Year 12 Maths Advanced MAV-12-06 Checkpoint 1

Checkpoint Quiz 1, Applications of Calculus

This checkpoint covers Optimisation with Exponentials and Growth and Decay Models (Lessons 1-2).

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Quick Review

Exponential optimisation: Differentiate using the product or quotient rule, then factor out $e^x$ or $e^{-x}$, it is never zero, so only the remaining polynomial factor needs to be solved.
Second derivative test: At a stationary point $x = c$ where $f'(c) = 0$, if $f''(c) < 0$ the point is a local maximum; if $f''(c) > 0$ it is a local minimum.
Growth and decay model: If a quantity's rate of change is proportional to its current amount, $\dfrac{dN}{dt} = kN$, then $N(t) = N_0 e^{kt}$. A positive $k$ gives growth; a negative $k$ gives decay.
Half-life and doubling time: $t_{1/2} = \dfrac{\ln 2}{|k|}$ for decay, $t_{double} = \dfrac{\ln 2}{k}$ for growth. Both depend only on $k$, not on the initial amount.

5 random questions from the Checkpoint 1 bank, feedback shown immediately

Great work finishing Checkpoint 1.

If you struggled with any questions, go back to Lessons 1-2 to review the concepts before moving on to Motion Applications.

Tick when you're ready to continue to Lesson 3.