When growth is continuous rather than in steps, the base is e. The formula is the same shape as compound interest, with the rate times the time sitting in the exponent. Answers are left exact so nothing is rounded away.
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Pick more than one for a sheet that mixes them.
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$1000 grows continuously at 5% a year. Find the value after 4 years, exactly
The exponent is the rate times the time.
5 percent of 4 years is 0.2, or one fifth.
The answer is 1000e51.
Where students go wrong
Putting the percent in as a whole number. Five percent is 0.05 in the exponent, not 5, and getting that wrong makes the answer astronomically too big.
Also called natural base, e to the rt or continuous growth.
What you can put on this worksheet
Continuous growth
$600 grows continuously at 5% a year. Find its exact value after 4 years.→ $600e0.2
Equations with e and ln
Solve exactly: ln(x+1)=4→ {e4−1}
Simplify with e, growth or decay
Simplify: 4e3x⋅2ex→ 8e4x
Continuous decay and growth, doubling time, two data points
A town's population grows continuously at 1% per year, starting from 686 million. How many million after 34 years? Round to the nearest tenth.→ 963.8 million
Questions about these worksheets
Yes. Every natural base e sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the natural base e questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 4: continuous growth, equations with e and ln, simplify with e, growth or decay and continuous decay and growth, doubling time, two data points. Tick as many as you want and set how many of each, or let it spread them evenly.
Putting the percent in as a whole number. Five percent is 0.05 in the exponent, not 5, and getting that wrong makes the answer astronomically too big.