Rates, local models and error
Differentiation & Approximation
Differentiate functions, model rates of change, approximate values, test limits, propagate measurement error, and search for optima. The tools are built around checks and warnings, not blind symbolic output.
A practical route through differentiation
- Start with rules and templates so the structure of the derivative is clear.
- Move into motion, numerical derivatives, and Taylor approximations when the derivative has a job to do.
- Use error, limit, and optimisation tools when the result needs interpretation, not just calculation.
Differentiation
Differentiation rules
Start with the rules: powers, trig functions, products, quotients, and chains.
Differentiation
Rates of change
Use derivatives to connect displacement, velocity, acceleration, and engineering motion.
Differentiation
Taylor and approximation
Build local polynomial models and see how far they can be trusted.
Differentiation
Numerical differentiation
Estimate derivatives from nearby values when an exact formula is unavailable.
Differentiation
Limits and errors
Use derivatives carefully for indeterminate limits and measurement uncertainty.
Differentiation
Optimisation
Classify stationary points and search for best values on an interval.
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