Implicit Di↵erentiation; Taylor approximations of higher degrees
@z
1. The equation x2 y 2 + y 2 z 2 + x2 z 2 = 9 describes the surface shown. Find @x
at the point (1, 1, 2).
z
x
y
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2. What shape does the equation x2 + y 2 /4 + z 2 /9 = 6 describe? Compute @z/@y at the point (2, 2, 3).
3. In the previous two problems, you could solve for z (picking the correct sign for the square root) and
find the derivative that way. Try it and make sure you get the same answers!
4. Find @y/@x at the point (1, 0, 1) on the surface given by y 3 ln(x) +
2
z
+ exy = 0.
x
5. Change of topic: in this problem we find the Taylor polynomial of degree 2 for a function f (x, y) of
two variables, using what we know about single-variable calculus.
Assume our center is at (x, y) = (x0 , y0 ), where we assume f and its derivatives are easy to evaluate at
the center, and we assume the center is close to our point of interest. Assume we want to approximate
f (x, y) with x = x0 + h1 , y = y0 + h2 , so with h1 , h2 small.
(a) Let F (t) = f (x0 + th1 , y0 + th2 ). Which value of t corresponds to the chosen center for f — this
is the center for the Taylor expansion of F . Which value of t corresponds to the point (x, y) we
are interested in?
(b) Write out the second degree Taylor polynomial for F (t) at the center for F . F is a function of one
variable so you know how to do this! Evaluate this polynomial at the value of t we are interested
in.
(c) Find the required derivatives of F and evaluate them at the center for F .
(d) Put everything together and give the degree 2 Taylor approximation of f close to the center
(x0 , y0 ).
(e) How do you think this would generalize to higher degrees?
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