General remarksA 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]
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 functionGraph of the exponential function (denoted by or ) :
On the graph we can see that :
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 (), if u is a function then :
The 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 :
Natural logarithm functionIt is the reciprocal function of the exponential function, that means that for every number a, and for every number a > 0, . Its domain is : ( is a nonsense because is never -2).
Remember the graph :
You can read below that :
The derivative of the logarithm function is the inverse function 1/x. Usually if u is a function and if , then :
The function ln transforms products into sums. For every number a and b, we have
With this important property, note that . 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 :
This formula, useless in mathematics, is sometimes useful in physics.