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Plotting a function graph/ cos2n

Graphing y = cos2n

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(n) = cos(2*n)
f(n)=cos⁡(2n)f{\left(n \right)} = \cos{\left(2 n \right)} 
f = cos(2*n)
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 N at f = 0
so we need to solve the equation:
cos⁡(2n)=0\cos{\left(2 n \right)} = 0
Solve this equation
The points of intersection with the axis N:

Analytical solution
n1=π4n_{1} = \frac{\pi}{4}
n2=3π4n_{2} = \frac{3 \pi}{4}
Numerical solution
n1=−55.7632696012188n_{1} = -55.7632696012188
n2=2.35619449019234n_{2} = 2.35619449019234
n3=87.1791961371168n_{3} = 87.1791961371168
n4=19.6349540849362n_{4} = 19.6349540849362
n5=5.49778714378214n_{5} = 5.49778714378214
n6=−46.3384916404494n_{6} = -46.3384916404494
n7=−27.4889357189107n_{7} = -27.4889357189107
n8=99.7455667514759n_{8} = 99.7455667514759
n9=33.7721210260903n_{9} = 33.7721210260903
n10=−54.1924732744239n_{10} = -54.1924732744239
n11=38.484510006475n_{11} = 38.484510006475
n12=96.6039740978861n_{12} = 96.6039740978861
n13=−32.2013246992954n_{13} = -32.2013246992954
n14=66.7588438887831n_{14} = 66.7588438887831
n15=−12461.9126586273n_{15} = -12461.9126586273
n16=−71.4712328691678n_{16} = -71.4712328691678
n17=27.4889357189107n_{17} = 27.4889357189107
n18=−41.6261026600648n_{18} = -41.6261026600648
n19=62.0464549083984n_{19} = 62.0464549083984
n20=91.8915851175014n_{20} = 91.8915851175014
n21=25.9181393921158n_{21} = 25.9181393921158
n22=8.63937979737193n_{22} = 8.63937979737193
n23=−19.6349540849362n_{23} = -19.6349540849362
n24=−85.6083998103219n_{24} = -85.6083998103219
n25=−82.4668071567321n_{25} = -82.4668071567321
n26=1973.70558461779n_{26} = 1973.70558461779
n27=24.3473430653209n_{27} = 24.3473430653209
n28=76.1836218495525n_{28} = 76.1836218495525
n29=−35.3429173528852n_{29} = -35.3429173528852
n30=−49.4800842940392n_{30} = -49.4800842940392
n31=82.4668071567321n_{31} = 82.4668071567321
n32=−93.4623814442964n_{32} = -93.4623814442964
n33=−99.7455667514759n_{33} = -99.7455667514759
n34=77.7544181763474n_{34} = 77.7544181763474
n35=162.577419823272n_{35} = 162.577419823272
n36=46.3384916404494n_{36} = 46.3384916404494
n37=−5.49778714378214n_{37} = -5.49778714378214
n38=16.4933614313464n_{38} = 16.4933614313464
n39=−11.7809724509617n_{39} = -11.7809724509617
n40=−3.92699081698724n_{40} = -3.92699081698724
n41=22.776546738526n_{41} = 22.776546738526
n42=−18.0641577581413n_{42} = -18.0641577581413
n43=−98.174770424681n_{43} = -98.174770424681
n44=−79.3252145031423n_{44} = -79.3252145031423
n45=−62.0464549083984n_{45} = -62.0464549083984
n46=74.6128255227576n_{46} = 74.6128255227576
n47=−13.3517687777566n_{47} = -13.3517687777566
n48=−24.3473430653209n_{48} = -24.3473430653209
n49=18.0641577581413n_{49} = 18.0641577581413
n50=60.4756585816035n_{50} = 60.4756585816035
n51=11.7809724509617n_{51} = 11.7809724509617
n52=69.9004365423729n_{52} = 69.9004365423729
n53=88.7499924639117n_{53} = 88.7499924639117
n54=55.7632696012188n_{54} = 55.7632696012188
n55=47.9092879672443n_{55} = 47.9092879672443
n56=−63.6172512351933n_{56} = -63.6172512351933
n57=54.1924732744239n_{57} = 54.1924732744239
n58=−47.9092879672443n_{58} = -47.9092879672443
n59=−68.329640215578n_{59} = -68.329640215578
n60=40.0553063332699n_{60} = 40.0553063332699
n61=−84.037603483527n_{61} = -84.037603483527
n62=−57.3340659280137n_{62} = -57.3340659280137
n63=−25.9181393921158n_{63} = -25.9181393921158
n64=−10.2101761241668n_{64} = -10.2101761241668
n65=−91.8915851175014n_{65} = -91.8915851175014
n66=−69.9004365423729n_{66} = -69.9004365423729
n67=41.6261026600648n_{67} = 41.6261026600648
n68=90.3207887907066n_{68} = 90.3207887907066
n69=−60.4756585816035n_{69} = -60.4756585816035
n70=3.92699081698724n_{70} = 3.92699081698724
n71=32.2013246992954n_{71} = 32.2013246992954
n72=−2.35619449019234n_{72} = -2.35619449019234
n73=−90.3207887907066n_{73} = -90.3207887907066
n74=384.059701901352n_{74} = 384.059701901352
n75=−77.7544181763474n_{75} = -77.7544181763474
n76=−38.484510006475n_{76} = -38.484510006475
n77=30.6305283725005n_{77} = 30.6305283725005
n78=49.4800842940392n_{78} = 49.4800842940392
n79=85.6083998103219n_{79} = 85.6083998103219
n80=−76.1836218495525n_{80} = -76.1836218495525
n81=98.174770424681n_{81} = 98.174770424681
n82=52.621676947629n_{82} = 52.621676947629
n83=44.7676953136546n_{83} = 44.7676953136546
n84=−40.0553063332699n_{84} = -40.0553063332699
n85=−16.4933614313464n_{85} = -16.4933614313464
n86=63.6172512351933n_{86} = 63.6172512351933
n87=68.329640215578n_{87} = 68.329640215578
n88=−33.7721210260903n_{88} = -33.7721210260903
n89=10.2101761241668n_{89} = 10.2101761241668
n90=84.037603483527n_{90} = 84.037603483527
The points of intersection with the Y axis coordinate
The graph crosses Y axis when n equals 0:
substitute n = 0 to cos(2*n).
cos⁡(0⋅2)\cos{\left(0 \cdot 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
ddnf(n)=0\frac{d}{d n} f{\left(n \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddnf(n)=\frac{d}{d n} f{\left(n \right)} =
the first derivative
−2sin⁡(2n)=0- 2 \sin{\left(2 n \right)} = 0
Solve this equation
The roots of this equation
n1=0n_{1} = 0
n2=π2n_{2} = \frac{\pi}{2}
The values of the extrema at the points:
(0, 1)

 pi     
(--, -1)
 2


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:
n1=π2n_{1} = \frac{\pi}{2}
Maxima of the function at points:
n1=0n_{1} = 0
Decreasing at intervals
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
[0,π2]\left[0, \frac{\pi}{2}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dn2f(n)=0\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dn2f(n)=\frac{d^{2}}{d n^{2}} f{\left(n \right)} =
the second derivative
−4cos⁡(2n)=0- 4 \cos{\left(2 n \right)} = 0
Solve this equation
The roots of this equation
n1=π4n_{1} = \frac{\pi}{4}
n2=3π4n_{2} = \frac{3 \pi}{4}

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
[π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]
Convex at the intervals
(−∞,π4]∪[3π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at n->+oo and n->-oo
lim⁡n→−∞cos⁡(2n)=⟨−1,1⟩\lim_{n \to -\infty} \cos{\left(2 n \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⁡n→∞cos⁡(2n)=⟨−1,1⟩\lim_{n \to \infty} \cos{\left(2 n \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(2*n), divided by n at n->+oo and n ->-oo
lim⁡n→−∞(cos⁡(2n)n)=0\lim_{n \to -\infty}\left(\frac{\cos{\left(2 n \right)}}{n}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡n→∞(cos⁡(2n)n)=0\lim_{n \to \infty}\left(\frac{\cos{\left(2 n \right)}}{n}\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(-n) и f = -f(-n).
So, check:
cos⁡(2n)=cos⁡(2n)\cos{\left(2 n \right)} = \cos{\left(2 n \right)}
- Yes
cos⁡(2n)=−cos⁡(2n)\cos{\left(2 n \right)} = - \cos{\left(2 n \right)}
- No
so, the function
is
even
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