VCE Methods Integral Calculus Application Task 5
Number of marks: 11
Reading time: 2 minutes
Writing time: 16 minutes
Section B – Calculator Allowed
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
The diagram below shows part of the graph of \( y = f(x) \), where \( f(x) = \frac{x^2}{12} \).
a. State the equation of the axis of symmetry of the graph of \( f \). 1 mark
b. State the derivative of \( f \) with respect to \( x \). 1 mark
The tangent to \( f \) at point M has gradient −2.
c. Find the equation of the tangent to \( f \) at point M. 2 marks
The diagram below shows part of the graph of \( y = f(x) \), the tangent to \( f \) at point M and the line perpendicular to the tangent at point M.
d. i. Find the equation of the line perpendicular to the tangent passing through point M. 1 mark
ii. The line perpendicular to the tangent at point M also cuts \( f \) at point N, as shown in the diagram above.
Find the area enclosed by this line and the curve \( y = f(x) \). 2 marks
e. Another parabola is defined by the rule \( g(x) = \frac{x^2}{4a} \), where \( a > 0 \).
A tangent to \( g \) and the line perpendicular to the tangent at \( x = -b \), where \( b > 0 \), are shown below.
Find the value of \( b \), in terms of \( a \), such that the shaded area is a minimum. 4 marks
End of examination questions
VCE is a registered trademark of the VCAA. The VCAA does not endorse or make any warranties regarding this study resource. Past VCE exams and related content can be accessed directly at www.vcaa.vic.edu.au
