Reconstruct a quantity from a rate
Reverse differentiation, fix the constant, then evaluate or solve with the model’s units and domain
\[Q^{\prime}(t)=r(t),\quad Q(t)=P(t)+C\]
- Use quantity data for C Substitute the measured quantity into P+C. Do not substitute it as a value of the rate
- Choose evaluation or solving A given time asks for evaluation. A given quantity level asks for an equation in time
- Check domain and units Convert time units first, reject roots outside the model and interpret the result in quantity units
- Prove the distance condition Use final minus initial position for distance only when velocity is nonnegative throughout the interval
Worked example
A reservoir has net rate −2t − 2 L/min and holds 100 L at t = 0. On 0 ≤ t ≤ 5, when does it hold 85 L?
\[Q^{\prime}(t)=-2t-2,\quad Q(0)=100\]
- \(\displaystyle Q(t)=-t^2-2t+C\)Reverse the rate rule
- \(\displaystyle C=100\)Use the measured initial volume
- \(\displaystyle -t^2-2t+100=85\)Set the quantity equal to 85.
- \(\displaystyle t^2+2t-15=(t-3)(t+5)=0\)Rearrange and factor
- \(\displaystyle t=3\text{ min}\)Reject −5 because it is outside the stated interval
Common mistakes
Evaluating the rate
✕Use r(3) to report the volume at 3 min.✓First recover Q and use Q(3)
The rate and the quantity have different meanings and unitsIgnoring the domain
✕Accept every real event root✓Retain only roots in the stated model interval
Algebra does not extend a physical model’s domainDifficulty levels
introConstant and linear net rates with initial values
standardQuadratic inflow and velocity-to-position evaluation
challengeExact event equations in filling and draining models
examNonnegative-velocity distance, converted time and model-domain judgement
Questions about this topic
Is total distance across a direction change included?+
No. This core packet uses distance only when velocity is proved nonnegative throughout the interval
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 7: Antiderivatives
In this chapter
7A Antiderivatives and arbitrary constants: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Recognise equivalent polynomial formsSimplify and combine algebraic fractionsAnti-differentiate a power of a linear function7C Integrating powers of linear functions: Recognise equivalent polynomial forms7C Integrating powers of linear functions: Solve a linear equationAnti-differentiate exponential and circular functionsAnti-differentiate by recognition from a supplied derivativeDetermine a polynomial from its derivative and a boundary conditionInfer primitive curves from a gradient functionReconstruct quantities from their ratesEvaluate and determine function rules7E Antiderivative families and applications: Solve a linear equationRound to decimal places and significant figuresScale a proportional quantity with the unitary method