## Why did calculus develop and how might we use it?

### Key questions

1. 1

What is the gradient of a curve and how might it be useful?

2. 2

What do we mean by the area ‘under’ a curve and what might it represent?

3. 3

What do we mean by the rate of change of a physical quantity and how do we represent this?

4. 4

How can we make deeper connections between the appearance of a graph and the features of its equation?

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#### Introducing...

Resource type Title
Building blocks A tangent is ...
Building blocks Is the Serpentine Lake really 40 acres?
Building blocks Mapping a derivative
Problem requiring decisions Discussing distance

#### Developing...

Resource type Title
Building blocks Speed vs velocity
Building blocks Walk-sorting
Many ways problem Rectangles in triangles
Problem requiring decisions Approximating areas
Food for thought Average speed
Food for thought Problem areas
Investigation Underneath the arches
Bigger picture Newton and Leibniz
Bigger picture Why are gradients important in the real world?

#### Review questions

Title Ref
At what time would these two trains meet? R5782
Can the trapezium rule help us find the area under $y = x\ln x$? R7903
Can we apply the trapezium rule to $f(x) = x/2 - \lfloor x/2 \rfloor$? R8140
Can we approximate the area under this sine curve? R7912
Can we calculate the total time taken for this car journey? R7250
Can we use the trapezium rule to estimate this integral? R6592
What is the acceleration of this particle? R9586
What is the stopping distance for this car? R6841
What's the average speed of the second car? R7140
When does the trapezium rule give the exact answer here? R5642
When is this parabola's turning point nearest to the origin? R9506