Amounts and rates in an exponential model
Read the initial amount, differentiate with the inner coefficient, then evaluate or solve the requested quantity
\[A=A_0e^{kt},\quad A^{\prime}=kA,\quad t=\frac{\ln(B/A_0)}{k}\]
- Amount versus rate A has amount units; A′ has amount per time units. k has inverse-time units
- Use a ratio for two observations Cancel A₀ with A(t₂)/A(t₁), take ln and divide by t₂ − t₁
- Check domains and signs Positive k gives growth; negative k gives decay. Positive amounts are required for logarithms; time must satisfy the stated model domain
- Round once Keep exact fractions and logarithms until a decimal answer is requested. Retain calculator precision before the final rounding
Worked example
A mass is M(t) = 180e^(−t/30) grams, with t in minutes. Find its signed rate at 6 min and the exact time when it reaches 90 g.
\[M(t)=180e^{-t/30},\quad t\ge0\]
- \(\displaystyle M^{\prime}(t)=180(-1/30)e^{-t/30}=-6e^{-t/30}\)Differentiate with the inner coefficient
- \(\displaystyle M^{\prime}(6)=-6e^{-1/5}\approx-4.91\text{ g/min}\)At 6 min the mass is decreasing at about 4.91 g/min.
- \(\displaystyle 180e^{-t/30}=90\quad\Rightarrow\quad e^{-t/30}=1/2\)Use the amount equation for the target mass
- \(\displaystyle t=-30\ln(1/2)=30\ln2\text{ min}\)Take ln and divide by −1/30. This positive time is in the domain
Common mistakes
Missing the inner coefficient
✕\(\displaystyle \frac{d}{dt}(180e^{-t/30})=180e^{-t/30}\)✓\(\displaystyle \frac{d}{dt}(180e^{-t/30})=-6e^{-t/30}\)
Multiply the outside coefficient by −1/30; retain the exponentialUsing additive changes for k
✕\(\displaystyle k=\frac{A(t_2)-A(t_1)}{t_2-t_1}\)✓\(\displaystyle k=\frac{\ln(A(t_2)/A(t_1))}{t_2-t_1}\)
An exponential has a constant relative rate. A difference quotient gives an average amount-per-time rate, not k.Giving a positive derivative for decay
✕M′(6) = 4.91 g/min.✓M′(6) ≈ −4.91 g/min, or decreasing at about 4.91 g/min.
Keep the negative sign for a signed derivative; use a positive magnitude only with decreasingDifficulty levels
introRead the initial amount and recognise the derivative rule
standardRecover exact model coefficients, evaluate the initial amount, or calculate a rounded rate at a nonzero time
challengeRecover k from two observations or recognise an exact logarithmic target time
examCalculate a contextual rate and interpret its sign and units
Questions about this topic
Does selecting a logarithmic expression prove I can construct the whole model?+
No. The choices check recognition; the numeric steps check individual values. Full independent modelling needs additional evidence
Does the lesson cover calculator procedures?+
It practises evaluating and rounding a stated expression. It does not assess a complete calculator workflow or all technology outcomes
Why is a negative target time sometimes rejected?+
It may solve the algebraic equation but lie outside the stated physical time domain
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 5: Differentiation
In this chapter
Evaluate limits and test continuityGradient of a tangent from first principlesDifferentiate power functions and evaluate a derivativeExpand and factor polynomial expressionsPascal rows and binomial coefficient structure5A Limits, first principles and the power rule: Simplify and combine algebraic fractionsEvaluate a polynomial or recover a coefficient parameterSubstitute values into a formula5A Limits, first principles and the power rule: Solve a linear equationTest or construct a smooth piecewise joinRead and construct a gradient function from a function graphAnalyse a cubic graph or determine its ruleEvaluate and determine function rulesEquation of a tangent or normal lineRecover a polynomial rule from derivative and tangent conditionsFind the gradient of a line through two pointsFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a pair of simultaneous linear equationsDetermine parameters in a polynomial identity5C Tangents, normals and derivative conditions: Solve a linear equationDifferentiate natural exponential functions and solve simple derivative conditionsExponential growth and decay rate models5D Exponential derivatives and rate models: Simplify and combine algebraic fractions5D Exponential derivatives and rate models: Round to decimal places and significant figuresDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsGeometric optimisation with a trigonometric modelSinusoidal models and their rates of changeSolve a practical right-angled-triangle problemPerimeter and area of standard shapes5E Trigonometric differentiation and rates: Round to decimal places and significant figuresApproximate a cubic root using Newton's method