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Plotting a function graph/ cos(3*x^2)

Graphing y = cos(3*x^2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /   2\
f(x) = cos\3*x /
f(x)=cos⁡(3x2)f{\left(x \right)} = \cos{\left(3 x^{2} \right)} 
f = cos(3*x^2)
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⁡(3x2)=0\cos{\left(3 x^{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−2π2x_{1} = - \frac{\sqrt{2} \sqrt{\pi}}{2}
x2=2π2x_{2} = \frac{\sqrt{2} \sqrt{\pi}}{2}
x3=−6π6x_{3} = - \frac{\sqrt{6} \sqrt{\pi}}{6}
x4=6π6x_{4} = \frac{\sqrt{6} \sqrt{\pi}}{6}
Numerical solution
x1=26.1999617329753x_{1} = 26.1999617329753
x2=−30.60585253747x_{2} = -30.60585253747
x3=−89.9800414929694x_{3} = -89.9800414929694
x4=−13.7102866651208x_{4} = -13.7102866651208
x5=93.948414357351x_{5} = 93.948414357351
x6=60.363265976822x_{6} = 60.363265976822
x7=60.1024787579736x_{7} = 60.1024787579736
x8=−57.3656543990854x_{8} = -57.3656543990854
x9=67.9068068509646x_{9} = 67.9068068509646
x10=−49.7393628296741x_{10} = -49.7393628296741
x11=−77.7223203081172x_{11} = -77.7223203081172
x12=−19.750470572134x_{12} = -19.750470572134
x13=52.1846455567799x_{13} = 52.1846455567799
x14=98.146047677336x_{14} = 98.146047677336
x15=−58.6563513564935x_{15} = -58.6563513564935
x16=−51.7109170966877x_{16} = -51.7109170966877
x17=−18.0320503450535x_{17} = -18.0320503450535
x18=20.2995624680893x_{18} = 20.2995624680893
x19=−39.7717475236186x_{19} = -39.7717475236186
x20=39.0141210418004x_{20} = 39.0141210418004
x21=−43.5064314071822x_{21} = -43.5064314071822
x22=−64.4005116556858x_{22} = -64.4005116556858
x23=−47.8398395496908x_{23} = -47.8398395496908
x24=99.0065284613886x_{24} = 99.0065284613886
x25=−81.8373505111073x_{25} = -81.8373505111073
x26=−1.91446896793552x_{26} = -1.91446896793552
x27=48.1017981926755x_{27} = 48.1017981926755
x28=94.0709461331545x_{28} = 94.0709461331545
x29=41.0033616725036x_{29} = 41.0033616725036
x30=13.0047070139335x_{30} = 13.0047070139335
x31=4.40149459810381x_{31} = 4.40149459810381
x32=−25.9186656503111x_{32} = -25.9186656503111
x33=−33.1358813437885x_{33} = -33.1358813437885
x34=−36.3893020472789x_{34} = -36.3893020472789
x35=58.6384955413594x_{35} = 58.6384955413594
x36=72.6742103463123x_{36} = 72.6742103463123
x37=77.9980395502133x_{37} = 77.9980395502133
x38=−35.7359400839582x_{38} = -35.7359400839582
x39=−63.1277298901097x_{39} = -63.1277298901097
x40=−75.7641958769718x_{40} = -75.7641958769718
x41=42.3476776981238x_{41} = 42.3476776981238
x42=6.09717254696628x_{42} = 6.09717254696628
x43=−3.61800627279134x_{43} = -3.61800627279134
x44=−291.991711229956x_{44} = -291.991711229956
x45=35.4564614733551x_{45} = 35.4564614733551
x46=22.1969918646198x_{46} = 22.1969918646198
x47=2.1708037636748x_{47} = 2.1708037636748
x48=−41.8751990977799x_{48} = -41.8751990977799
x49=71.4462347734842x_{49} = 71.4462347734842
x50=−49.7498885652336x_{50} = -49.7498885652336
x51=92.2726378186076x_{51} = 92.2726378186076
x52=−99.7914080124338x_{52} = -99.7914080124338
x53=−16.0012437412711x_{53} = -16.0012437412711
x54=−10.0003683001815x_{54} = -10.0003683001815
x55=−50.3150084966481x_{55} = -50.3150084966481
x56=−73.8463353718807x_{56} = -73.8463353718807
x57=32.4008571657385x_{57} = 32.4008571657385
x58=45.4027353749805x_{58} = 45.4027353749805
x59=24.2703239069462x_{59} = 24.2703239069462
x60=54.2218325491167x_{60} = 54.2218325491167
x61=18.0610641214116x_{61} = 18.0610641214116
x62=45.6672084017408x_{62} = 45.6672084017408
x63=32.0433643512704x_{63} = 32.0433643512704
x64=64.1561374692884x_{64} = 64.1561374692884
x65=−85.7123936220412x_{65} = -85.7123936220412
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(3*x^2).
cos⁡(3⋅02)\cos{\left(3 \cdot 0^{2} \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−6xsin⁡(3x2)=0- 6 x \sin{\left(3 x^{2} \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−πx_{2} = - \sqrt{\pi}
x3=πx_{3} = \sqrt{\pi}
x4=−3π3x_{4} = - \frac{\sqrt{3} \sqrt{\pi}}{3}
x5=3π3x_{5} = \frac{\sqrt{3} \sqrt{\pi}}{3}
x6=−6π3x_{6} = - \frac{\sqrt{6} \sqrt{\pi}}{3}
x7=6π3x_{7} = \frac{\sqrt{6} \sqrt{\pi}}{3}
The values of the extrema at the points:
(0, 1)

    ____     
(-\/ pi, -1)

   ____     
(\/ pi, -1)

    ___   ____      
 -\/ 3 *\/ pi       
(--------------, -1)
       3            

   ___   ____     
 \/ 3 *\/ pi      
(------------, -1)
      3           

    ___   ____     
 -\/ 6 *\/ pi      
(--------------, 1)
       3           

   ___   ____    
 \/ 6 *\/ pi     
(------------, 1)
      3


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=−πx_{1} = - \sqrt{\pi}
x2=πx_{2} = \sqrt{\pi}
x3=−3π3x_{3} = - \frac{\sqrt{3} \sqrt{\pi}}{3}
x4=3π3x_{4} = \frac{\sqrt{3} \sqrt{\pi}}{3}
Maxima of the function at points:
x4=−6π3x_{4} = - \frac{\sqrt{6} \sqrt{\pi}}{3}
x4=6π3x_{4} = \frac{\sqrt{6} \sqrt{\pi}}{3}
Decreasing at intervals
[π,∞)\left[\sqrt{\pi}, \infty\right)
Increasing at intervals
(−∞,−π]\left(-\infty, - \sqrt{\pi}\right]
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
−6(6x2cos⁡(3x2)+sin⁡(3x2))=0- 6 \left(6 x^{2} \cos{\left(3 x^{2} \right)} + \sin{\left(3 x^{2} \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−69.8678137362531x_{1} = -69.8678137362531
x2=2.40191871479273x_{2} = 2.40191871479273
x3=−85.7551445152571x_{3} = -85.7551445152571
x4=76.4316135434824x_{4} = 76.4316135434824
x5=75.6881378653139x_{5} = 75.6881378653139
x6=54.8172727490775x_{6} = 54.8172727490775
x7=52.1846457522451x_{7} = 52.1846457522451
x8=22.1969944045103x_{8} = 22.1969944045103
x9=98.1727185535889x_{9} = 98.1727185535889
x10=−101.130879998117x_{10} = -101.130879998117
x11=−6.97823953408788x_{11} = -6.97823953408788
x12=61.6676407995246x_{12} = 61.6676407995246
x13=−12.2158103839531x_{13} = -12.2158103839531
x14=−72.6453857432282x_{14} = -72.6453857432282
x15=82.0737367025503x_{15} = 82.0737367025503
x16=24.7826845646599x_{16} = 24.7826845646599
x17=2.6105429490573x_{17} = 2.6105429490573
x18=43.6626060960023x_{18} = 43.6626060960023
x19=−66.8969262930126x_{19} = -66.8969262930126
x20=5.46324002172655x_{20} = 5.46324002172655
x21=−89.0793997015776x_{21} = -89.0793997015776
x22=−8.83271926754254x_{22} = -8.83271926754254
x23=−45.4142665251604x_{23} = -45.4142665251604
x24=14.6696746701966x_{24} = 14.6696746701966
x25=−41.7499743214192x_{25} = -41.7499743214192
x26=6.18256937073897x_{26} = 6.18256937073897
x27=−11.5096007578645x_{27} = -11.5096007578645
x28=−22.8479339480409x_{28} = -22.8479339480409
x29=32.8980048102268x_{29} = 32.8980048102268
x30=34.0400372847106x_{30} = 34.0400372847106
x31=22.5480491479443x_{31} = 22.5480491479443
x32=−29.8260688357005x_{32} = -29.8260688357005
x33=−7.55468331206371x_{33} = -7.55468331206371
x34=−28.9169035525869x_{34} = -28.9169035525869
x35=0x_{35} = 0
x36=−1.26699675319841x_{36} = -1.26699675319841
x37=0.782431509921076x_{37} = 0.782431509921076
x38=−79.8686665208779x_{38} = -79.8686665208779
x39=−79.1839336396597x_{39} = -79.1839336396597
x40=2.17350961003933x_{40} = 2.17350961003933
x41=−49.7498887908243x_{41} = -49.7498887908243
x42=32.3846939231418x_{42} = 32.3846939231418
x43=−97.9217257983412x_{43} = -97.9217257983412
x44=−23.7689540620825x_{44} = -23.7689540620825
x45=−96.0592589414969x_{45} = -96.0592589414969
x46=24.2487426370705x_{46} = 24.2487426370705
x47=−8.15462322537082x_{47} = -8.15462322537082
x48=−43.6506125221568x_{48} = -43.6506125221568
x49=64.1642983755252x_{49} = 64.1642983755252
x50=1.26699675319841x_{50} = 1.26699675319841
x51=5.16774784826217x_{51} = 5.16774784826217
x52=−4.40182028923456x_{52} = -4.40182028923456
x53=20.2995657888476x_{53} = 20.2995657888476
x54=3.31671887860803x_{54} = 3.31671887860803
x55=53.5318234159474x_{55} = 53.5318234159474
x56=56.269012907544x_{56} = 56.269012907544
x57=−19.7504741776293x_{57} = -19.7504741776293
x58=−5.46324002172655x_{58} = -5.46324002172655
x59=−3.76046478766038x_{59} = -3.76046478766038
x60=94.1432763442015x_{60} = 94.1432763442015
x61=−4.15715938453264x_{61} = -4.15715938453264
x62=−93.8870885345791x_{62} = -93.8870885345791
x63=−25.8984577604517x_{63} = -25.8984577604517
x64=−12.8426613732715x_{64} = -12.8426613732715
x65=−1.91840470339041x_{65} = -1.91840470339041
x66=85.9259352130434x_{66} = 85.9259352130434
x67=6.09729508962876x_{67} = 6.09729508962876
x68=10.9978078035033x_{68} = 10.9978078035033
x69=6.59241812112581x_{69} = 6.59241812112581
x70=38.7447767660209x_{70} = 38.7447767660209

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
[85.9259352130434,∞)\left[85.9259352130434, \infty\right)
Convex at the intervals
(−∞,−96.0592589414969]\left(-\infty, -96.0592589414969\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⁡(3x2)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(3 x^{2} \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⁡(3x2)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(3 x^{2} \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(3*x^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(3x2)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x^{2} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(3x2)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(3 x^{2} \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⁡(3x2)=cos⁡(3x2)\cos{\left(3 x^{2} \right)} = \cos{\left(3 x^{2} \right)}
- Yes
cos⁡(3x2)=−cos⁡(3x2)\cos{\left(3 x^{2} \right)} = - \cos{\left(3 x^{2} \right)}
- No
so, the function
is
even
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