201-DDB Final Exam | May 2010 Calculus III Page 1
(6) 1. For the function f(x) = xln(x):
(a) ▯nd the 4 degree Taylor polynomial T (4) around x = 1.
(b) Use Taylor’s Inequality (or Lagrange’s Remainder) to estimate the error in using T (x) to4
approximate f(x) on the interval [0:5;1:5].
(4) 2. Find the Maclaurin series for the function f(x) = p . What is its radius of convergence?
1 + x 3
x t dt
(8) 3. Let g(x) = 4
0 1 + t
(a) Find the Maclaurin series for g(x), and its radius of convergence;
Z 1=2 2
t dt ▯4
(b) approximate 1 + t4 within an error of ▯10 (and justify your answer).
(c) Find g (7(0).
(6) 4. Sketch (on the same axes) the graphs of r = 3sin▯ and r = 1 + sin▯.
(a) Find all points of intersection.
(b) Set up (but do not evaluate) the integrals needed to ▯nd
i. the area of the region common to both (i.e. inside) r = 3sin▯ and r = 1 + sin▯, and
ii. the perimeter (length) of r = 1 + sin▯.
(7) 5. Suppose that a plane curve C given by parametric equations in t passes through the point (0;2) at
t = 1, and satis▯es dt = t and dt= 1 ▯ t2.
(a) Find the parametric equations for C (i.e. for x and y).
(b) Find the Cartesian equation for C by eliminating the parameter t.
(c) Find the length of C from t = 1 to t = 3.
(10) 6. A space curve C is de▯ned by the vector equation r(t) = ht ;3t ;6ti.
(a) Compute the velocity v, acceleration a, and speed v of a point moving along C.
(b) Find the tangential and normal components of acceleration a ;a , the unit tangent vector T
and the unit normal vector N.
Simplify your answers.
(9) 7. Identify and sketch the following. Show all your work.
(a) The surface z = r .2
(b) The surface ▯ = 4cos’.
p 2 2
(c) The graph of the function z = 4 ▯