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Aus Online Mathematik Brückenkurs 2

The derivative f(−5) gives the function's instantaneous rate of change at the point x=−5, i.e. it is a measure of how much the function's value changes in the vicinity of x=−5.

In the graph of the function, this derivative is equal to the slope of the tangent to the graph of the function at the point x=−5.

1.1 - Figure - Solution - The graph of f(x) in exercise 1.1:1a with the tangent line at x = -5

The red tangent line has the equation y = kx + m, where k = f'(-5).

Because the tangent is sloping upwards, it has a positive gradient and therefore f(−5)0.

At the point x=1, the tangent slopes downwards and this means that f(1)0.

1.1 - Figure - Solution - The graph of f(x) in exercise 1.1:1a with the tangent line at x = 1

The red tangent line has the equation y = kx + m, where k = f'(1).