Mister Exam
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-
How to use it?
- Graphing y =:
- x^4
- x^4-4x^3-8x^2+1
- x^4-2x^2+7
-
x^2-1
-
Identical expressions
- sinh(tanh(x))
- hyperbolic sinus of e of ( hyperbolic tangent of gent of (x))
- sinhtanhx
Graphing y = sinh(tanh(x))
The solution
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f(x) = sinh(tanh(x))
$$f{\left(x \right)} = \sinh{\left(\tanh{\left(x \right)} \right)}$$
f = sinh(tanh(x))
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:
$$\sinh{\left(\tanh{\left(x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\sinh{\left(\tanh{\left(x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
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
$$\left(1 - \tanh^{2}{\left(x \right)}\right) \cosh{\left(\tanh{\left(x \right)} \right)} = 0$$
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
$$\left(1 - \tanh^{2}{\left(x \right)}\right) \cosh{\left(\tanh{\left(x \right)} \right)} = 0$$
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
$$\left(\left(\tanh^{2}{\left(x \right)} - 1\right) \sinh{\left(\tanh{\left(x \right)} \right)} + 2 \cosh{\left(\tanh{\left(x \right)} \right)} \tanh{\left(x \right)}\right) \left(\tanh^{2}{\left(x \right)} - 1\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -22.36613259083$$
$$x_{2} = -72$$
$$x_{3} = -62$$
$$x_{4} = -44$$
$$x_{5} = -54$$
$$x_{6} = 31.8870724794909$$
$$x_{7} = -98$$
$$x_{8} = 96$$
$$x_{9} = 64$$
$$x_{10} = 28.6760419286256$$
$$x_{11} = 20.6753972494872$$
$$x_{12} = -31.976245996576$$
$$x_{13} = 54$$
$$x_{14} = 62$$
$$x_{15} = 40$$
$$x_{16} = -48$$
$$x_{17} = -96$$
$$x_{18} = 0$$
$$x_{19} = -40$$
$$x_{20} = 18.6753976927522$$
$$x_{21} = 100$$
$$x_{22} = 78$$
$$x_{23} = -30.3746613927776$$
$$x_{24} = -32$$
$$x_{25} = 80$$
$$x_{26} = -20.3661326061554$$
$$x_{27} = -52$$
$$x_{28} = 32.468253968254$$
$$x_{29} = 70$$
$$x_{30} = 16.6754218978551$$
$$x_{31} = -100$$
$$x_{32} = -16.3661781103476$$
$$x_{33} = 32$$
$$x_{34} = 44$$
$$x_{35} = 60$$
$$x_{36} = 90$$
$$x_{37} = -32.2038153090912$$
$$x_{38} = 76$$
$$x_{39} = 72$$
$$x_{40} = 86$$
$$x_{41} = 56$$
$$x_{42} = -58$$
$$x_{43} = -31.7529322617035$$
$$x_{44} = 32.000015780841$$
$$x_{45} = -86$$
$$x_{46} = -56$$
$$x_{47} = 42$$
$$x_{48} = -26.3661315069563$$
$$x_{49} = 31.7312936927894$$
$$x_{50} = 98$$
$$x_{51} = 22.6753972443119$$
$$x_{52} = -80$$
$$x_{53} = -76$$
$$x_{54} = 84$$
$$x_{55} = -84$$
$$x_{56} = 94$$
$$x_{57} = 58$$
$$x_{58} = 31.7787220654169$$
$$x_{59} = -38$$
$$x_{60} = 34$$
$$x_{61} = 30.6502781482389$$
$$x_{62} = -18.3661334245818$$
$$x_{63} = -82$$
$$x_{64} = -92$$
$$x_{65} = 38$$
$$x_{66} = -42$$
$$x_{67} = 88$$
$$x_{68} = -78$$
$$x_{69} = -36$$
$$x_{70} = -60$$
$$x_{71} = -28.3660709532537$$
$$x_{72} = 92$$
$$x_{73} = -64$$
$$x_{74} = -88$$
$$x_{75} = 24.6753973204929$$
$$x_{76} = -24.3661326142577$$
$$x_{77} = -66$$
$$x_{78} = 36$$
$$x_{79} = -46$$
$$x_{80} = 26.6753908017134$$
$$x_{81} = 66$$
$$x_{82} = -34$$
$$x_{83} = 52$$
$$x_{84} = -94$$
$$x_{85} = -90$$
$$x_{86} = 68$$
$$x_{87} = 82$$
$$x_{88} = 74$$
$$x_{89} = -68$$
$$x_{90} = -74$$
$$x_{91} = 46$$
$$x_{92} = 50$$
$$x_{93} = -31.3350186083881$$
$$x_{94} = -70$$
$$x_{95} = -50$$
$$x_{96} = 48$$
$$x_{97} = -31.5673076923077$$
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\left(\left(\tanh^{2}{\left(x \right)} - 1\right) \sinh{\left(\tanh{\left(x \right)} \right)} + 2 \cosh{\left(\tanh{\left(x \right)} \right)} \tanh{\left(x \right)}\right) \left(\tanh^{2}{\left(x \right)} - 1\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -22.36613259083$$
$$x_{2} = -72$$
$$x_{3} = -62$$
$$x_{4} = -44$$
$$x_{5} = -54$$
$$x_{6} = 31.8870724794909$$
$$x_{7} = -98$$
$$x_{8} = 96$$
$$x_{9} = 64$$
$$x_{10} = 28.6760419286256$$
$$x_{11} = 20.6753972494872$$
$$x_{12} = -31.976245996576$$
$$x_{13} = 54$$
$$x_{14} = 62$$
$$x_{15} = 40$$
$$x_{16} = -48$$
$$x_{17} = -96$$
$$x_{18} = 0$$
$$x_{19} = -40$$
$$x_{20} = 18.6753976927522$$
$$x_{21} = 100$$
$$x_{22} = 78$$
$$x_{23} = -30.3746613927776$$
$$x_{24} = -32$$
$$x_{25} = 80$$
$$x_{26} = -20.3661326061554$$
$$x_{27} = -52$$
$$x_{28} = 32.468253968254$$
$$x_{29} = 70$$
$$x_{30} = 16.6754218978551$$
$$x_{31} = -100$$
$$x_{32} = -16.3661781103476$$
$$x_{33} = 32$$
$$x_{34} = 44$$
$$x_{35} = 60$$
$$x_{36} = 90$$
$$x_{37} = -32.2038153090912$$
$$x_{38} = 76$$
$$x_{39} = 72$$
$$x_{40} = 86$$
$$x_{41} = 56$$
$$x_{42} = -58$$
$$x_{43} = -31.7529322617035$$
$$x_{44} = 32.000015780841$$
$$x_{45} = -86$$
$$x_{46} = -56$$
$$x_{47} = 42$$
$$x_{48} = -26.3661315069563$$
$$x_{49} = 31.7312936927894$$
$$x_{50} = 98$$
$$x_{51} = 22.6753972443119$$
$$x_{52} = -80$$
$$x_{53} = -76$$
$$x_{54} = 84$$
$$x_{55} = -84$$
$$x_{56} = 94$$
$$x_{57} = 58$$
$$x_{58} = 31.7787220654169$$
$$x_{59} = -38$$
$$x_{60} = 34$$
$$x_{61} = 30.6502781482389$$
$$x_{62} = -18.3661334245818$$
$$x_{63} = -82$$
$$x_{64} = -92$$
$$x_{65} = 38$$
$$x_{66} = -42$$
$$x_{67} = 88$$
$$x_{68} = -78$$
$$x_{69} = -36$$
$$x_{70} = -60$$
$$x_{71} = -28.3660709532537$$
$$x_{72} = 92$$
$$x_{73} = -64$$
$$x_{74} = -88$$
$$x_{75} = 24.6753973204929$$
$$x_{76} = -24.3661326142577$$
$$x_{77} = -66$$
$$x_{78} = 36$$
$$x_{79} = -46$$
$$x_{80} = 26.6753908017134$$
$$x_{81} = 66$$
$$x_{82} = -34$$
$$x_{83} = 52$$
$$x_{84} = -94$$
$$x_{85} = -90$$
$$x_{86} = 68$$
$$x_{87} = 82$$
$$x_{88} = 74$$
$$x_{89} = -68$$
$$x_{90} = -74$$
$$x_{91} = 46$$
$$x_{92} = 50$$
$$x_{93} = -31.3350186083881$$
$$x_{94} = -70$$
$$x_{95} = -50$$
$$x_{96} = 48$$
$$x_{97} = -31.5673076923077$$
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
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:
$$\sinh{\left(\tanh{\left(x \right)} \right)} = - \sinh{\left(\tanh{\left(x \right)} \right)}$$
- No
$$\sinh{\left(\tanh{\left(x \right)} \right)} = \sinh{\left(\tanh{\left(x \right)} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\sinh{\left(\tanh{\left(x \right)} \right)} = - \sinh{\left(\tanh{\left(x \right)} \right)}$$
- No
$$\sinh{\left(\tanh{\left(x \right)} \right)} = \sinh{\left(\tanh{\left(x \right)} \right)}$$
- Yes
so, the function
is
odd
The graph