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Plotting a function graph/ cos(x)+sin(x)

Graphing y = cos(x)+sin(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(x) + sin(x)
f(x)=sin⁡(x)+cos⁡(x)f{\left(x \right)} = \sin{\left(x \right)} + \cos{\left(x \right)} 
f = sin(x) + cos(x)
The graph of the function
0.00.51.01.52.02.53.03.54.04.55.05.56.05-5
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:
sin⁡(x)+cos⁡(x)=0\sin{\left(x \right)} + \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π4x_{1} = - \frac{\pi}{4}
Numerical solution
x1=18.0641577581413x_{1} = 18.0641577581413
x2=52.621676947629x_{2} = 52.621676947629
x3=8.63937979737193x_{3} = 8.63937979737193
x4=49.4800842940392x_{4} = 49.4800842940392
x5=−76.1836218495525x_{5} = -76.1836218495525
x6=27.4889357189107x_{6} = 27.4889357189107
x7=58.9048622548086x_{7} = 58.9048622548086
x8=43.1968989868597x_{8} = 43.1968989868597
x9=5.49778714378214x_{9} = 5.49778714378214
x10=93.4623814442964x_{10} = 93.4623814442964
x11=−79.3252145031423x_{11} = -79.3252145031423
x12=2.35619449019234x_{12} = 2.35619449019234
x13=−73.0420291959627x_{13} = -73.0420291959627
x14=24.3473430653209x_{14} = 24.3473430653209
x15=33.7721210260903x_{15} = 33.7721210260903
x16=−25.9181393921158x_{16} = -25.9181393921158
x17=11.7809724509617x_{17} = 11.7809724509617
x18=−60.4756585816035x_{18} = -60.4756585816035
x19=71.4712328691678x_{19} = 71.4712328691678
x20=80.8960108299372x_{20} = 80.8960108299372
x21=65.1880475619882x_{21} = 65.1880475619882
x22=−13.3517687777566x_{22} = -13.3517687777566
x23=351.072979038659x_{23} = 351.072979038659
x24=−51.0508806208341x_{24} = -51.0508806208341
x25=68.329640215578x_{25} = 68.329640215578
x26=−19.6349540849362x_{26} = -19.6349540849362
x27=−10.2101761241668x_{27} = -10.2101761241668
x28=74.6128255227576x_{28} = 74.6128255227576
x29=−16.4933614313464x_{29} = -16.4933614313464
x30=−69.9004365423729x_{30} = -69.9004365423729
x31=46.3384916404494x_{31} = 46.3384916404494
x32=87.1791961371168x_{32} = 87.1791961371168
x33=99.7455667514759x_{33} = 99.7455667514759
x34=−98.174770424681x_{34} = -98.174770424681
x35=90.3207887907066x_{35} = 90.3207887907066
x36=−82.4668071567321x_{36} = -82.4668071567321
x37=−57.3340659280137x_{37} = -57.3340659280137
x38=−66.7588438887831x_{38} = -66.7588438887831
x39=−54.1924732744239x_{39} = -54.1924732744239
x40=36.9137136796801x_{40} = 36.9137136796801
x41=−95.0331777710912x_{41} = -95.0331777710912
x42=−35.3429173528852x_{42} = -35.3429173528852
x43=−7.06858347057703x_{43} = -7.06858347057703
x44=−3.92699081698724x_{44} = -3.92699081698724
x45=−85.6083998103219x_{45} = -85.6083998103219
x46=62.0464549083984x_{46} = 62.0464549083984
x47=−47.9092879672443x_{47} = -47.9092879672443
x48=−32.2013246992954x_{48} = -32.2013246992954
x49=30.6305283725005x_{49} = 30.6305283725005
x50=21.2057504117311x_{50} = 21.2057504117311
x51=−44.7676953136546x_{51} = -44.7676953136546
x52=40.0553063332699x_{52} = 40.0553063332699
x53=−91.8915851175014x_{53} = -91.8915851175014
x54=−0.785398163397448x_{54} = -0.785398163397448
x55=−29.0597320457056x_{55} = -29.0597320457056
x56=84.037603483527x_{56} = 84.037603483527
x57=77.7544181763474x_{57} = 77.7544181763474
x58=96.6039740978861x_{58} = 96.6039740978861
x59=−63.6172512351933x_{59} = -63.6172512351933
x60=14.9225651045515x_{60} = 14.9225651045515
x61=−88.7499924639117x_{61} = -88.7499924639117
x62=−22.776546738526x_{62} = -22.776546738526
x63=−107.59954838545x_{63} = -107.59954838545
x64=−38.484510006475x_{64} = -38.484510006475
x65=55.7632696012188x_{65} = 55.7632696012188
x66=−41.6261026600648x_{66} = -41.6261026600648
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x) + sin(x).
sin⁡(0)+cos⁡(0)\sin{\left(0 \right)} + \cos{\left(0 \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
−sin⁡(x)+cos⁡(x)=0- \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
The values of the extrema at the points:
 pi    ___ 
(--, \/ 2 )
 4


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:
The function has no minima
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]\left(-\infty, \frac{\pi}{4}\right]
Increasing at intervals
[π4,∞)\left[\frac{\pi}{4}, \infty\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
−(sin⁡(x)+cos⁡(x))=0- (\sin{\left(x \right)} + \cos{\left(x \right)}) = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\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]\left(-\infty, - \frac{\pi}{4}\right]
Convex at the intervals
[−π4,∞)\left[- \frac{\pi}{4}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x)+cos⁡(x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(sin⁡(x)+cos⁡(x))=⟨−2,2⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x) + sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)+cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)+cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \cos{\left(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:
sin⁡(x)+cos⁡(x)=−sin⁡(x)+cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)} = - \sin{\left(x \right)} + \cos{\left(x \right)}
- No
sin⁡(x)+cos⁡(x)=sin⁡(x)−cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)} = \sin{\left(x \right)} - \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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