DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1063

FINAL EXAMINATION
APRIL 1996
TIME: 2 HOURS

ATTEMPT AT LEAST FOUR QUESTIONS.
CALCULATORS PERMITTED.
NOTE THE LIST OF FORMULAS ON THE LAST PAGE.

MARKS
(25)1.(a)Evaluate the following integrals exactly:
.
(b)(i)Use the two-term Gaussian quadrature formula to obtain an
approximation to
[This is the total arc length of a lemniscate.]
(ii)What answer does the trapezoidal rule give for this integral? Why?
(iii)What answer does the mid-point rule give for L?
(iv)The exact value of . Which of and gives the better approximation? Why?
(25)2.(a)Consider a simple electric circuit in which the current x at time , is governed by the differential equation
with initial value .
(i)Find the solution .
(ii)Compute .
(iii)Sketch x as a function of t.
(b)(i)Use the Taylor formula to approximate by a polynomial of degree 4, near x = 0.
(ii)Approximate the integral
by replacing by its fourth order Taylor polynomial to simplify the integrand and then integrating over the interval .
[The exact value .]
(25)3.(a)(i) Sketch (roughly) the curves and for x > 0, and compute the area of the finite region between these curves.
(ii)Compute the area of the finite region between and for any fixed integer .
(iii)What is ?
(b)Show that the function defined on by
satisfies the conditions
(i) ;
(ii) and .
(25)4.(a)Consider the Cornu spiral, defined parametrically by
Show that the curvature is proportional to the arc length parameter , specifically . [Jakob Bernoulli discovered this spiral in seeking curves with this property.]
(b)(i)Find the four complex fourth roots of unity (solutions z to ) and compute the sum of these roots.
(ii)Use the Euler formula to prove that
and .
(iii)Define functions for integers and . Show that and are polynomials in x, specifically,
(25)5.(a)Define the Riemann integral of a function f over an interval .
(b)State the Fundamental Theorem of Calculus.
(c)Prove that if f is defined and has continuous derivatives of all orders on some open interval I containing the origin, then for any ,
(d)(i)Suppose that f and its derivative are defined and continuous on the interval , that and that . Prove that
.
(ii)Evaluate .

FORMULAS

1.
2.
3.
4.Arc length and curvature for a parametrically defined curve,
(i)total length
(ii)arc length parameter
(iii)curvature
5.Two-term Gaussian quadrature: .
6.Two-subinterval mid-point quadrature:
7.Two-subinterval trapezoidal quadrature: