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).