Continuous and discrete.
AskSia uses P(t) = P0 e^(rt) for continuous compounding and P(t) = P0 (1+r)^t for discrete steps, choosing based on the problem.
Two models
AI Solver by AskSia
Exponential Growth Solver: growth and decay, modeled
Type your population, money, or substance problem. AskSia picks the right model (continuous P0 e^(rt) or discrete P0 (1+r)^t), solves for whichever variable you need (future value, rate, time, or starting value), and shows the logarithm step when t is the unknown. Doubling time and half-life come as one-line shortcuts.
Works with word problems, equations, code, and science prompts.
∫ 3x² · sin(x) dx
SubjectsCalculusAlgebraPhysicsChemistryBiologyCSStatisticsEcon
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asksia.ai/solver
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Problem
Find the area enclosed between y = x² and y = 2x in the first quadrant.
1
Find the points of intersection
Set the curves equal: x² = 2x → x(x - 2) = 0, so x = 0 and x = 2.
2
Identify which curve is on top
On (0, 2), 2x > x², so y = 2x is the upper curve.
3
Set up the area integral
A = ∫₀² (2x - x²) dx
4
Integrate term-by-term
A = [ x² - x³/3 ]₀²
Final Answer
A = 4/3 sq. units
Quick Answer
What is exponential growth?
Exponential growth is growth where the rate of change is proportional to the current value, producing a curve that climbs faster and faster. The two standard models are continuous, P(t) = P0 * e^(rt), and discrete, P(t) = P0 * (1+r)^t, where P0 is the initial value, r is the growth rate per unit time, and t is time. Exponential decay uses negative r (or 1-r in discrete form). Doubling time for continuous growth is ln(2)/r; half-life for decay is ln(2)/r as well (positive r in the decay formula).
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solution accuracy
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problems solved
~1.5s
avg solve time
A+
study-ready explanations
Why AskSia Solver
Built for population, money, and decay problems.
Continuous and discrete models, every formula step shown.
Logarithm step shown.
When time t is the unknown, AskSia takes the natural log (or log base appropriate to the model) and isolates t with every algebra step labeled.
Transparent
Doubling time and half-life.
Shortcuts ln(2)/r computed automatically when the question asks for them, with the derivation available on toggle.
Shortcuts
Rate conversion.
Continuous rate r_c and discrete rate r_d are related by r_d = e^(r_c) - 1. AskSia converts both ways when needed.
Rate-aware
Graph included.
The growth or decay curve appears with starting value, doubling point, and any specific times you asked about marked on the plot.
Visual
Word problem parsing.
Population biology, compound interest, radioactive decay, viral growth: AskSia identifies the model from the words used.
Word problems
How It Works
Solve exponential growth in three steps.
Step 01
Enter what you know.
Starting value, ending value, growth rate, time, or any subset. AskSia identifies which variable is the unknown.
Input mode
Snap a Photo
Textbook, handwriting, screenshot
Paste Text
Word problem or equation
Calculator
LaTeX-ready equation editor
Step 02
AskSia applies the right formula.
Continuous or discrete, growth or decay. The formula appears with values substituted and any logarithm step shown.
1
Set curves equal
x² = 2x → x = 0, x = 2
2
Set up the integral
A = ∫₀² (2x - x²) dx
3
Evaluate
A = [x² - x³/3]₀² = 4/3
Step 03
Read the answer.
Final answer in the requested units (years, hours, dollars, grams). The growth curve appears with the key points labeled.
Auto-generated diagram
Region between y = 2x and y = x² — area = 4/3
Available On
Solve anywhere
you study.
Every solve syncs across Web, iOS, and Android — start it at your desk, finish on your phone.
Web App
Full study studio
Split-panel interface with the worked solution on the left, the auto-generated diagram and AI tutor chat on the right.
Drag & drop image upload + LaTeX equation editor
Auto-generated diagrams render alongside steps
Side-panel AI tutor chat for hints and alt methods
Export to PDF, DOCX, Notion, or Google Docs
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Calculus
98% verified
1.4s
Step 4 of 4 · Evaluate
A = [x² - x³/3]₀² = 4/3
Mobile App
Snap & solve, anywhere
Open the camera, frame the problem, and the worked solution plus diagram appear in seconds.
One-tap snap-and-solve on iOS and Android
Pinch-to-zoom diagrams, swipe between steps
Auto-sync solves with your Web library
Offline review of saved solutions and flashcards
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98%
1.4s
Area between y=2x & y=x²
A = 4/3 sq. units ✓
Use Cases
Where exponential growth and decay apply.
Population biology.
Bacterial populations doubling every 20 minutes, animal populations growing at 3% per year. AskSia computes future populations and doubling times.
Biology
Compound interest.
Continuous compounding uses A = P e^(rt). Annual compounding uses A = P(1+r)^t. AskSia distinguishes and applies correctly.
Finance
Radioactive decay.
Half-life problems for carbon-14 dating, medical isotopes, or chemistry homework. AskSia uses P0 * (1/2)^(t/h).
Chemistry
Viral spread models.
Early-phase epidemic growth follows exponentials. AskSia fits the model and predicts case counts under continued growth.
Public health
Drug concentration.
Pharmacology problems with first-order elimination follow exponential decay. AskSia computes half-life and remaining concentration.
Pharma
SAT, ACT, AP word problems.
Exponential growth and decay appear regularly. AskSia parses the prompt and solves with the steps the test expects.
Test prep
Compare
AskSia vs. ChatGPT,
Photomath & Symbolab.
General chatbots hallucinate. Photo solvers stop at math. AskSia is built for actual coursework with verified accuracy, visual learning, and every subject.
| Feature | AskSia Solver | ChatGPT | Photo Solvers |
|---|---|---|---|
| Solution accuracy | ✓ 98% | ~70-85%, hallucinations | ~90%, math only |
| Auto-generated diagrams | ✓ Every solve | Inconsistent / broken | Graphs only, math-only |
| Step-by-step explanations | ✓ Numbered + plain English | Inconsistent depth | ✓ Math steps |
| Subject coverage | ✓ Math, Physics, Chem, Bio, CS, Econ | ✓ Wide but unverified | Math only |
| Photo input | ✓ Handwriting + diagrams + code | Photos OK, weak on handwriting | ✓ Math photos only |
| Answer verification | ✓ Self-checked before display | No verification | Math engine only |
| Tutor follow-ups | ✓ Hints, alt methods, ELI5 | ✓ General chat | Not available |
| Practice and flashcards | ✓ One-tap from any solve | Manual prompting | Not available |
| Code debugging | ✓ Python, Java, C++, SQL... | ✓ Yes | Not available |
| Free to start | ✓ Daily solves, no card | Limited model access | Steps locked behind paywall |
FAQ
Frequently asked questions.
How do I know whether to use continuous or discrete growth?
Look at how the rate is described. 'Compounded continuously' or 'instantaneous growth rate' or 'natural growth rate r' all signal continuous: use P(t) = P0 e^(rt). 'Compounded annually,' 'grows by 5% per year,' or 'doubles every 7 years' signals discrete: use P(t) = P0 (1+r)^t with r as a fraction. AskSia checks the wording of word problems and picks the right model.
How is doubling time computed?
For continuous growth, set 2 P0 = P0 e^(rt) and solve for t: t = ln(2)/r. For discrete growth, set 2 P0 = P0 (1+r)^t and solve: t = ln(2)/ln(1+r). AskSia applies whichever formula matches the model and shows the logarithm derivation if you want to see it.
How is half-life related to the decay rate?
Half-life is the time for the quantity to drop to half its current value. For continuous decay P(t) = P0 e^(-rt), set P0/2 = P0 e^(-r*h) and solve: h = ln(2)/r. So the half-life is ln(2)/r when r is the decay constant. AskSia computes either from the other and uses whichever is provided in the problem.
What is the difference between exponential and linear growth?
Linear growth adds a constant amount each unit of time: P(t) = P0 + kt. Exponential growth multiplies by a constant factor each unit of time: P(t) = P0 (1+r)^t. Exponentials overtake linears no matter how small the rate or how large the additive constant, because exponentials are eventually faster. AskSia identifies which model fits a data set if you provide one.
How accurate is AskSia?
AskSia is engineered for accuracy on standard high school and college coursework. Accuracy comes from subject-specialized models, a symbolic verification pass that catches arithmetic errors, and a self-check step that re-derives the answer before showing it to you.
Can I get practice problems and flashcards?
Yes. After any solve, ask Sia to generate similar practice problems at SAT, ACT, AP, IB, or college difficulty, or build a flashcard set on the underlying concept in one tap. Useful for exam prep and spaced repetition before a quiz, midterm, or final.
How much does AskSia cost?
AskSia has a free plan that includes daily solves across all subjects. AskSia Pro and Super include unlimited solves, advanced subjects, the full AI tutor companion, exports, and priority response speed. See pricing for details.
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From doubling time to drug concentration.
Join 2M+ students using AskSia to model exponential growth and decay with every step shown. Continuous, discrete, half-life, doubling time, and a graph of every model.
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