Mister Exam
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-
How to use it?
- Graphing y =:
- x^4-4x^3-8x^2+1
- x^4-2x^2+7
- x^4-2x^2+2
-
x^2-1
-
Identical expressions
- tgx/ four - one
- tgx divide by 4 minus 1
- tgx divide by four minus one
- tgx divide by 4-1
-
Similar expressions
- tgx/4+1
- (-2x^2+82x)/tg((x/4)-1)
Graphing y = tgx/4-1
The solution
You have entered
[src]
tan(x)
f(x) = ------ - 1
4
$$f{\left(x \right)} = \frac{\tan{\left(x \right)}}{4} - 1$$
f = tan(x)/4 - 1
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\frac{\tan{\left(x \right)}}{4} - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(4 \right)}$$
Numerical solution
$$x_{1} = 86.1488193105924$$
$$x_{2} = 10.7505956244374$$
$$x_{3} = 20.1753735852068$$
$$x_{4} = -67.7892207153074$$
$$x_{5} = 57.8744854282843$$
$$x_{6} = 42.1665221603353$$
$$x_{7} = 35.8833368531558$$
$$x_{8} = 64.1576707354639$$
$$x_{9} = -74.072406022487$$
$$x_{10} = -8.09896029710135$$
$$x_{11} = -36.3732941794095$$
$$x_{12} = 95.5735972713618$$
$$x_{13} = -271.992743198644$$
$$x_{14} = 13.8921882780272$$
$$x_{15} = -152.612222362232$$
$$x_{16} = 92.432004617772$$
$$x_{17} = -23.8069235650503$$
$$x_{18} = -1.81577498992176$$
$$x_{19} = -45.7980721401789$$
$$x_{20} = -39.5148868329993$$
$$x_{21} = -96.0635545976156$$
$$x_{22} = -52.0812574473585$$
$$x_{23} = -89.780369290436$$
$$x_{24} = 7.60900297084762$$
$$x_{25} = -30.0901088722299$$
so we need to solve the equation:
$$\frac{\tan{\left(x \right)}}{4} - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(4 \right)}$$
Numerical solution
$$x_{1} = 86.1488193105924$$
$$x_{2} = 10.7505956244374$$
$$x_{3} = 20.1753735852068$$
$$x_{4} = -67.7892207153074$$
$$x_{5} = 57.8744854282843$$
$$x_{6} = 42.1665221603353$$
$$x_{7} = 35.8833368531558$$
$$x_{8} = 64.1576707354639$$
$$x_{9} = -74.072406022487$$
$$x_{10} = -8.09896029710135$$
$$x_{11} = -36.3732941794095$$
$$x_{12} = 95.5735972713618$$
$$x_{13} = -271.992743198644$$
$$x_{14} = 13.8921882780272$$
$$x_{15} = -152.612222362232$$
$$x_{16} = 92.432004617772$$
$$x_{17} = -23.8069235650503$$
$$x_{18} = -1.81577498992176$$
$$x_{19} = -45.7980721401789$$
$$x_{20} = -39.5148868329993$$
$$x_{21} = -96.0635545976156$$
$$x_{22} = -52.0812574473585$$
$$x_{23} = -89.780369290436$$
$$x_{24} = 7.60900297084762$$
$$x_{25} = -30.0901088722299$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\tan^{2}{\left(x \right)}}{4} + \frac{1}{4} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\tan^{2}{\left(x \right)}}{4} + \frac{1}{4} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{4} - 1\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{4} - 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{4} - 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{4} - 1\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x)/4 - 1, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{\tan{\left(x \right)}}{4} - 1}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{\tan{\left(x \right)}}{4} - 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{\tan{\left(x \right)}}{4} - 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{\tan{\left(x \right)}}{4} - 1}{x}\right)$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\frac{\tan{\left(x \right)}}{4} - 1 = - \frac{\tan{\left(x \right)}}{4} - 1$$
- No
$$\frac{\tan{\left(x \right)}}{4} - 1 = \frac{\tan{\left(x \right)}}{4} + 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\tan{\left(x \right)}}{4} - 1 = - \frac{\tan{\left(x \right)}}{4} - 1$$
- No
$$\frac{\tan{\left(x \right)}}{4} - 1 = \frac{\tan{\left(x \right)}}{4} + 1$$
- No
so, the function
not is
neither even, nor odd