M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY
Interphase Calculus III Worksheet
Instructor: Samuel S. Watson
29 June 2015
Topics: Taylor Series, Integration by parts, Integration by trigonometric substitution
1. Find the linear function
L( x ) which best approximates the graph of the function f ( x ) = sin( x )
√
at the point (π/4, 2/2). Sketch graphs of f and L.
2. Find the derivative of f at x = π/4 and the derivative of L at x = π/4.
√ 2
2
π
π 2
3. Find c2 so that Q( x ) =
+
x−
+ c2 x −
has the same second derivative as f
2
2
4
4
at x = π/4.
√
4. Consider an arbitrary function f and a cubic function C ( x ) = c0 + c1 ( x − x0 ) + c2 ( x − x0 )2 +
c3 ( x − x0 )3 . Solve for c0 , c1 , c2 , and c3 such that the derivatives of order up to three at x0 of f and
C are the same.
5. Find the Taylor series for the function f ( x ) = log( x ) at x = 1 (note: log denotes the natural
logarithm, throughout this course). Use your answer to approximate log(1.001) to four significant
figures (without a calculator!).
6. Integrate both sides of the product rule ( f g)0 = f 0 g + f g0 and solve for
Z
f 0 g. This is the
integration by parts formula.
7. Write
Z
log x dx as
8. Integrate
Z
Z
( x )0 log x dx and apply the integration by parts formula to integrate.
e x x4 dx using integration by parts.
9. Write
x2
1
as a sum of two fractions whose denominators are linear functions.
− 2x − 35
10. Use your answer to the previous question to find
11. Integrate
Z
1
.
x ( x + 1)2
Z
dx
.
x2 − 2x − 35