Mister Exam
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-
How to use it?
- Graphing y =:
- -x^2+2x+4
- -x^2+2x-3
- -x^2+4x-2
- (x^2-2)/x
- Derivative of:
-
cos(x^2+1)
-
Identical expressions
- cos(x^ two + one)
- co sinus of e of (x squared plus 1)
- co sinus of e of (x to the power of two plus one)
- cos(x2+1)
- cosx2+1
- cos(x²+1)
- cos(x to the power of 2+1)
- cosx^2+1
-
Similar expressions
- cos(x^2-1)
Graphing y = cos(x^2+1)
The solution
You have entered
[src]
/ 2 \ f(x) = cos\x + 1/
$$f{\left(x \right)} = \cos{\left(x^{2} + 1 \right)}$$
f = cos(x^2 + 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:
$$\cos{\left(x^{2} + 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \sqrt{-1 + \frac{\pi}{2}}$$
$$x_{2} = \sqrt{-1 + \frac{\pi}{2}}$$
$$x_{3} = - \sqrt{-1 + \frac{3 \pi}{2}}$$
$$x_{4} = \sqrt{-1 + \frac{3 \pi}{2}}$$
Numerical solution
$$x_{1} = 84.9333122453667$$
$$x_{2} = -41.9132567296254$$
$$x_{3} = -16.644246749033$$
$$x_{4} = 32.1582769960991$$
$$x_{5} = -73.640747663388$$
$$x_{6} = -57.8749390069097$$
$$x_{7} = -99.7654064438133$$
$$x_{8} = 46.1921227928991$$
$$x_{9} = -43.9261731275435$$
$$x_{10} = -5.92691450219105$$
$$x_{11} = -50.0755602454076$$
$$x_{12} = -96.9709111461837$$
$$x_{13} = 59.665695947035$$
$$x_{14} = 20.4548923061022$$
$$x_{15} = -31.8145191820136$$
$$x_{16} = 78.5135375758217$$
$$x_{17} = 12.9257575007059$$
$$x_{18} = -66.7954516746784$$
$$x_{19} = 80.0196015007748$$
$$x_{20} = -19.7516653822388$$
$$x_{21} = -33.8247028402835$$
$$x_{22} = 4.03469448592379$$
$$x_{23} = 11.5116327155433$$
$$x_{24} = -114.802229848274$$
$$x_{25} = 18.2641621107379$$
$$x_{26} = 48.5465270283997$$
$$x_{27} = -41.9507173042021$$
$$x_{28} = -69.8531616131421$$
$$x_{29} = 24.2497138652827$$
$$x_{30} = -85.2839831937226$$
$$x_{31} = -1.92675607703328$$
$$x_{32} = -4.03469448592379$$
$$x_{33} = -86.1270634089674$$
$$x_{34} = 8.15746541838705$$
$$x_{35} = 86.0358244464732$$
$$x_{36} = -46.8337748695583$$
$$x_{37} = -0.755510639762867$$
$$x_{38} = 1.92675607703328$$
$$x_{39} = -71.4758732095229$$
$$x_{40} = -27.7535762822793$$
$$x_{41} = -26.000254168923$$
$$x_{42} = 96.0922036651632$$
$$x_{43} = 10.9522288673679$$
$$x_{44} = -79.9999689480515$$
$$x_{45} = 87.178457346758$$
$$x_{46} = -45.8850506617565$$
$$x_{47} = -29.7741485963766$$
$$x_{48} = -22.0085452717717$$
$$x_{49} = -9.73748303898336$$
$$x_{50} = 58.8705995657963$$
$$x_{51} = -10.9522288673679$$
$$x_{52} = 42.2492061885877$$
$$x_{53} = 71.9359013065742$$
$$x_{54} = 94.1268403774928$$
$$x_{55} = 51.8933107486239$$
$$x_{56} = 28.6448333720713$$
$$x_{57} = -63.4672559531993$$
$$x_{58} = -83.9099513348776$$
$$x_{59} = -7.76279954301288$$
$$x_{60} = -89.8929758176897$$
$$x_{61} = 58.1727262919919$$
$$x_{62} = 70.1672750254846$$
$$x_{63} = -54.6071627589596$$
$$x_{64} = 31.1660540698697$$
$$x_{65} = -77.5877717208378$$
$$x_{66} = -71.9795601688171$$
$$x_{67} = -81.8056425222188$$
$$x_{68} = 30.4006420406246$$
$$x_{69} = -97.1651003208669$$
$$x_{70} = -91.8805216382807$$
$$x_{71} = -15.8713014152582$$
$$x_{72} = -13.7501402008191$$
$$x_{73} = 6.18626770919853$$
so we need to solve the equation:
$$\cos{\left(x^{2} + 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \sqrt{-1 + \frac{\pi}{2}}$$
$$x_{2} = \sqrt{-1 + \frac{\pi}{2}}$$
$$x_{3} = - \sqrt{-1 + \frac{3 \pi}{2}}$$
$$x_{4} = \sqrt{-1 + \frac{3 \pi}{2}}$$
Numerical solution
$$x_{1} = 84.9333122453667$$
$$x_{2} = -41.9132567296254$$
$$x_{3} = -16.644246749033$$
$$x_{4} = 32.1582769960991$$
$$x_{5} = -73.640747663388$$
$$x_{6} = -57.8749390069097$$
$$x_{7} = -99.7654064438133$$
$$x_{8} = 46.1921227928991$$
$$x_{9} = -43.9261731275435$$
$$x_{10} = -5.92691450219105$$
$$x_{11} = -50.0755602454076$$
$$x_{12} = -96.9709111461837$$
$$x_{13} = 59.665695947035$$
$$x_{14} = 20.4548923061022$$
$$x_{15} = -31.8145191820136$$
$$x_{16} = 78.5135375758217$$
$$x_{17} = 12.9257575007059$$
$$x_{18} = -66.7954516746784$$
$$x_{19} = 80.0196015007748$$
$$x_{20} = -19.7516653822388$$
$$x_{21} = -33.8247028402835$$
$$x_{22} = 4.03469448592379$$
$$x_{23} = 11.5116327155433$$
$$x_{24} = -114.802229848274$$
$$x_{25} = 18.2641621107379$$
$$x_{26} = 48.5465270283997$$
$$x_{27} = -41.9507173042021$$
$$x_{28} = -69.8531616131421$$
$$x_{29} = 24.2497138652827$$
$$x_{30} = -85.2839831937226$$
$$x_{31} = -1.92675607703328$$
$$x_{32} = -4.03469448592379$$
$$x_{33} = -86.1270634089674$$
$$x_{34} = 8.15746541838705$$
$$x_{35} = 86.0358244464732$$
$$x_{36} = -46.8337748695583$$
$$x_{37} = -0.755510639762867$$
$$x_{38} = 1.92675607703328$$
$$x_{39} = -71.4758732095229$$
$$x_{40} = -27.7535762822793$$
$$x_{41} = -26.000254168923$$
$$x_{42} = 96.0922036651632$$
$$x_{43} = 10.9522288673679$$
$$x_{44} = -79.9999689480515$$
$$x_{45} = 87.178457346758$$
$$x_{46} = -45.8850506617565$$
$$x_{47} = -29.7741485963766$$
$$x_{48} = -22.0085452717717$$
$$x_{49} = -9.73748303898336$$
$$x_{50} = 58.8705995657963$$
$$x_{51} = -10.9522288673679$$
$$x_{52} = 42.2492061885877$$
$$x_{53} = 71.9359013065742$$
$$x_{54} = 94.1268403774928$$
$$x_{55} = 51.8933107486239$$
$$x_{56} = 28.6448333720713$$
$$x_{57} = -63.4672559531993$$
$$x_{58} = -83.9099513348776$$
$$x_{59} = -7.76279954301288$$
$$x_{60} = -89.8929758176897$$
$$x_{61} = 58.1727262919919$$
$$x_{62} = 70.1672750254846$$
$$x_{63} = -54.6071627589596$$
$$x_{64} = 31.1660540698697$$
$$x_{65} = -77.5877717208378$$
$$x_{66} = -71.9795601688171$$
$$x_{67} = -81.8056425222188$$
$$x_{68} = 30.4006420406246$$
$$x_{69} = -97.1651003208669$$
$$x_{70} = -91.8805216382807$$
$$x_{71} = -15.8713014152582$$
$$x_{72} = -13.7501402008191$$
$$x_{73} = 6.18626770919853$$
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
$$- 2 \left(2 x^{2} \cos{\left(x^{2} + 1 \right)} + \sin{\left(x^{2} + 1 \right)}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 43.9619214434102$$
$$x_{2} = -75.4945028307817$$
$$x_{3} = -89.8405386523628$$
$$x_{4} = 9.73775378723$$
$$x_{5} = 10.6617320557088$$
$$x_{6} = 89.6830421633998$$
$$x_{7} = 5.65706155582907$$
$$x_{8} = 24.2497313967903$$
$$x_{9} = 25.5123743271578$$
$$x_{10} = 70.1672757491473$$
$$x_{11} = -56.9446623400667$$
$$x_{12} = -43.8545979379999$$
$$x_{13} = -76.2604673739696$$
$$x_{14} = 20.0672842611393$$
$$x_{15} = -1.01229453086059$$
$$x_{16} = 2.63173856168775$$
$$x_{17} = 98.5774666205898$$
$$x_{18} = 56.0269757933469$$
$$x_{19} = -33.8247093003621$$
$$x_{20} = 46.1581070900829$$
$$x_{21} = -41.8757660486458$$
$$x_{22} = 13.5198337885067$$
$$x_{23} = 80.0196019886973$$
$$x_{24} = 56.6404188852419$$
$$x_{25} = 1.01229453086059$$
$$x_{26} = 51.0387235155318$$
$$x_{27} = -57.8749402965492$$
$$x_{28} = 97.2458983016984$$
$$x_{29} = 18.2642031441833$$
$$x_{30} = -29.8794857718858$$
$$x_{31} = -45.679191739084$$
$$x_{32} = -91.8805219605876$$
$$x_{33} = 9.06963525509434$$
$$x_{34} = -7.76333386150031$$
$$x_{35} = 6.1873231744953$$
$$x_{36} = -48.3844761967088$$
$$x_{37} = 79.2305054751583$$
$$x_{38} = -21.8653590295624$$
$$x_{39} = -4.03849070189123$$
$$x_{40} = -50.6680588642432$$
$$x_{41} = -15.8713639465704$$
$$x_{42} = 1.96005320703295$$
$$x_{43} = 49.823982117835$$
$$x_{44} = -13.7502363642665$$
$$x_{45} = 42.2492095035932$$
$$x_{46} = -93.8929168735086$$
$$x_{47} = 86.0358248390295$$
$$x_{48} = -26.3006065129735$$
$$x_{49} = -27.7535879768054$$
$$x_{50} = 28.3693344458994$$
$$x_{51} = -9.73775378723$$
$$x_{52} = -54.6933923768852$$
$$x_{53} = -31.8145269456219$$
$$x_{54} = -75.6192403644685$$
$$x_{55} = 13.6355217026362$$
$$x_{56} = 64.0584961091231$$
$$x_{57} = -19.7516978256737$$
$$x_{58} = 3.16943334466592$$
$$x_{59} = -99.765406695581$$
$$x_{60} = -59.639364691151$$
$$x_{61} = 6.67565073578945$$
$$x_{62} = -37.3555657543961$$
$$x_{63} = 4.03849070189123$$
$$x_{64} = -51.1923750202349$$
$$x_{65} = 36.6765977670048$$
$$x_{66} = -86.0540803701783$$
$$x_{67} = -71.3438924742338$$
$$x_{68} = -69.8531623466111$$
$$x_{69} = -81.7864391605895$$
$$x_{70} = 66.4181224916093$$
$$x_{71} = 51.6809901131857$$
$$x_{72} = 21.3565640340599$$
$$x_{73} = -83.9099517580327$$
$$x_{74} = -11.7815311084128$$
$$x_{75} = -6.67565073578945$$
$$x_{76} = -37.8982715587281$$
$$x_{77} = 100.001301305042$$
$$x_{78} = -3.16943334466592$$
$$x_{79} = 106.6890857783$$
$$x_{80} = 3.62975193593284$$
$$x_{81} = -71.9577347198112$$
$$x_{82} = 82.2460932410883$$
$$x_{83} = 8.15792589208374$$
$$x_{84} = 58.1727275619276$$
$$x_{85} = 71.9359019781622$$
$$x_{86} = 94.0934585963188$$
$$x_{87} = 47.3342009703147$$
$$x_{88} = 32.1582845133909$$
$$x_{89} = -1.96005320703295$$
$$x_{90} = -18.3500053177433$$
$$x_{91} = 20.2232307965203$$
$$x_{92} = 8.53430091745704$$
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
$$\left[97.2458983016984, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -91.8805219605876\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 2 \left(2 x^{2} \cos{\left(x^{2} + 1 \right)} + \sin{\left(x^{2} + 1 \right)}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 43.9619214434102$$
$$x_{2} = -75.4945028307817$$
$$x_{3} = -89.8405386523628$$
$$x_{4} = 9.73775378723$$
$$x_{5} = 10.6617320557088$$
$$x_{6} = 89.6830421633998$$
$$x_{7} = 5.65706155582907$$
$$x_{8} = 24.2497313967903$$
$$x_{9} = 25.5123743271578$$
$$x_{10} = 70.1672757491473$$
$$x_{11} = -56.9446623400667$$
$$x_{12} = -43.8545979379999$$
$$x_{13} = -76.2604673739696$$
$$x_{14} = 20.0672842611393$$
$$x_{15} = -1.01229453086059$$
$$x_{16} = 2.63173856168775$$
$$x_{17} = 98.5774666205898$$
$$x_{18} = 56.0269757933469$$
$$x_{19} = -33.8247093003621$$
$$x_{20} = 46.1581070900829$$
$$x_{21} = -41.8757660486458$$
$$x_{22} = 13.5198337885067$$
$$x_{23} = 80.0196019886973$$
$$x_{24} = 56.6404188852419$$
$$x_{25} = 1.01229453086059$$
$$x_{26} = 51.0387235155318$$
$$x_{27} = -57.8749402965492$$
$$x_{28} = 97.2458983016984$$
$$x_{29} = 18.2642031441833$$
$$x_{30} = -29.8794857718858$$
$$x_{31} = -45.679191739084$$
$$x_{32} = -91.8805219605876$$
$$x_{33} = 9.06963525509434$$
$$x_{34} = -7.76333386150031$$
$$x_{35} = 6.1873231744953$$
$$x_{36} = -48.3844761967088$$
$$x_{37} = 79.2305054751583$$
$$x_{38} = -21.8653590295624$$
$$x_{39} = -4.03849070189123$$
$$x_{40} = -50.6680588642432$$
$$x_{41} = -15.8713639465704$$
$$x_{42} = 1.96005320703295$$
$$x_{43} = 49.823982117835$$
$$x_{44} = -13.7502363642665$$
$$x_{45} = 42.2492095035932$$
$$x_{46} = -93.8929168735086$$
$$x_{47} = 86.0358248390295$$
$$x_{48} = -26.3006065129735$$
$$x_{49} = -27.7535879768054$$
$$x_{50} = 28.3693344458994$$
$$x_{51} = -9.73775378723$$
$$x_{52} = -54.6933923768852$$
$$x_{53} = -31.8145269456219$$
$$x_{54} = -75.6192403644685$$
$$x_{55} = 13.6355217026362$$
$$x_{56} = 64.0584961091231$$
$$x_{57} = -19.7516978256737$$
$$x_{58} = 3.16943334466592$$
$$x_{59} = -99.765406695581$$
$$x_{60} = -59.639364691151$$
$$x_{61} = 6.67565073578945$$
$$x_{62} = -37.3555657543961$$
$$x_{63} = 4.03849070189123$$
$$x_{64} = -51.1923750202349$$
$$x_{65} = 36.6765977670048$$
$$x_{66} = -86.0540803701783$$
$$x_{67} = -71.3438924742338$$
$$x_{68} = -69.8531623466111$$
$$x_{69} = -81.7864391605895$$
$$x_{70} = 66.4181224916093$$
$$x_{71} = 51.6809901131857$$
$$x_{72} = 21.3565640340599$$
$$x_{73} = -83.9099517580327$$
$$x_{74} = -11.7815311084128$$
$$x_{75} = -6.67565073578945$$
$$x_{76} = -37.8982715587281$$
$$x_{77} = 100.001301305042$$
$$x_{78} = -3.16943334466592$$
$$x_{79} = 106.6890857783$$
$$x_{80} = 3.62975193593284$$
$$x_{81} = -71.9577347198112$$
$$x_{82} = 82.2460932410883$$
$$x_{83} = 8.15792589208374$$
$$x_{84} = 58.1727275619276$$
$$x_{85} = 71.9359019781622$$
$$x_{86} = 94.0934585963188$$
$$x_{87} = 47.3342009703147$$
$$x_{88} = 32.1582845133909$$
$$x_{89} = -1.96005320703295$$
$$x_{90} = -18.3500053177433$$
$$x_{91} = 20.2232307965203$$
$$x_{92} = 8.53430091745704$$
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
$$\left[97.2458983016984, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -91.8805219605876\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cos{\left(x^{2} + 1 \right)} = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty} \cos{\left(x^{2} + 1 \right)} = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty} \cos{\left(x^{2} + 1 \right)} = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty} \cos{\left(x^{2} + 1 \right)} = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x^2 + 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x^{2} + 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x^{2} + 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x^{2} + 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x^{2} + 1 \right)}}{x}\right) = 0$$
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:
$$\cos{\left(x^{2} + 1 \right)} = \cos{\left(x^{2} + 1 \right)}$$
- Yes
$$\cos{\left(x^{2} + 1 \right)} = - \cos{\left(x^{2} + 1 \right)}$$
- No
so, the function
is
even
So, check:
$$\cos{\left(x^{2} + 1 \right)} = \cos{\left(x^{2} + 1 \right)}$$
- Yes
$$\cos{\left(x^{2} + 1 \right)} = - \cos{\left(x^{2} + 1 \right)}$$
- No
so, the function
is
even