Taylor Series & The Total Differential
Reading Assignment: 4.1
Suggested problems: 3, 9, 11, 12, 23, 25, 31
Write the Taylor expansion to degree three about the given point for each function.
1) f ( x, y )  3( x  1) 2  2 y, ( x0 , y0 )  (0, 0)
2) g ( x, y )  x 2  xy  y 2 , ( x0 , y0 )  (1, 2)
3) h(x, y) = ex ln(1 + y), (x0, y0) = (0, 0)
4) Multiply out the expressions from problem 2) and collect like terms. Does this result make
sense? Why or why not?
5) Write out the Maclaurin series for ex and for ln(y + 1) to third degree. Multiply them together
term by term. Is your answer similar to 3)? How?
6) Suppose that x 2  y 3  z 4  1 and z 3  zx  xy  3 .
a. Take the total differential of both these surfaces.
b. The two given surfaces intersect in a curve along which y is a function of x. Find
( x, y, z )  (1,1,1) . You should use your result from a.
dy
at
dx
7) Use differentials to find the approximate amount of copper used in the four sides and bottom
of a rectangular copper tank that is 6 feet long, 4 feet wide and 3 feet deep inside if the sheet
copper is ¼ inch thick.