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VCE Methods Differential Calculus Application Task 5
Adapted for Year 11. Unit 1 & 2.
Number of marks: 6
Reading time: 1 minute
Writing time: 10 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.
Consider the function \( g : \mathbb{R} \rightarrow \mathbb{R} \), \( g(x) = 2^x + 5 \).
a. State the value of \( \lim_{x \to -\infty} g(x) \). 1 mark
b. The derivative, \( g'(x) \), can be expressed in the form \( g'(x) = k \times 2^x \). Find the real number \( k \). 1 mark
c. i. Let \( a \) be a real number. Find, in terms of \( a \), the equation of the tangent to \( g \) at the point \( (a, g(a)) \). 1 mark
ii. Hence, or otherwise, find the equation of the tangent to \( g \) that passes through the origin, correct to three decimal places. 2 marks
Let \( h : \mathbb{R} \rightarrow \mathbb{R} \), \( h(x) = 2^x - x^2 \).
d. Find the coordinates of the point of inflection for \( h \), correct to two decimal places. 1 mark
End of examination questions
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