**
MATH 1063**

FINAL EXAMINATION APRIL 1996 | TIME: 2 HOURS |

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 at time , is governed
by the differential equation
x | |

with initial value . | ||||

(i) | Find the solution . | |||

(ii) | Compute . | |||

(iii) | Sketch as a function of x.
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 , and compute the area of the finite region
between these curves.
x > 0 |

(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 to
) and compute the sum of
these roots.
z | ||

(ii) | Use the Euler formula to prove that and . | |||

(iii) | Define functions for integers and . Show that
and are polynomials in , specifically,
x | |||

(25) | 5. | (a) | Define the Riemann integral of a function
over an interval .
f | |

(b) | State the Fundamental Theorem of Calculus. | |||

(c) | Prove that if is defined and has continuous derivatives of
all orders on some open interval f containing the origin,
then for any ,
I | |||

(d) | (i) | Suppose that and its derivative are defined and
continuous on the interval , that and that
. Prove thatf. | ||

(ii) | Evaluate . |

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: | |