Calculus unit 1

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Formulas

Calculus

Functions and Their Graphs

Functions: $f$, $g$, $y$, $u$
Argument (independent variable): $x$
Set of natural numbers: $N$
Set of real numbers: $R$
The base of natural logarithms: $e$

Natural numbers: $n$
Integers: $k$
Real numbers: $a$, $b$, $c$, $d$
Angle: $\alpha$
Period of a function: $T$

1. The concept of function is one of the most important in mathematics. It is defined as follows. Let two sets $X$ and $Y$ be given. If for every element $x$ in the set $X$ there is exactly one element (an image) $y=f\left(x\right)$ in the set $Y$, then it is said that the function $f$ is defined on the set $X$. The element $x$ is called the independent variable, and respectively, the output $y$ of the function is called the dependent variable. If we consider the number sets $X\subset R$, $Y\subset R$ (where $R$ is the set of real numbers), then the function $y=f\left(x\right)$ can be represented as a graph in a Cartesian coordinate system $Oxy$.
2. Even function
   $f\left(-x\right)=f\left(x\right)$
3. Odd function
   $f\left(-x\right)=-f\left(x\right)$
4. Periodic function
   $f\left(x+kT\right)=f\left(x\right),$
   where $k$ is an integer, $T$ is the period of the function.
5. Inverse function
   Given a function $y=f\left(x\right)$. To find its inverse function of it, it is necessary solve the equation $y=f\left(x\right)$ for $x$ and then switch the variables $x$ and $y$. The inverse function is often denoted as $y=f^{-1}\left(x\right)$. The graphs of the original and inverse functions are symmetric about the line $y=x$.



6. Composite function
   Suppose that a function $y=f\left(u\right)$ depends on an intermediate variable $u$, which in turn is a function of the independent variable $x$: $u=g\left(x\right)$. In this case, the relationship between $y$ and $x$ represents a “function of a function” or a composite function, which can be written as $y=f\left(g\left(x\right)\right)$. The two-layer composite functions can be easily generalized to an arbitrary number of “layers”.
7. Linear function
   $y=ax+b,$ $x\in R$.
   Here the number $a$ is called the slope of the straight line. It is equal to the tangent of the angle between the straight line and the positive direction of the $x$-axis: $a=\tan \alpha$. The number $b$ is the $y$-intercept.



8. Quadratic function
   The simplest quadratic function has the form
   $y=x^2,$ $x\in R.$
   In general, a quadratic function is described by the formula
   $y=a{x}^2+bx+c,$ $x\in R,$
   where $a$, $b$, $c$ are real numbers (in this case $a\ne 0.$) The graph of a quadratic function is called a parabola. The direction of the branches of the parabola depends on the sign of the coefficient $a$. If $a>0$, the parabola is concave upwards. If $a<0$, the parabola is concave downwards.


9. Cubic function
   The simplest cubic function is given by
   $y=x^3,$ $x\in R.$
   In general, a cubic function is described by the formula
   $y=a{x}^3+b{x}^2+cx+d,$ $x\in R,$
   where $a$, $b$, $c$, $d$ are real numbers $\left(a\ne 0\right).$ The graph of a cubic function is called a cubic parabola. When $a>0$, the cubic function is increasing, and when $a<0,$ the cubic function is, respectively, decreasing.



10. Power function
    $y=x^n,$ $x\in R,$ $n\in N$.


11. Square root function
    $y=\sqrt{x},$ $x\in \left[0,\infty \right).$



12. Exponential functions
    $y=a^x,$ $x\in R,$ $a>0,$ $a\ne 1,$
    $y=e^x$ when $a=e\approx 2.71828182846\dots$
    An exponential function increases when $a>1$ and decreases when $0<a<1$.



13. Logarithmic functions
    $y={\log }_{a}x,$ $x\in \left(0,\infty \right),$ $a>0,$ $a\ne 1,$
    $y=\ln x$, when $a=e,x\in \left(0,\infty \right).$
    A logarithmic function increases if $a>1$ and decreases if $0<a<1$.



14. Hyperbolic sine function
    $y=\sinh x=$ $\frac{e^x-e^{-x}}{2},$ $x\in R.$



15. Hyperbolic cosine function
    $y=\cosh x=$ $\frac{e^x+e^{-x}}{2},$ $x\in R.$



16. Hyperbolic tangent function
    $y=\tanh x=$ $\frac{\sinh x}{\cosh x}=$ $\frac{e^x-e^{-x}}{e^x+e^{-x}},$ $x\in R.$



17. Hyperbolic cotangent function
    $y=\coth x=$ $\frac{\cosh x}{\sinh x}=$ $\frac{e^x+e^{-x}}{e^x-e^{-x}},$ $x\in R,$ $x\ne 0.$



18. Hyperbolic secant function
    $y=\mathrm{sech}x=$ $\frac{1}{\cosh x}=$ $\frac{2}{e^x+e^{-x}},$ $x\in R.$



19. Hyperbolic cosecant function
    $y=\mathrm{csch}x=$ $\frac{1}{\sinh x}=$ $\frac{2}{e^x-e^{-x}},$ $x\in R,$ $x\ne 0.$



20. Inverse hyperbolic sine function
    $y=\mathrm{arcsinh}x,$ $x\in R.$



21. Inverse hyperbolic cosine function
    $y=\mathrm{arccosh}x,$ $x\in \left[1,\infty \right).$



22. Inverse hyperbolic tangent function
    $y=\mathrm{arctanh}x,$ $x\in \left(-1,1\right).$



23. Inverse hyperbolic cotangent function
    $y=\mathrm{arccoth}x,$ $x\in \left(-\infty ,-1\right)\cup \left(1,\infty \right).$



24. Inverse hyperbolic secant function
    $y=\mathrm{arcsech}x,$ $x\in \left(0,1\right].$



25. Inverse hyperbolic cosecant function
    $y=\mathrm{arccsch}x,$ $x\in R,x\ne 0.$

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