Description of Bottleneck
Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. They do not have an idea of "derivative=slope" or in case they have it they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value"
Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. They should be able to draw for a given graph of function a triangle to illustrate the changes in values and the slope
Concrete problems may be, for example:
- A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
- B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
- C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N
Description of mental tasks needed to overcome the bottleneck
- While thinking about derivative, mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis)
- He clearly understands that the story is not about calculating one value, but about the process and how function value depends on argument value
- He knows, that he can look onto small local changes if he wants to calculate the concrete change rate, and he knows where to place the changes into the graph of function.
- He can forget, that the derivative should be calculated as a limit, if the fact that derivative is a quotient df/dx is enough to deduce needed information or is the only possibility. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values df and dx
- He is not afraid of being unprecise, because he knows, that the approximation by tangent is allowed
Modelling the tasks
People interested in this Bottleneck
Inna Mikhailova
