may 2010 cal 3.pdf

2 Pages

Mathematics & Statistics (Sci)
Course Code
MATH 222
Christa Scholtz

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201-DDB Final Exam | May 2010 Calculus III Page 1 (Marks) (6) 1. For the function f(x) = xln(x): th (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]. 1 (4) 2. Find the Maclaurin series for the function f(x) = p . What is its radius of convergence? 1 + x 3 Z 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). 0 (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 dy 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 T N 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 ▯
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