Loading module…

What you will learn in Applications of Derivatives

1. Tangents and Normals
Two lines at a point, and the one case where neither can be written as y = mx + c.
2. Derivatives as Rates in Context
One derivative read three ways, with the units computed rather than typed.
3. Related Rates
A geometric relationship differentiated with respect to time, and one rate driving another.
4. Increasing and Decreasing Functions
The sign of the derivative as a strip, and the stationary point that is not a turning point.
5. Critical Points, Local and Absolute Extrema
The closed-interval method, with the endpoint check enforced rather than remembered.
6. The Second-Derivative Test
A faster test that sometimes says nothing, and what to do when it does.
7. Optimisation
Constraint, objective, reduce, then differentiate — in that order and not before.
8. Concavity and Points of Inflection
Which way a curve bends, and the trap that f″ = 0 is enough.
9. Systematic Curve Sketching
The module's synthesis: nine rows, filled in, and then the truth revealed.
10. Differentials, Approximation and Error
Replacing a curve by its tangent, and measuring where that stops working.
11. Position, Velocity and Acceleration
One function differentiated twice, laid alongside the formulas Physics teaches without calculus — both labelled, neither replacing the other.
12. Marginal Cost, Revenue and Elasticity
The derivative in economics, where profit is maximised at an intersection.

Loading module…

1. Tangents and Normals
Two lines at a point, and the one case where neither can be written as y = mx + c.
2. Derivatives as Rates in Context
One derivative read three ways, with the units computed rather than typed.
3. Related Rates
A geometric relationship differentiated with respect to time, and one rate driving another.
4. Increasing and Decreasing Functions
The sign of the derivative as a strip, and the stationary point that is not a turning point.
5. Critical Points, Local and Absolute Extrema
The closed-interval method, with the endpoint check enforced rather than remembered.
6. The Second-Derivative Test
A faster test that sometimes says nothing, and what to do when it does.
7. Optimisation
Constraint, objective, reduce, then differentiate — in that order and not before.
8. Concavity and Points of Inflection
Which way a curve bends, and the trap that f″ = 0 is enough.
9. Systematic Curve Sketching
The module's synthesis: nine rows, filled in, and then the truth revealed.
10. Differentials, Approximation and Error
Replacing a curve by its tangent, and measuring where that stops working.
11. Position, Velocity and Acceleration
One function differentiated twice, laid alongside the formulas Physics teaches without calculus — both labelled, neither replacing the other.
12. Marginal Cost, Revenue and Elasticity
The derivative in economics, where profit is maximised at an intersection.