Mister Exam
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-
How to use it?
- Graphing y =:
- ((x^2+x-6)(x^2-2x-3))/(x^2-9)
- x/(x²-1)
-
|x+7|
- (x+5)^2
-
Identical expressions
- 3cos7x-11sin2x
- 3 co sinus of e of 7x minus 11 sinus of 2x
-
Similar expressions
- 3cos7x+11sin2x
Graphing y = 3cos7x-11sin2x
The solution
You have entered
[src]
f(x) = 3*cos(7*x) - 11*sin(2*x)
$$f{\left(x \right)} = - 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}$$
f = -11*sin(2*x) + 3*cos(7*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:
$$- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 62.934949974151$$
$$x_{2} = -34.6606160918429$$
$$x_{3} = 44.0853940526123$$
$$x_{4} = 3.03849575123464$$
$$x_{5} = -1.5707963267949$$
$$x_{6} = 18.9526528238939$$
$$x_{7} = -39.2699081698724$$
$$x_{8} = -98.9601685880785$$
$$x_{9} = -53.5101720133816$$
$$x_{10} = -92.6769832808989$$
$$x_{11} = -51.8362787842316$$
$$x_{12} = 2159.84494934298$$
$$x_{13} = -9.52787486312453$$
$$x_{14} = 21.8880516727734$$
$$x_{15} = -72.3597279349204$$
$$x_{16} = 51.8362787842316$$
$$x_{17} = 97.2862753589284$$
$$x_{18} = -97.4924691636387$$
$$x_{19} = -87.8614973981591$$
$$x_{20} = -22.0942454774837$$
$$x_{21} = 10.9955742875643$$
$$x_{22} = 7.85398163397448$$
$$x_{23} = 149.225651045515$$
$$x_{24} = 29.845130209103$$
$$x_{25} = 94.3508765100489$$
$$x_{26} = 67.5442420521806$$
$$x_{27} = -6.18008840482443$$
$$x_{28} = -15.8110601703041$$
$$x_{29} = 88.0676912028694$$
$$x_{30} = -20.4203522483337$$
$$x_{31} = -45.553093477052$$
$$x_{32} = 37.8022087454327$$
$$x_{33} = 59.5871635158509$$
$$x_{34} = -66.0765426277408$$
$$x_{35} = 14.1371669411541$$
$$x_{36} = 53.3039782086713$$
$$x_{37} = -25.0296443263632$$
$$x_{38} = -58.1194640914112$$
$$x_{39} = -78.6429132421$$
$$x_{40} = 64.4026493985908$$
$$x_{41} = -56.4455708622611$$
$$x_{42} = 73.8274273593601$$
$$x_{43} = -80.1106126665397$$
$$x_{44} = 56.6517646669714$$
$$x_{45} = 36.1283155162826$$
$$x_{46} = -7.85398163397448$$
$$x_{47} = -59.7933573205612$$
$$x_{48} = 81.7845058956898$$
$$x_{49} = 12.6694675167143$$
$$x_{50} = 28.171236979953$$
$$x_{51} = 92.6769832808989$$
$$x_{52} = 50.3685793597918$$
$$x_{53} = 100.634061817229$$
$$x_{54} = -31.3128296335428$$
$$x_{55} = -28.3774307846633$$
$$x_{56} = 23.5619449019235$$
$$x_{57} = -95.8185759344887$$
$$x_{58} = 126695.719330296$$
$$x_{59} = -73.8274273593601$$
$$x_{60} = 86.3937979737193$$
$$x_{61} = 78.4367194373897$$
$$x_{62} = -40.9438013990225$$
$$x_{63} = -246.615023306799$$
$$x_{64} = -50.1623855550815$$
$$x_{65} = 42.4115008234622$$
$$x_{66} = 15.6048663655938$$
$$x_{67} = -89.5353906273091$$
$$x_{68} = -69.0119414766203$$
$$x_{69} = 0.103096902355152$$
$$x_{70} = -10.9955742875643$$
$$x_{71} = -94.1446827053386$$
$$x_{72} = -26.7035375555132$$
$$x_{73} = -84.9260985492796$$
$$x_{74} = 58.1194640914112$$
$$x_{75} = 72.1535341302101$$
$$x_{76} = 65.8703488230305$$
$$x_{77} = 34.4544222871326$$
$$x_{78} = 9.32168105841423$$
$$x_{79} = -43.879200247902$$
$$x_{80} = -29.845130209103$$
$$x_{81} = -83.2522053201295$$
$$x_{82} = -14.1371669411541$$
$$x_{83} = 6.38628220953474$$
$$x_{84} = 48.6946861306418$$
$$x_{85} = -64.4026493985908$$
$$x_{86} = 80.1106126665397$$
$$x_{87} = 4.71238898038469$$
$$x_{88} = -36.1283155162826$$
so we need to solve the equation:
$$- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 62.934949974151$$
$$x_{2} = -34.6606160918429$$
$$x_{3} = 44.0853940526123$$
$$x_{4} = 3.03849575123464$$
$$x_{5} = -1.5707963267949$$
$$x_{6} = 18.9526528238939$$
$$x_{7} = -39.2699081698724$$
$$x_{8} = -98.9601685880785$$
$$x_{9} = -53.5101720133816$$
$$x_{10} = -92.6769832808989$$
$$x_{11} = -51.8362787842316$$
$$x_{12} = 2159.84494934298$$
$$x_{13} = -9.52787486312453$$
$$x_{14} = 21.8880516727734$$
$$x_{15} = -72.3597279349204$$
$$x_{16} = 51.8362787842316$$
$$x_{17} = 97.2862753589284$$
$$x_{18} = -97.4924691636387$$
$$x_{19} = -87.8614973981591$$
$$x_{20} = -22.0942454774837$$
$$x_{21} = 10.9955742875643$$
$$x_{22} = 7.85398163397448$$
$$x_{23} = 149.225651045515$$
$$x_{24} = 29.845130209103$$
$$x_{25} = 94.3508765100489$$
$$x_{26} = 67.5442420521806$$
$$x_{27} = -6.18008840482443$$
$$x_{28} = -15.8110601703041$$
$$x_{29} = 88.0676912028694$$
$$x_{30} = -20.4203522483337$$
$$x_{31} = -45.553093477052$$
$$x_{32} = 37.8022087454327$$
$$x_{33} = 59.5871635158509$$
$$x_{34} = -66.0765426277408$$
$$x_{35} = 14.1371669411541$$
$$x_{36} = 53.3039782086713$$
$$x_{37} = -25.0296443263632$$
$$x_{38} = -58.1194640914112$$
$$x_{39} = -78.6429132421$$
$$x_{40} = 64.4026493985908$$
$$x_{41} = -56.4455708622611$$
$$x_{42} = 73.8274273593601$$
$$x_{43} = -80.1106126665397$$
$$x_{44} = 56.6517646669714$$
$$x_{45} = 36.1283155162826$$
$$x_{46} = -7.85398163397448$$
$$x_{47} = -59.7933573205612$$
$$x_{48} = 81.7845058956898$$
$$x_{49} = 12.6694675167143$$
$$x_{50} = 28.171236979953$$
$$x_{51} = 92.6769832808989$$
$$x_{52} = 50.3685793597918$$
$$x_{53} = 100.634061817229$$
$$x_{54} = -31.3128296335428$$
$$x_{55} = -28.3774307846633$$
$$x_{56} = 23.5619449019235$$
$$x_{57} = -95.8185759344887$$
$$x_{58} = 126695.719330296$$
$$x_{59} = -73.8274273593601$$
$$x_{60} = 86.3937979737193$$
$$x_{61} = 78.4367194373897$$
$$x_{62} = -40.9438013990225$$
$$x_{63} = -246.615023306799$$
$$x_{64} = -50.1623855550815$$
$$x_{65} = 42.4115008234622$$
$$x_{66} = 15.6048663655938$$
$$x_{67} = -89.5353906273091$$
$$x_{68} = -69.0119414766203$$
$$x_{69} = 0.103096902355152$$
$$x_{70} = -10.9955742875643$$
$$x_{71} = -94.1446827053386$$
$$x_{72} = -26.7035375555132$$
$$x_{73} = -84.9260985492796$$
$$x_{74} = 58.1194640914112$$
$$x_{75} = 72.1535341302101$$
$$x_{76} = 65.8703488230305$$
$$x_{77} = 34.4544222871326$$
$$x_{78} = 9.32168105841423$$
$$x_{79} = -43.879200247902$$
$$x_{80} = -29.845130209103$$
$$x_{81} = -83.2522053201295$$
$$x_{82} = -14.1371669411541$$
$$x_{83} = 6.38628220953474$$
$$x_{84} = 48.6946861306418$$
$$x_{85} = -64.4026493985908$$
$$x_{86} = 80.1106126665397$$
$$x_{87} = 4.71238898038469$$
$$x_{88} = -36.1283155162826$$
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
$$- 21 \sin{\left(7 x \right)} - 22 \cos{\left(2 x \right)} = 0$$
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
$$- 21 \sin{\left(7 x \right)} - 22 \cos{\left(2 x \right)} = 0$$
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
$$44 \sin{\left(2 x \right)} - 147 \cos{\left(7 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
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
$$\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$44 \sin{\left(2 x \right)} - 147 \cos{\left(7 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
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
$$\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -14, 14\right\rangle$$
$$\lim_{x \to \infty}\left(- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -14, 14\right\rangle$$
$$\lim_{x \to -\infty}\left(- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -14, 14\right\rangle$$
$$\lim_{x \to \infty}\left(- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}\right) = \left\langle -14, 14\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -14, 14\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*cos(7*x) - 11*sin(2*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}}{x}\right) = 0$$
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:
$$- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)} = 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}$$
- No
$$- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)} = - 11 \sin{\left(2 x \right)} - 3 \cos{\left(7 x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)} = 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)}$$
- No
$$- 11 \sin{\left(2 x \right)} + 3 \cos{\left(7 x \right)} = - 11 \sin{\left(2 x \right)} - 3 \cos{\left(7 x \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph