FUTA MTS 102 Introductory Mathematics II: Study Guide, Practice

FUTA MTS 102, Introductory Mathematics II, is the 100-level calculus course in the Department of Mathematical Sciences at the Federal University of Technology, Akure. It runs in the second semester and takes students from functions and limits to differentiation, integration and their applications. This guide maps every MTS 102 topic to free practice, so you can learn each idea by solving problems with instant feedback.

What FUTA MTS 102 covers

The department's course synopsis groups MTS 102 into seven parts:

• Functions of a real variable: definitions, types of function and their graphs.
• Limits and continuity: the idea of a limit, rates of change, differentiation from first principles and continuity.
• Techniques of differentiation: sums and differences, products, quotients, second and higher derivatives, function of a function (the chain rule), inverse functions, implicit functions and parametric equations.
• Applications of differentiation: kinematics, tangents and normals to a curve, and maximum and minimum values.
• Curve sketching: turning points and maximum and minimum values of a curve.
• Integration: methods of integration, rational algebraic fractions, substitution, partial fractions and trigonometric functions.
• Applications of integration: geometry and mechanics, areas and volumes.

That is a complete Calculus 1 course, with integration by partial fractions added from what is usually early Calculus 2.

MTS 102 topic map

How MTS 102 compares with other Nigerian first calculus courses

Many Nigerian universities teach a similar first calculus course at 100 level, with different codes and emphases.

If you are studying from another university's materials, the core ideas match, so the practice lessons on this page serve all of them.

Example problems at the MTS 102 level

1. The quotient rule. Differentiate y = \dfrac{x}{x + 1}.

2. Implicit differentiation. Find the gradient of x^2 + y^2 = 25 at (3, 4).

3. Kinematics. A particle moves with s = t^3 - 6t^2 + 9t. Then

so the particle is momentarily at rest at t = 1 and t = 3.

4. Partial fractions. Since \dfrac{1}{x(x + 1)} = \dfrac{1}{x} - \dfrac{1}{x + 1},

5. An area. The area between y = 4 - x^2 and the x-axis for 0 \le x \le 2 is

How to do well in FUTA MTS 102

• Build the differentiation rules one at a time. Master the power rule, then products, quotients and the chain rule, before moving on to implicit and parametric differentiation.
• Connect calculus to motion. In kinematics questions, velocity is \frac{ds}{dt} and acceleration is \frac{dv}{dt}. Writing these down first prevents most errors.
• Practice partial fractions separately. Decomposing the fraction is algebra, not calculus, and it is where most marks are lost. Use the partial fractions guide.
• Keep the standard results nearby. The calculus formula sheet has every derivative and integral you need for MTS 102.
• Rehearse with timed problems from practice exams, and read common calculus mistakes before each test.

Tip: after finding a turning point, always check which kind it is with the second derivative or a sign change in \frac{dy}{dx}. Questions on curve sketching almost always ask you to classify them.