IB Mathematics SL 5.4 Tangents and normal AI SL Paper 1- Exam Style Questions- New Syllabus

(a)
Differentiate \( f(x) = \frac{2}{x} + 3x^2 – 3 \):
\[ \begin{aligned} f(x) &= 2x^{-1} + 3x^2 – 3 \\ f'(x) &= 2 \cdot (-1)x^{-2} + 3 \cdot 2x – 0 \\ &= -2x^{-2} + 6x \\ &= -\frac{2}{x^2} + 6x \end{aligned} \] A1(M1)A1
[3 marks]

(b)
Find the gradient at \( x = 1 \):
\[ \begin{aligned} f'(x) &= -\frac{2}{x^2} + 6x \\ f'(1) &= -\frac{2}{1^2} + 6 \cdot 1 = -2 + 6 = 4 \end{aligned} \] A1
Find the perpendicular gradient:
\[ m_{\perp} = -\frac{1}{4} \] M1
Find the equation of the normal at \( (1, 2) \):
\[ \begin{aligned} y – 2 &= -\frac{1}{4}(x – 1) \\ y – 2 &= -\frac{1}{4}x + \frac{1}{4} \\ y &= -\frac{1}{4}x + \frac{1}{4} + 2 \\ y &= -\frac{1}{4}x + \frac{9}{4} \\ \frac{1}{4}x + y – \frac{9}{4} &= 0 \\ x + 4y – 9 &= 0 \end{aligned} \] M1, A1
Final equation: \( x + 4y – 9 = 0 \). [4 marks]