Grade 11 · Exponential calibration · 95 of 600
Exponential calibration · Initial value a=2.5; Exponential rate b=0
Evaluate at t=2.4 time units. Find the signal value, tangent slope, and multiplicative change from f(0). Givens: Initial value a=2.5; Exponential rate b=0.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
An exponential calibration signal is f(t)=2.5e^((0)t), with t measured in time units. Evaluate at t=2.4 time units. Find the signal value, tangent slope, and multiplicative change from f(0).
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A starting point
Evaluate e^(kt), multiply by the initial amount, then multiply that amount by k to get its rate.
Work through the reasoning
Step 1
Identify the model and target
An exponential calibration signal is f(t)=2.5e^((0)t), with t measured in time units. Evaluate at t=2.4 time units. The governing relation is f(t)=a e^(bt); f′(t)=b f(t). Evaluate e^(kt), multiply by the initial amount, then multiply that amount by k to get its rate.
Step 2
Substitute and calculate
Substitution gives y=2.5 exp((0)(2.4))=2.5. The rate is k y=(0)(2.5)=0, rather than just the amount y.
Step 3
Check the mathematical meaning
Dividing y by 2.5 gives e^((0)(2.4))=1. The rate has zero sign because k=0; an amount and its rate have different units. Time / input: 2.4; Current amount: 2.5; Instantaneous rate: 0; Growth multiplier: 1. Decimal values are rounded, so use unrounded intermediate values.
The answer
Substitution gives y=2.5 exp((0)(2.4))=2.5. The rate is k y=(0)(2.5)=0, rather than just the amount y. Dividing y by 2.5 gives e^((0)(2.4))=1. The rate has zero sign because k=0; an amount and its rate have different units. Animation check: Time / input: 2.4; Current amount: 2.5; Instantaneous rate: 0; Growth multiplier: 1. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
The initial amount and proportional growth rate play different roles. Multiplying a scales every output, but changing b changes each time interval's multiplicative factor. A negative rate produces decay without making a positive starting amount negative. An exponential calibration signal is f(t)=2.5e^((0)t), with t measured in time units. Evaluate at t=2.4 time units. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
f(t)=a e^(bt); f′(t)=b f(t)
03 · Reflect and transfer
Explain what changes and why.
Change the initial amount while holding the rate constant. Which readout stays unchanged, and why?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.