That was a brief introduction to the subject.
So that you understand it completely, watch more clearly explained math videos here:
Extreme value problems
Extreme value tasks include all tasks in which something should be the largest or the smallest (a triangular area, a volume, a distance). There are currently several standard tasks of such a maximization (or minimization). These extreme values are calculated here.
Calculate the intersection angle between functions
The mutual position of two functions can be traced back to two important special cases: 1. both functions touch each other, 2. both functions are perpendicular to each other (orthogonal). If both are not the case, there is some cutting angle. (It can of course also be the case that both functions GAR do not intersect, but mathematically this is not necessarily the most interesting case.)
Moving, mirroring, stretching functions
Two functions are called “similar” if one can be converted into the other by stretching, mirroring or shifting (= converting). The stretching factor indicates how many times a function is stretched. There is a mathematical procedure each for stretching, mirroring and shifting functions, which is worth remembering.
Function group, function group: what it is and how to calculate with it
A function group or function group is simply a function in which a parameter occurs. (With a function “f (x)”, “x” always means “variable”, every other letter is called “parameter” and is treated like a number). Since you could use an infinite number of values for the parameter, you have an infinite number of curves, all of which look similar and are called family of curves. The functional investigation is of course the same in principle as with functions without parameters (just a little uglier).
Continuity and differentiability of functions
A function is continuous if it does NOT jump, i.e. if it runs continuously, if you can draw it without lifting the pen. A function can be differentiated (sometimes you can also see the notation “differentiable”) if it has NO kink, i.e. if it runs smoothly everywhere. One can also say that a function is differentiable if the function AND the first derivative are continuous. (The function is twice differentiable if the function, first and second derivative is continuous).
An inequality does not have an equal sign, but an inequality sign, i.e. a “less than” or a “greater than” sign (or “less than or equal to” or “greater than or equal to”). Treat inequalities just like equations, except that the inequality sign flips when you multiply by a negative number or divide by a negative number.
Diagrams of functions
There are essentially three types of questions relating to diagrams in the four quadrants: 1. Different diagrams and various functional equations are given and the individual diagrams must be assigned to the individual functions. 2. only a diagram is given and you have to find the functional equation that fits it. (Sometimes a function is given depending on several parameters and now you have to determine the parameters). 3. The diagram of a function is given and one must draw the diagram of the derivative function or the antiderivative.
If you solve a function equation for “x”, you get the inverse function (sometimes also called “inverse function”). (If you insert a number in the function for “y” and then solve for “x”, you have already done this a thousand times. If you reverse the function (invert) the only new thing is that you do not insert anything for “y”, but let it stand.)
How to calculate with GTR and CAS
A graphical pocket calculator (GTR) or a computer algebra system (CAS) of course allows calculations that are never possible by hand (or at least not in the short time). Here are a few examples of such calculations. As a pupil / student it is your task to know how to use the GTR / CAS (i.e.: calculate zeros, solve equations, calculate high, low and inflection points, calculate tangent equations, calculate areas or integrals). If you don’t know how to do one or the other, you will find links to short operating instructions for all common GTR and CAS models from Casio and TI (= Texas Instruments) on the start page.