11th Grade - Functions

11th Grade lesson

2 - Functions

General remarks

A function is continuous if you can draw its graph without lifting the pencil. Consider an increasing continuous function on an interval [a,b] and let y be a number on the y axis such that f(a)<y<f(b). In that case, the equation f(x)=y has only one solution in the interval [a,b]
continuous function
It's obvious on the drawing and this property is called the intermediate value theorem. You can also use it in the case of a decreasing function. In the problems it is often asked to give an approximate value of the solution of the equation f(x)=y. To do this, you need to use a table values or an equation solver on your calculationator.

Exponential function

Graph of the exponential function (denoted by or exponential function) :

exponential function

On the graph we can see that :
exponential function

The main property of this function, is that the exponential function is always equal to its derivative : the slope of its tangent at x = a is always exp(a). Thanks to the derivation formula of a composition of functions that you know (formules derivation composee function), if u is a function then :
equation math derivation

The notation exponential notation for the exponential function is not innocent, since it is actually a power function. The number e is about 2.7. You can apply the power formulas to the exponential function :
exponential function

Natural logarithm function

It is the reciprocal function of the exponential function, that means that for every number a,  logarithmic function and for every number a > 0,  logarithmic function. Its domain is :  logarithmic function ensemble definition ( logarithmic function is a nonsense because equation logarithmic function is never -2).

Remember the graph :
graph of logarithmic function

You can read below that :
property logarithmic

The derivative of the logarithm function is the inverse function 1/x. Usually if u is a function and if equation math, then :
property ln

The function ln transforms products into sums. For every number a and b, we have
property ln

With this important property, note that math power equation. When there is a power inside of the function ln, you can move it right in front of the function ln. You can also learn for your own culture that the decimal logarithm function is defined by :
logarithmic decimal

This formula, useless in mathematics, is sometimes useful in physics.

>>> Antiderivatives lesson >>>

11th Grade function

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