A Level calculus

Calculus is about local change and accumulated change.

This prototype brings together the core visual ideas behind A Level calculus: tangent gradients, derivatives, definite integrals and numerical area estimates.

It is designed as a revision lab: change the function, move the tangent, adjust the interval and explain what the values mean.

Differentiation

The derivative gives the gradient of the tangent.

Move the point along the curve and watch the tangent change. This is the visual meaning behind differentiation.

x value

1.5

f(x)

1.75

f'(x)

Integration

A definite integral measures signed area.

Compare the exact area with a trapezium-rule estimate. More strips usually help, but the shape of the curve still matters.

Exact area

7.5

Trapezium estimate

7.625

Absolute error

0.125

Guided tasks

Move the tangent point until the gradient is close to zero. What does that suggest about the curve?

Increase the number of trapezia. Does the estimate always improve by the same amount?

Choose the sine function. Where does the tangent have positive gradient, negative gradient and zero gradient?

Linked tools

Function TransformationsExplore transformations before differentiation or sketching.Area ApproximationCompare rectangles, midpoint rule and trapezia.Slope FieldsBridge into differential equations and slope fields.

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Future extensions

This prototype can grow into a richer classroom and exploration tool.

Add exam-style tangent and normal questions with editable student answers.

Add optimisation scenarios where students identify stationary points.

Add parametric differentiation and area under parametric curves.

Add numerical methods cards for trapezium rule, Newton-Raphson and iteration.