Grade 11 · Exponential calibration · 339 of 600
Exponential calibration · Initial value a=1.5; Exponential rate b=0.3
Evaluate at t=2.4 time units. Find the signal value, tangent slope, and multiplicative change from f(0). Givens: Initial value a=1.5; Exponential rate b=0.3.
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)=1.5e^((0.3)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).
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
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)=1.5e^((0.3)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=1.5 exp((0.3)(2.4))=3.08165. The rate is k y=(0.3)(3.08165)=0.924495, rather than just the amount y.
Step 3
Check the mathematical meaning
Dividing y by 1.5 gives e^((0.3)(2.4))=2.054433. The rate has positive sign because k=0.3; an amount and its rate have different units. Time / input: 2.4; Current amount: 3.082; Instantaneous rate: 0.924; Growth multiplier: 2.054. Decimal values are rounded, so use unrounded intermediate values.
The answer
Substitution gives y=1.5 exp((0.3)(2.4))=3.08165. The rate is k y=(0.3)(3.08165)=0.924495, rather than just the amount y. Dividing y by 1.5 gives e^((0.3)(2.4))=2.054433. The rate has positive sign because k=0.3; an amount and its rate have different units. Animation check: Time / input: 2.4; Current amount: 3.082; Instantaneous rate: 0.924; Growth multiplier: 2.054. 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.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.
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)=1.5e^((0.3)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.