Mathematics I
Semester I · BCA 104 · 3 credits
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chapters done
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The syllabus asks for
- Functions, domain and range
- Standard families of functions
- Even, odd and one-to-one functions
- Composition and inverse
- Graph transformations
- Eight ways functions are used in computing
Key points
- A function pairs each input with exactly one output; domain and range come from restrictions of the formula.
- Even and odd symmetry, composition and inverses are the algebra of functions.
- Transformations shift, stretch and reflect graphs; knowing them saves plotting.
- Exponent, logarithm and trigonometric rules are the toolkit used in every later unit.
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The syllabus asks for
- The idea of a limit
- Techniques for finding limits
- Limits at infinity and infinite limits
- Continuity
- Eight uses of limits in computing and daily life
Key points
- A limit describes the approached value and ignores the value at the point; both one-sided limits must agree.
- For (0)/(0), factor, rationalise or use a standard limit; for x → ∞ divide by the highest power.
- Continuity means the graph can be drawn without lifting the pen; the three kinds of break are removable, jump and infinite.
- The IVT guarantees roots of continuous functions between sign changes.
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The syllabus asks for
- The derivative from first principles
- Rules of differentiation
- Implicit differentiation
- Logarithmic differentiation
- Parametric differentiation
- Higher-order derivatives
- Eight uses of derivatives
Key points
- The derivative is the limit of difference quotients, equal to the slope of the tangent.
- A small table plus four rules (sum, product, quotient, chain) handles nearly every function.
- Implicit, logarithmic and parametric methods cover curves not written as y = f (x).
- Higher derivatives and nth-derivative formulas appear in series and in applications.
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The syllabus asks for
- Tangent and normal
- Rates of change and related rates
- Increasing, decreasing, maxima and minima
- Optimisation
- Mean value theorems
- L'Hôpital's rule
- Taylor and Maclaurin series
- Eight uses of the applications of derivatives
Key points
- The derivative gives tangents, rates, and the shape of a graph (increasing, decreasing, concave).
- Optimisation reduces a real problem to one variable and tests critical points and endpoints.
- Rolle and the MVT link the average slope to an instantaneous slope; L'Hôpital turns indeterminate limits into derivative quotients.
- Taylor series replace a function by polynomials and enable numerical approximation.
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The syllabus asks for
- Indefinite integrals
- Integration by substitution
- Integration by parts
- Partial fractions
- The definite integral and the Fundamental Theorem
- Applications
- Improper integrals
- Eight uses of integration
Key points
- Integration is reverse differentiation; always add C for indefinite integrals and verify by differentiating.
- Substitution undoes the chain rule, parts undoes the product rule, partial fractions handle rational functions.
- The Fundamental Theorem lets us compute definite integrals from antiderivatives.
- Areas, volumes, averages and total change are all definite integrals; improper integrals are limits.
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The syllabus asks for
- Basic terms
- Separable equations
- Homogeneous equations
- Linear first-order equations
- Exact equations
- Second-order linear equations with constant coefficients
- Modelling with differential equations
- Eight uses of differential equations
Key points
- The order sets the number of constants; initial conditions pick a particular solution.
- Separable, homogeneous, linear and exact first-order equations each have a systematic method.
- Constant-coefficient second-order equations reduce to an auxiliary quadratic whose roots give exponentials, polynomial-times-exponential, or oscillations.
- Growth, decay and cooling are first-order linear models with exponential solutions.
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TU assessment: 60% final examination set by the university, 40% internal (assignments, lab work, attendance). Confirm the split in your college's notice.