Mister Exam

Other calculators

Plotting a function graph/ cos(x^2+1)

Graphing y = cos(x^2+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          / 2    \
f(x) = cos\x  + 1/
f(x)=cos⁡(x2+1)f{\left(x \right)} = \cos{\left(x^{2} + 1 \right)} 
f = cos(x^2 + 1)
The graph of the function
02468-8-6-4-2-10102-2
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⁡(x2+1)=0\cos{\left(x^{2} + 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−−1+π2x_{1} = - \sqrt{-1 + \frac{\pi}{2}}
x2=−1+π2x_{2} = \sqrt{-1 + \frac{\pi}{2}}
x3=−−1+3π2x_{3} = - \sqrt{-1 + \frac{3 \pi}{2}}
x4=−1+3π2x_{4} = \sqrt{-1 + \frac{3 \pi}{2}}
Numerical solution
x1=84.9333122453667x_{1} = 84.9333122453667
x2=−41.9132567296254x_{2} = -41.9132567296254
x3=−16.644246749033x_{3} = -16.644246749033
x4=32.1582769960991x_{4} = 32.1582769960991
x5=−73.640747663388x_{5} = -73.640747663388
x6=−57.8749390069097x_{6} = -57.8749390069097
x7=−99.7654064438133x_{7} = -99.7654064438133
x8=46.1921227928991x_{8} = 46.1921227928991
x9=−43.9261731275435x_{9} = -43.9261731275435
x10=−5.92691450219105x_{10} = -5.92691450219105
x11=−50.0755602454076x_{11} = -50.0755602454076
x12=−96.9709111461837x_{12} = -96.9709111461837
x13=59.665695947035x_{13} = 59.665695947035
x14=20.4548923061022x_{14} = 20.4548923061022
x15=−31.8145191820136x_{15} = -31.8145191820136
x16=78.5135375758217x_{16} = 78.5135375758217
x17=12.9257575007059x_{17} = 12.9257575007059
x18=−66.7954516746784x_{18} = -66.7954516746784
x19=80.0196015007748x_{19} = 80.0196015007748
x20=−19.7516653822388x_{20} = -19.7516653822388
x21=−33.8247028402835x_{21} = -33.8247028402835
x22=4.03469448592379x_{22} = 4.03469448592379
x23=11.5116327155433x_{23} = 11.5116327155433
x24=−114.802229848274x_{24} = -114.802229848274
x25=18.2641621107379x_{25} = 18.2641621107379
x26=48.5465270283997x_{26} = 48.5465270283997
x27=−41.9507173042021x_{27} = -41.9507173042021
x28=−69.8531616131421x_{28} = -69.8531616131421
x29=24.2497138652827x_{29} = 24.2497138652827
x30=−85.2839831937226x_{30} = -85.2839831937226
x31=−1.92675607703328x_{31} = -1.92675607703328
x32=−4.03469448592379x_{32} = -4.03469448592379
x33=−86.1270634089674x_{33} = -86.1270634089674
x34=8.15746541838705x_{34} = 8.15746541838705
x35=86.0358244464732x_{35} = 86.0358244464732
x36=−46.8337748695583x_{36} = -46.8337748695583
x37=−0.755510639762867x_{37} = -0.755510639762867
x38=1.92675607703328x_{38} = 1.92675607703328
x39=−71.4758732095229x_{39} = -71.4758732095229
x40=−27.7535762822793x_{40} = -27.7535762822793
x41=−26.000254168923x_{41} = -26.000254168923
x42=96.0922036651632x_{42} = 96.0922036651632
x43=10.9522288673679x_{43} = 10.9522288673679
x44=−79.9999689480515x_{44} = -79.9999689480515
x45=87.178457346758x_{45} = 87.178457346758
x46=−45.8850506617565x_{46} = -45.8850506617565
x47=−29.7741485963766x_{47} = -29.7741485963766
x48=−22.0085452717717x_{48} = -22.0085452717717
x49=−9.73748303898336x_{49} = -9.73748303898336
x50=58.8705995657963x_{50} = 58.8705995657963
x51=−10.9522288673679x_{51} = -10.9522288673679
x52=42.2492061885877x_{52} = 42.2492061885877
x53=71.9359013065742x_{53} = 71.9359013065742
x54=94.1268403774928x_{54} = 94.1268403774928
x55=51.8933107486239x_{55} = 51.8933107486239
x56=28.6448333720713x_{56} = 28.6448333720713
x57=−63.4672559531993x_{57} = -63.4672559531993
x58=−83.9099513348776x_{58} = -83.9099513348776
x59=−7.76279954301288x_{59} = -7.76279954301288
x60=−89.8929758176897x_{60} = -89.8929758176897
x61=58.1727262919919x_{61} = 58.1727262919919
x62=70.1672750254846x_{62} = 70.1672750254846
x63=−54.6071627589596x_{63} = -54.6071627589596
x64=31.1660540698697x_{64} = 31.1660540698697
x65=−77.5877717208378x_{65} = -77.5877717208378
x66=−71.9795601688171x_{66} = -71.9795601688171
x67=−81.8056425222188x_{67} = -81.8056425222188
x68=30.4006420406246x_{68} = 30.4006420406246
x69=−97.1651003208669x_{69} = -97.1651003208669
x70=−91.8805216382807x_{70} = -91.8805216382807
x71=−15.8713014152582x_{71} = -15.8713014152582
x72=−13.7501402008191x_{72} = -13.7501402008191
x73=6.18626770919853x_{73} = 6.18626770919853
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x^2 + 1).
cos⁡(02+1)\cos{\left(0^{2} + 1 \right)}
The result:
f(0)=cos⁡(1)f{\left(0 \right)} = \cos{\left(1 \right)}
The point:
(0, cos(1))
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−2(2x2cos⁡(x2+1)+sin⁡(x2+1))=0- 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
x1=43.9619214434102x_{1} = 43.9619214434102
x2=−75.4945028307817x_{2} = -75.4945028307817
x3=−89.8405386523628x_{3} = -89.8405386523628
x4=9.73775378723x_{4} = 9.73775378723
x5=10.6617320557088x_{5} = 10.6617320557088
x6=89.6830421633998x_{6} = 89.6830421633998
x7=5.65706155582907x_{7} = 5.65706155582907
x8=24.2497313967903x_{8} = 24.2497313967903
x9=25.5123743271578x_{9} = 25.5123743271578
x10=70.1672757491473x_{10} = 70.1672757491473
x11=−56.9446623400667x_{11} = -56.9446623400667
x12=−43.8545979379999x_{12} = -43.8545979379999
x13=−76.2604673739696x_{13} = -76.2604673739696
x14=20.0672842611393x_{14} = 20.0672842611393
x15=−1.01229453086059x_{15} = -1.01229453086059
x16=2.63173856168775x_{16} = 2.63173856168775
x17=98.5774666205898x_{17} = 98.5774666205898
x18=56.0269757933469x_{18} = 56.0269757933469
x19=−33.8247093003621x_{19} = -33.8247093003621
x20=46.1581070900829x_{20} = 46.1581070900829
x21=−41.8757660486458x_{21} = -41.8757660486458
x22=13.5198337885067x_{22} = 13.5198337885067
x23=80.0196019886973x_{23} = 80.0196019886973
x24=56.6404188852419x_{24} = 56.6404188852419
x25=1.01229453086059x_{25} = 1.01229453086059
x26=51.0387235155318x_{26} = 51.0387235155318
x27=−57.8749402965492x_{27} = -57.8749402965492
x28=97.2458983016984x_{28} = 97.2458983016984
x29=18.2642031441833x_{29} = 18.2642031441833
x30=−29.8794857718858x_{30} = -29.8794857718858
x31=−45.679191739084x_{31} = -45.679191739084
x32=−91.8805219605876x_{32} = -91.8805219605876
x33=9.06963525509434x_{33} = 9.06963525509434
x34=−7.76333386150031x_{34} = -7.76333386150031
x35=6.1873231744953x_{35} = 6.1873231744953
x36=−48.3844761967088x_{36} = -48.3844761967088
x37=79.2305054751583x_{37} = 79.2305054751583
x38=−21.8653590295624x_{38} = -21.8653590295624
x39=−4.03849070189123x_{39} = -4.03849070189123
x40=−50.6680588642432x_{40} = -50.6680588642432
x41=−15.8713639465704x_{41} = -15.8713639465704
x42=1.96005320703295x_{42} = 1.96005320703295
x43=49.823982117835x_{43} = 49.823982117835
x44=−13.7502363642665x_{44} = -13.7502363642665
x45=42.2492095035932x_{45} = 42.2492095035932
x46=−93.8929168735086x_{46} = -93.8929168735086
x47=86.0358248390295x_{47} = 86.0358248390295
x48=−26.3006065129735x_{48} = -26.3006065129735
x49=−27.7535879768054x_{49} = -27.7535879768054
x50=28.3693344458994x_{50} = 28.3693344458994
x51=−9.73775378723x_{51} = -9.73775378723
x52=−54.6933923768852x_{52} = -54.6933923768852
x53=−31.8145269456219x_{53} = -31.8145269456219
x54=−75.6192403644685x_{54} = -75.6192403644685
x55=13.6355217026362x_{55} = 13.6355217026362
x56=64.0584961091231x_{56} = 64.0584961091231
x57=−19.7516978256737x_{57} = -19.7516978256737
x58=3.16943334466592x_{58} = 3.16943334466592
x59=−99.765406695581x_{59} = -99.765406695581
x60=−59.639364691151x_{60} = -59.639364691151
x61=6.67565073578945x_{61} = 6.67565073578945
x62=−37.3555657543961x_{62} = -37.3555657543961
x63=4.03849070189123x_{63} = 4.03849070189123
x64=−51.1923750202349x_{64} = -51.1923750202349
x65=36.6765977670048x_{65} = 36.6765977670048
x66=−86.0540803701783x_{66} = -86.0540803701783
x67=−71.3438924742338x_{67} = -71.3438924742338
x68=−69.8531623466111x_{68} = -69.8531623466111
x69=−81.7864391605895x_{69} = -81.7864391605895
x70=66.4181224916093x_{70} = 66.4181224916093
x71=51.6809901131857x_{71} = 51.6809901131857
x72=21.3565640340599x_{72} = 21.3565640340599
x73=−83.9099517580327x_{73} = -83.9099517580327
x74=−11.7815311084128x_{74} = -11.7815311084128
x75=−6.67565073578945x_{75} = -6.67565073578945
x76=−37.8982715587281x_{76} = -37.8982715587281
x77=100.001301305042x_{77} = 100.001301305042
x78=−3.16943334466592x_{78} = -3.16943334466592
x79=106.6890857783x_{79} = 106.6890857783
x80=3.62975193593284x_{80} = 3.62975193593284
x81=−71.9577347198112x_{81} = -71.9577347198112
x82=82.2460932410883x_{82} = 82.2460932410883
x83=8.15792589208374x_{83} = 8.15792589208374
x84=58.1727275619276x_{84} = 58.1727275619276
x85=71.9359019781622x_{85} = 71.9359019781622
x86=94.0934585963188x_{86} = 94.0934585963188
x87=47.3342009703147x_{87} = 47.3342009703147
x88=32.1582845133909x_{88} = 32.1582845133909
x89=−1.96005320703295x_{89} = -1.96005320703295
x90=−18.3500053177433x_{90} = -18.3500053177433
x91=20.2232307965203x_{91} = 20.2232307965203
x92=8.53430091745704x_{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
[97.2458983016984,∞)\left[97.2458983016984, \infty\right)
Convex at the intervals
(−∞,−91.8805219605876]\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→−∞cos⁡(x2+1)=⟨−1,1⟩\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=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(x2+1)=⟨−1,1⟩\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=⟨−1,1⟩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→−∞(cos⁡(x2+1)x)=0\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→∞(cos⁡(x2+1)x)=0\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⁡(x2+1)=cos⁡(x2+1)\cos{\left(x^{2} + 1 \right)} = \cos{\left(x^{2} + 1 \right)}
- Yes
cos⁡(x2+1)=−cos⁡(x2+1)\cos{\left(x^{2} + 1 \right)} = - \cos{\left(x^{2} + 1 \right)}
- No
so, the function
is
even
    © Mister Exam – Calculator