Mister Exam

Other calculators

Plotting a function graph/ cos(5x)*cos(x)

Graphing y = cos(5x)*cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = cos(5*x)*cos(x)
f(x)=cos⁡(x)cos⁡(5x)f{\left(x \right)} = \cos{\left(x \right)} \cos{\left(5 x \right)} 
f = cos(x)*cos(5*x)
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⁡(x)cos⁡(5x)=0\cos{\left(x \right)} \cos{\left(5 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π2x_{1} = - \frac{\pi}{2}
x2=−π10x_{2} = - \frac{\pi}{10}
x3=π10x_{3} = \frac{\pi}{10}
x4=π2x_{4} = \frac{\pi}{2}
Numerical solution
x1=64.4026493525391x_{1} = 64.4026493525391
x2=93.9336203423348x_{2} = 93.9336203423348
x3=−17.9070781254618x_{3} = -17.9070781254618
x4=68.1725605828985x_{4} = 68.1725605828985
x5=29.8451302545263x_{5} = 29.8451302545263
x6=−83.8805238508475x_{6} = -83.8805238508475
x7=−65.6592864600267x_{7} = -65.6592864600267
x8=−85.7654794430014x_{8} = -85.7654794430014
x9=4.08407044966673x_{9} = 4.08407044966673
x10=61.8893752757189x_{10} = 61.8893752757189
x11=−38.0132711084365x_{11} = -38.0132711084365
x12=34.2433599241287x_{12} = 34.2433599241287
x13=98.3318500573605x_{13} = 98.3318500573605
x14=83.8805238508475x_{14} = 83.8805238508475
x15=−23.5619448943193x_{15} = -23.5619448943193
x16=−71.9424717672063x_{16} = -71.9424717672063
x17=−5.96902604182061x_{17} = -5.96902604182061
x18=90.1637091580271x_{18} = 90.1637091580271
x19=−97.7035315266426x_{19} = -97.7035315266426
x20=86.3937979237947x_{20} = 86.3937979237947
x21=100.216805649514x_{21} = 100.216805649514
x22=−45.5530934328517x_{22} = -45.5530934328517
x23=70.0575161750524x_{23} = 70.0575161750524
x24=−55.6061899685393x_{24} = -55.6061899685393
x25=2.19911485751286x_{25} = 2.19911485751286
x26=−60.0044196835651x_{26} = -60.0044196835651
x27=44.2964564156161x_{27} = 44.2964564156161
x28=−9.73893722612836x_{28} = -9.73893722612836
x29=−14.1371668872424x_{29} = -14.1371668872424
x30=46.18141200777x_{30} = 46.18141200777
x31=−63.7743308678728x_{31} = -63.7743308678728
x32=−67.5442419493278x_{32} = -67.5442419493278
x33=60.0044196835651x_{33} = 60.0044196835651
x34=−29.8451301142458x_{34} = -29.8451301142458
x35=78.2256570743859x_{35} = 78.2256570743859
x36=−53.7212343763855x_{36} = -53.7212343763855
x37=−87.6504350351552x_{37} = -87.6504350351552
x38=92.0486647501809x_{38} = 92.0486647501809
x39=5.96902604182061x_{39} = 5.96902604182061
x40=−93.9336203423348x_{40} = -93.9336203423348
x41=81.9955682586936x_{41} = 81.9955682586936
x42=−33.6150413934108x_{42} = -33.6150413934108
x43=−61.8893752757189x_{43} = -61.8893752757189
x44=76.340701482232x_{44} = 76.340701482232
x45=42.4115007836476x_{45} = 42.4115007836476
x46=58.7477826221291x_{46} = 58.7477826221291
x47=38.0132711084365x_{47} = 38.0132711084365
x48=−21.6769893097696x_{48} = -21.6769893097696
x49=17.9070781254618x_{49} = 17.9070781254618
x50=−36.1283154601932x_{50} = -36.1283154601932
x51=−58.1194640341413x_{51} = -58.1194640341413
x52=−80.1106126089377x_{52} = -80.1106126089377
x53=−51.8362786978016x_{53} = -51.8362786978016
x54=32.3584043319749x_{54} = 32.3584043319749
x55=24.1902634326414x_{55} = 24.1902634326414
x56=48.0663675999238x_{56} = 48.0663675999238
x57=−39.8982267005904x_{57} = -39.8982267005904
x58=−48.0663675999238x_{58} = -48.0663675999238
x59=39.2699081498428x_{59} = 39.2699081498428
x60=12.2522113490002x_{60} = 12.2522113490002
x61=−81.9955682586936x_{61} = -81.9955682586936
x62=16.0221225333079x_{62} = 16.0221225333079
x63=20.4203522181614x_{63} = 20.4203522181614
x64=−95.8185758695698x_{64} = -95.8185758695698
x65=−70.0575161750524x_{65} = -70.0575161750524
x66=−27.9601746169492x_{66} = -27.9601746169492
x67=−92.0486647501809x_{67} = -92.0486647501809
x68=26.0752190247953x_{68} = 26.0752190247953
x69=−77.5973385436679x_{69} = -77.5973385436679
x70=−1.57079634217526x_{70} = -1.57079634217526
x71=22.3053078404875x_{71} = 22.3053078404875
x72=−73.827427283678x_{72} = -73.827427283678
x73=66.2876049907446x_{73} = 66.2876049907446
x74=49.9513231920777x_{74} = 49.9513231920777
x75=−31.7300858012569x_{75} = -31.7300858012569
x76=−49.9513231920777x_{76} = -49.9513231920777
x77=56.2345084992573x_{77} = 56.2345084992573
x78=−43.6681378848981x_{78} = -43.6681378848981
x79=10.3672557568463x_{79} = 10.3672557568463
x80=−41.7831822927443x_{80} = -41.7831822927443
x81=−4.08407044966673x_{81} = -4.08407044966673
x82=27.9601746169492x_{82} = 27.9601746169492
x83=39.8982267005904x_{83} = 39.8982267005904
x84=88.2787535658732x_{84} = 88.2787535658732
x85=−16.0221225333079x_{85} = -16.0221225333079
x86=−75.712382951514x_{86} = -75.712382951514
x87=95.8185759450348x_{87} = 95.8185759450348
x88=71.9424717672063x_{88} = 71.9424717672063
x89=7.85398168446658x_{89} = 7.85398168446658
x90=−19.7920337176157x_{90} = -19.7920337176157
x91=−89.5353904622891x_{91} = -89.5353904622891
x92=−11.6238928182822x_{92} = -11.6238928182822
x93=73.8274273864408x_{93} = 73.8274273864408
x94=−7.85398154718208x_{94} = -7.85398154718208
x95=0.314159265358979x_{95} = 0.314159265358979
x96=51.8362788222469x_{96} = 51.8362788222469
x97=54.3495529071034x_{97} = 54.3495529071034
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(5*x)*cos(x).
cos⁡(0)cos⁡(0⋅5)\cos{\left(0 \right)} \cos{\left(0 \cdot 5 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(cos⁡(x)cos⁡(5x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\cos{\left(x \right)} \cos{\left(5 x \right)}\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⁡(x)cos⁡(5x))=⟨−1,1⟩\lim_{x \to \infty}\left(\cos{\left(x \right)} \cos{\left(5 x \right)}\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(5*x)*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)cos⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \cos{\left(5 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)cos⁡(5x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(5 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:
cos⁡(x)cos⁡(5x)=cos⁡(x)cos⁡(5x)\cos{\left(x \right)} \cos{\left(5 x \right)} = \cos{\left(x \right)} \cos{\left(5 x \right)}
- Yes
cos⁡(x)cos⁡(5x)=−cos⁡(x)cos⁡(5x)\cos{\left(x \right)} \cos{\left(5 x \right)} = - \cos{\left(x \right)} \cos{\left(5 x \right)}
- No
so, the function
is
even
    © Mister Exam – Calculator