UNIBEN MTH 218 Mathematical Methods I: Study Guide and Practice

UNIBEN MTH 218, Mathematical Methods I, is the main 200-level calculus course at the University of Benin. It is a 3-credit core course in the first semester of 200 level, with MTH 110 and MTH 125 as its listed prerequisites, and it builds on the calculus of MTH 112. This guide maps every MTH 218 topic to free practice, so you can learn each method by solving problems with instant feedback.

What UNIBEN MTH 218 covers

The department's description opens with techniques of integration: substitution, integration by parts and partial fractions. It continues with differentiation topics, including reduction formulae, partial differentiation with applications, and classifying the critical points of functions of two variables, followed by Lagrange multipliers.

Next come coordinate systems (changing from Cartesian to polar, spherical and cylindrical coordinates), Taylor's and Maclaurin's series, nth-order derivatives and Leibniz's rule with applications to differential equations. The course ends with complex numbers and hyperbolic functions: De Moivre's theorem, roots of complex numbers and polynomials, the exponential form, and functions of a complex variable.

In standard terms, MTH 218 combines the techniques and series of Calculus 2 with the partial derivative and optimization part of Calculus 3, plus complex numbers.

MTH 218 topic map

MTH 218 in the UNIBEN sequence

MTH 218 and MTH 213 Vector Analysis run in the same semester, and together they cover most of a standard Calculus 3 course. If your 100-level calculus is rusty, start with the MTH 112 guide.

Example problems at the MTH 218 level

1. Partial fractions. Since \dfrac{1}{x^2 - 1} = \dfrac{1}{2}\left(\dfrac{1}{x - 1} - \dfrac{1}{x + 1}\right),

2. Classifying a critical point. For f(x, y) = x^2 + y^2 - 2x + 4y, set f_x = 2x - 2 = 0 and f_y = 2y + 4 = 0, giving (1, -2). With f_{xx} = 2, f_{yy} = 2 and f_{xy} = 0,

so (1, -2) is a minimum, with f(1, -2) = -5.

3. Changing coordinates. The Cartesian point (1, \sqrt{3}) has r = \sqrt{1 + 3} = 2 and \tan\theta = \sqrt{3}, so in polar form it is (2, \frac{\pi}{3}).

4. Roots of unity with De Moivre. The solutions of z^3 = 1 are z = \cos\frac{2k\pi}{3} + i\sin\frac{2k\pi}{3} for k = 0, 1, 2:

How to do well in UNIBEN MTH 218

• Choose the integration method before you start. Ask whether the integrand suits substitution, parts or partial fractions. Our choosing a strategy lesson trains exactly this decision.
• Learn the second-derivative test for two variables. Most critical point questions come down to computing D = f_{xx}f_{yy} - f_{xy}^2 correctly.
• Memorize key series. The Maclaurin series for e^x, \sin x, \cos x and \ln(1 + x) are on the calculus formula sheet and come up repeatedly.
• Watch your algebra. Partial fraction decompositions and complex roots are full of sign traps; see common calculus mistakes.
• Practice with timed sets from practice exams before each test.

Tip: hyperbolic functions behave like trig functions with small sign changes: \cosh^2 x - \sinh^2 x = 1 instead of \cos^2 x + \sin^2 x = 1. Learn the definitions in terms of e^x and every identity follows.