Mathematics (Scientific Branch) - الرياضيات
Mathematics for the scientific branch - الرياضيات - tests limits and continuity, differentiation and its applications, integration, exponential and logarithmic functions, analytic geometry, complex numbers, and sequences. Nearly every item is procedural: the method is standard, so marks are won on algebra and lost on slips of sign, domain, or notation.
Core rules
- A limit exists only when the left-hand and right-hand limits agree; factor, rationalise, or simplify before substituting. L'Hôpital is valid only for the indeterminate forms 0/0 and ∞/∞.
- Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x). Consequences: d/dx ln u = u′/u and d/dx e^u = u′e^u.
- Stationary points come from f′(x) = 0; classify them with the sign of f″ or with a sign table around each root.
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1, while ∫(1/x)dx = ln|x| + C. A definite integral equals F(b) − F(a).
- Circle: (x−h)² + (y−k)² = r². Parabola y² = 4ax has focus (a, 0) and directrix x = −a.
Worked example
Differentiate f(x) = x²e^{3x}. Set u = x² and v = e^{3x}, so u′ = 2x and v′ = 3e^{3x}. Product rule: f′(x) = u′v + uv′ = 2x·e^{3x} + x²·3e^{3x} = e^{3x}(2x + 3x²). Check: f′(0) = 0, so the graph has a horizontal tangent at x = 0, which is what the factor x² in front predicts.
Traps
- Dropping + C in an indefinite integral, or dropping the absolute-value bars in ln|x|.
- Using substitution on a definite integral but forgetting to change the limits (or to return to x).
- Reversing the quotient rule: it is (u′v − uv′)/v², not (uv′ − u′v)/v².
- Reporting roots of f′(x) = 0 that lie outside the domain (negative arguments of ln, zero denominators).
- Differentiating trigonometric functions with the calculator still in degree mode.