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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY
Interphase Calculus III Problem Set 1
Instructor: Samuel S. Watson
Due: 6 July 2015
1. Find the fourth-order Taylor polynomial of the function f ( x ) = sin( x2 ) at x = 0 two different
ways:
(a) explicitly calculate derivatives of f and use the formula for the Taylor coefficients of f in terms
of the derivatives of f , and
(b) substitute x2 into the Taylor series for sine at x = 0.
2. Use integration by parts to find
Z
√
x x + 7 dx.
Z
(log x )2 dx (note: log denotes the natural logarithm through-
3. Use integration by parts to find
out this course).
4. Find
Z √ 2
x −1
x
dx.
5. Use partial fractions to calculate
Z
1
x2
dx.
+
x 2 − 1 ( x + 1)3
6. Use dot products to show that the diagonals of a parallelogram have the same length if and
only if the parallelogram is a rectangle. (Hints: let a and b be vectors along two sides of the
parallelogram, and express vectors running along the diagonals in terms of a and b. Also, recall
that if c is a vector, then c · c equals the squared length of c.)
7. Find the equation of the plane passing through the points (1, 0, 0), (0, 2, 0), and (0, 0, −1).
8. Consider the parallelepiped spanned by (1, 4, −2), (3, 0, 1), and (−2, 2, 1). Find the perpendicular distance between the face F spanned by (1, 4, −2), and (3, 0, 1) and the face parallel to F (Hint:
use the triple scalar product).
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