Mister Exam
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- The intersection points of functions
-
How to use it?
- Graphing y =:
- log1/6x
-
sin(2x)+cos(3x)
-
y=3x^5-10x^3+15x
-
tgx
-
Identical expressions
- sin(2x)+cos(3x)
- sinus of (2x) plus co sinus of e of (3x)
- sin2x+cos3x
-
Similar expressions
- sin(2x)-cos(3x)
Graphing y = sin(2x)+cos(3x)
The solution
You have entered
[src]
f(x) = sin(2*x) + cos(3*x)
$$f{\left(x \right)} = \sin{\left(2 x \right)} + \cos{\left(3 x \right)}$$
f = sin(2*x) + cos(3*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:
$$\sin{\left(2 x \right)} + \cos{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{9 \pi}{10}$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = - \frac{\pi}{10}$$
$$x_{4} = \frac{3 \pi}{10}$$
$$x_{5} = \frac{\pi}{2}$$
$$x_{6} = \frac{7 \pi}{10}$$
Numerical solution
$$x_{1} = 53.7212343763855$$
$$x_{2} = -58.1194640914112$$
$$x_{3} = 26.0752190247953$$
$$x_{4} = 97.7035315266426$$
$$x_{5} = -34.2433599241287$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = 73.8274273593601$$
$$x_{8} = -44.2964564156161$$
$$x_{9} = -54.3495529071034$$
$$x_{10} = 16.0221225333079$$
$$x_{11} = -51.8362787842316$$
$$x_{12} = -4.08407044966673$$
$$x_{13} = -38.0132711084365$$
$$x_{14} = 21.0486707790516$$
$$x_{15} = -31.7300858012569$$
$$x_{16} = 9.73893722612836$$
$$x_{17} = 245.986704776081$$
$$x_{18} = 70.0575161750524$$
$$x_{19} = -93.3053018116169$$
$$x_{20} = 93.9336203423348$$
$$x_{21} = -78.2256570743859$$
$$x_{22} = 33.6150413934108$$
$$x_{23} = -45.553093477052$$
$$x_{24} = 56.2345084992573$$
$$x_{25} = -87.0221165044373$$
$$x_{26} = -98.3318500573605$$
$$x_{27} = 64.4026493985908$$
$$x_{28} = -65.6592864600267$$
$$x_{29} = 18.5353966561798$$
$$x_{30} = -10.3672557568463$$
$$x_{31} = -21.6769893097696$$
$$x_{32} = 58.1194640914112$$
$$x_{33} = -25.4469004940773$$
$$x_{34} = -11.6238928182822$$
$$x_{35} = 80.1106126665397$$
$$x_{36} = 49.9513231920777$$
$$x_{37} = 39.8982267005904$$
$$x_{38} = 51.8362787842316$$
$$x_{39} = 86.3937979737193$$
$$x_{40} = -55.6061899685393$$
$$x_{41} = -49.3230046613598$$
$$x_{42} = 60.0044196835651$$
$$x_{43} = 12.2522113490002$$
$$x_{44} = -80.1106126665397$$
$$x_{45} = -69.4291976443344$$
$$x_{46} = -0.314159265358979$$
$$x_{47} = 96.4468944652067$$
$$x_{48} = 2.19911485751286$$
$$x_{49} = 32.3584043319749$$
$$x_{50} = 31.101767270539$$
$$x_{51} = -81.9955682586936$$
$$x_{52} = 14.1371669411541$$
$$x_{53} = 95.8185759344887$$
$$x_{54} = -61.8893752757189$$
$$x_{55} = -24.1902634326414$$
$$x_{56} = 7.85398163397448$$
$$x_{57} = -92.0486647501809$$
$$x_{58} = 62.5176938064369$$
$$x_{59} = -56.8628270299753$$
$$x_{60} = 100.216805649514$$
$$x_{61} = 90.1637091580271$$
$$x_{62} = 5.96902604182061$$
$$x_{63} = 46.18141200777$$
$$x_{64} = -71.9424717672063$$
$$x_{65} = 52.4645973149496$$
$$x_{66} = 2363.4201532956$$
$$x_{67} = -89.5353906273091$$
$$x_{68} = -88.2787535658732$$
$$x_{69} = -95.8185759344887$$
$$x_{70} = -36.1283155162826$$
$$x_{71} = 89.5353906273091$$
$$x_{72} = -5.34070751110265$$
$$x_{73} = -17.9070781254618$$
$$x_{74} = 82.6238867894116$$
$$x_{75} = -14.1371669411541$$
$$x_{76} = 83.8805238508475$$
$$x_{77} = 77.5973385436679$$
$$x_{78} = -41.7831822927443$$
$$x_{79} = -19.1637151868977$$
$$x_{80} = -73.8274273593601$$
$$x_{81} = 29.845130209103$$
$$x_{82} = -27.9601746169492$$
$$x_{83} = -22.9336263712055$$
$$x_{84} = -1.5707963267949$$
$$x_{85} = 42.4115008234622$$
$$x_{86} = -85.7654794430014$$
$$x_{87} = -99.5884871187965$$
$$x_{88} = 76.340701482232$$
$$x_{89} = -48.0663675999238$$
$$x_{90} = 20.4203522483337$$
$$x_{91} = -29.845130209103$$
$$x_{92} = 22.3053078404875$$
$$x_{93} = -66.9159235214626$$
$$x_{94} = -75.712382951514$$
$$x_{95} = -7.85398163397448$$
$$x_{96} = 38.6415896391545$$
$$x_{97} = 66.2876049907446$$
so we need to solve the equation:
$$\sin{\left(2 x \right)} + \cos{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{9 \pi}{10}$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = - \frac{\pi}{10}$$
$$x_{4} = \frac{3 \pi}{10}$$
$$x_{5} = \frac{\pi}{2}$$
$$x_{6} = \frac{7 \pi}{10}$$
Numerical solution
$$x_{1} = 53.7212343763855$$
$$x_{2} = -58.1194640914112$$
$$x_{3} = 26.0752190247953$$
$$x_{4} = 97.7035315266426$$
$$x_{5} = -34.2433599241287$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = 73.8274273593601$$
$$x_{8} = -44.2964564156161$$
$$x_{9} = -54.3495529071034$$
$$x_{10} = 16.0221225333079$$
$$x_{11} = -51.8362787842316$$
$$x_{12} = -4.08407044966673$$
$$x_{13} = -38.0132711084365$$
$$x_{14} = 21.0486707790516$$
$$x_{15} = -31.7300858012569$$
$$x_{16} = 9.73893722612836$$
$$x_{17} = 245.986704776081$$
$$x_{18} = 70.0575161750524$$
$$x_{19} = -93.3053018116169$$
$$x_{20} = 93.9336203423348$$
$$x_{21} = -78.2256570743859$$
$$x_{22} = 33.6150413934108$$
$$x_{23} = -45.553093477052$$
$$x_{24} = 56.2345084992573$$
$$x_{25} = -87.0221165044373$$
$$x_{26} = -98.3318500573605$$
$$x_{27} = 64.4026493985908$$
$$x_{28} = -65.6592864600267$$
$$x_{29} = 18.5353966561798$$
$$x_{30} = -10.3672557568463$$
$$x_{31} = -21.6769893097696$$
$$x_{32} = 58.1194640914112$$
$$x_{33} = -25.4469004940773$$
$$x_{34} = -11.6238928182822$$
$$x_{35} = 80.1106126665397$$
$$x_{36} = 49.9513231920777$$
$$x_{37} = 39.8982267005904$$
$$x_{38} = 51.8362787842316$$
$$x_{39} = 86.3937979737193$$
$$x_{40} = -55.6061899685393$$
$$x_{41} = -49.3230046613598$$
$$x_{42} = 60.0044196835651$$
$$x_{43} = 12.2522113490002$$
$$x_{44} = -80.1106126665397$$
$$x_{45} = -69.4291976443344$$
$$x_{46} = -0.314159265358979$$
$$x_{47} = 96.4468944652067$$
$$x_{48} = 2.19911485751286$$
$$x_{49} = 32.3584043319749$$
$$x_{50} = 31.101767270539$$
$$x_{51} = -81.9955682586936$$
$$x_{52} = 14.1371669411541$$
$$x_{53} = 95.8185759344887$$
$$x_{54} = -61.8893752757189$$
$$x_{55} = -24.1902634326414$$
$$x_{56} = 7.85398163397448$$
$$x_{57} = -92.0486647501809$$
$$x_{58} = 62.5176938064369$$
$$x_{59} = -56.8628270299753$$
$$x_{60} = 100.216805649514$$
$$x_{61} = 90.1637091580271$$
$$x_{62} = 5.96902604182061$$
$$x_{63} = 46.18141200777$$
$$x_{64} = -71.9424717672063$$
$$x_{65} = 52.4645973149496$$
$$x_{66} = 2363.4201532956$$
$$x_{67} = -89.5353906273091$$
$$x_{68} = -88.2787535658732$$
$$x_{69} = -95.8185759344887$$
$$x_{70} = -36.1283155162826$$
$$x_{71} = 89.5353906273091$$
$$x_{72} = -5.34070751110265$$
$$x_{73} = -17.9070781254618$$
$$x_{74} = 82.6238867894116$$
$$x_{75} = -14.1371669411541$$
$$x_{76} = 83.8805238508475$$
$$x_{77} = 77.5973385436679$$
$$x_{78} = -41.7831822927443$$
$$x_{79} = -19.1637151868977$$
$$x_{80} = -73.8274273593601$$
$$x_{81} = 29.845130209103$$
$$x_{82} = -27.9601746169492$$
$$x_{83} = -22.9336263712055$$
$$x_{84} = -1.5707963267949$$
$$x_{85} = 42.4115008234622$$
$$x_{86} = -85.7654794430014$$
$$x_{87} = -99.5884871187965$$
$$x_{88} = 76.340701482232$$
$$x_{89} = -48.0663675999238$$
$$x_{90} = 20.4203522483337$$
$$x_{91} = -29.845130209103$$
$$x_{92} = 22.3053078404875$$
$$x_{93} = -66.9159235214626$$
$$x_{94} = -75.712382951514$$
$$x_{95} = -7.85398163397448$$
$$x_{96} = 38.6415896391545$$
$$x_{97} = 66.2876049907446$$
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
$$- 3 \sin{\left(3 x \right)} + 2 \cos{\left(2 x \right)} = 0$$
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
$$- 3 \sin{\left(3 x \right)} + 2 \cos{\left(2 x \right)} = 0$$
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
$$- (4 \sin{\left(2 x \right)} + 9 \cos{\left(3 x \right)}) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
$$x_{3} = - i \log{\left(- \frac{\sqrt{- 4 \sqrt{85} + 235}}{18} + \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{4} = - i \log{\left(\frac{\sqrt{- 4 \sqrt{85} + 235}}{18} + \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{5} = - i \log{\left(- \frac{\sqrt{4 \sqrt{85} + 235}}{18} + \frac{i \left(- \sqrt{85} + 2\right)}{18} \right)}$$
$$x_{6} = - i \log{\left(\frac{\sqrt{4 \sqrt{85} + 235}}{18} + \frac{i \left(- \sqrt{85} + 2\right)}{18} \right)}$$
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
$$\left[- \operatorname{atan}{\left(\frac{2 + \sqrt{85}}{\sqrt{- 4 \sqrt{85} + 235}} \right)} + \pi, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- (4 \sin{\left(2 x \right)} + 9 \cos{\left(3 x \right)}) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
$$x_{3} = - i \log{\left(- \frac{\sqrt{- 4 \sqrt{85} + 235}}{18} + \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{4} = - i \log{\left(\frac{\sqrt{- 4 \sqrt{85} + 235}}{18} + \frac{i \left(2 + \sqrt{85}\right)}{18} \right)}$$
$$x_{5} = - i \log{\left(- \frac{\sqrt{4 \sqrt{85} + 235}}{18} + \frac{i \left(- \sqrt{85} + 2\right)}{18} \right)}$$
$$x_{6} = - i \log{\left(\frac{\sqrt{4 \sqrt{85} + 235}}{18} + \frac{i \left(- \sqrt{85} + 2\right)}{18} \right)}$$
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
$$\left[- \operatorname{atan}{\left(\frac{2 + \sqrt{85}}{\sqrt{- 4 \sqrt{85} + 235}} \right)} + \pi, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} + \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} + \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} + \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 2\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} + \cos{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 2\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x) + cos(3*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} + \cos{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} + \cos{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} + \cos{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} + \cos{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\sin{\left(2 x \right)} + \cos{\left(3 x \right)} = - \sin{\left(2 x \right)} + \cos{\left(3 x \right)}$$
- No
$$\sin{\left(2 x \right)} + \cos{\left(3 x \right)} = \sin{\left(2 x \right)} - \cos{\left(3 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(2 x \right)} + \cos{\left(3 x \right)} = - \sin{\left(2 x \right)} + \cos{\left(3 x \right)}$$
- No
$$\sin{\left(2 x \right)} + \cos{\left(3 x \right)} = \sin{\left(2 x \right)} - \cos{\left(3 x \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph