·
give the equation of a line tangent to f(x) at the point (a, f(a)). The next term
gives a quadratic correction, followed by a cubic correction, and so forth, so we
might write
f(x near a) = (equation of a tangent line through a)
+ (quadratic correction at a)
+ (cubic correction at a)
+ (quartic correction at a) + · · ·
For a function of two variables z = f(x, y), the tangent line because a tangent plane;
the quadratic correction becomes a parabolid correction; and so forth. The explicit
result is the following.
Theorem 13.3 Let f(x, y) be infinitely differentiable in some open set J that contains
the point (a, b). Then the Taylor Series of f(x,y) about the point (a,b)
is
f(x, y) = f(a, b) + f x (a, b)(x − a) + f y (a, b)(y − b)
+ 1 [
fxx (a, b)(x − a) 2 + 2f xy (a, b)(x − a)(y − b) + f yy (a, b)(y − b) 2]
2
+ · · ·
for all points (x, y) ∈ J.
Revised December 6, 2006. Math 250, Fall 2006