function evaluation

Practise function evaluation with an original VectorPaths question, worked explanation and related VCE Methods practice.

1 practice marks

Original practice question

Find the value of the acute angle between the tangent to the graph of $h$ and the tangent to the graph of $h^{-1}$ at $x = 4$, given that $h(x) = 2x^{\frac{3}{2}}$ for $x \geq 0$.

Worked explanation

• Find gradient via derivative

  \[h'(x) = 3x^{\frac{1}{2}}, \quad h'(4) = 3 \times 2 = 6\]

• Inverse gradient

  \[(h^{-1})'(16) = \frac{1}{h'(4)} = \frac{1}{6}\]

• Find angle between tangents

  \[\tan(\theta) = \frac{|6 - \frac{1}{6}|}{1 + 6 \times \frac{1}{6}} = \frac{\frac{35}{6}}{2} = \frac{35}{12}\]

$\approx 73.7°$

Original VectorPaths practice; not an official VCAA question or marking rubric.