### 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: