Continuity and rate of change
See when a graph can be drawn without lifting the pen, find the k that makes a function continuous, and squeeze an average rate of change down to an instantaneous one.
- UK: Year 11
- France: grade 11
- Germany: grade 11
- Iran: grade 12
- Afghanistan: grade 12
- Syria: grade 12
- Palestine: grade 11
- Ukraine: grade 10
- Russia: grade 10
Guide
A function f is continuous at a if three things hold: f(a) exists, the limit of f as x → a exists, and the two are equal. That means the graph has no hole and no jump there.
The average rate of change over an interval is the gradient of the line joining two points of the graph: change in y divided by change in x. Shrinking the interval gives the rate of change at one instant.