Proportional exponential rate
A supplied exponential function has a proportional derivative
- Rate rule Differentiate the model to obtain the rate rule \(\displaystyle A(t)=A_0e^{kt},quad A'(t)=kA_0e^{kt}=kA(t)\)
- Units The exponent is dimensionless; k has inverse-time units
- Initial value At t=0, the exponential factor is 1.
Worked example
A(t)=80e^(−t/5) grams, with t in minutes. Give its rate equation and initial value
\[A\prime=-\frac15A\]
- \(\displaystyle A\prime=-16e^{-t/5}\ \mathrm{g/min}\)Differentiate the supplied function
- \(\displaystyle A(0)=80\ \mathrm g\)This does not require solving a new differential equation
Common mistakes
Missing the inner coefficient
✕\(\displaystyle \frac{d}{dx}\left(3e^{2x}\right)=3e^{2x}\)✓\(\displaystyle \frac{d}{dx}\left(3e^{2x}\right)=6e^{2x}\)
The coefficient of x in the exponent is an inner derivative factor and must multiply the exponentialDropping the exponential factor
✕\(\displaystyle \frac{d}{dx}e^{3x}=3\)✓\(\displaystyle \frac{d}{dx}e^{3x}=3e^{3x}\)
The exponential is its own derivative, and the coefficient 3 from the exponent is multiplied by itChanging the square middle sign
✕\(\displaystyle \left(2e^{x}+3\right)^{2}=4e^{2x}-12e^{x}+9\)✓\(\displaystyle \left(2e^{x}+3\right)^{2}=4e^{2x}+12e^{x}+9\)
The middle term is twice the product of the two terms, so its sign is positive when both terms are positiveDifficulty levels
introFind initial values and derivative rules for supplied exponential models
Questions about this topic
Does 5% growth per year mean a continuous rate coefficient of exactly 0.05?+
Only for a stated continuous rate. A discrete annual factor of 1.05 gives a continuous rate coefficient equal to the natural logarithm of 1.05 per year
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, Specialist Mathematics, chapter 6: Differential equations and rectilinear motion
In this chapter
6A Models, verification and direction fields: Form differential models, verify solutions and read direction fieldsExponential growth and decay rate modelsSeparate variables, impose initial values and retain solution domainsAnti-differentiate a power of a linear functionDifferentiate parametric curves and integrate vector motionSolve and interpret continuous logistic models6C Logistic growth and applications: Form differential models, verify solutions and read direction fieldsApproximate solutions with explicit Euler stepsBinary place value and algorithm tracesUse acceleration forms and differential equations in rectilinear motionAnalyse straight-line motion through displacement, velocity and acceleration