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Plotting a function graph/ sin(x)-x*cos(x)

Graphing y = sin(x)-x*cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sin(x) - x*cos(x)
f(x)=−xcos⁡(x)+sin⁡(x)f{\left(x \right)} = - x \cos{\left(x \right)} + \sin{\left(x \right)} 
f = -x*cos(x) + sin(x)
The graph of the function
02468-8-6-4-2-1010-2020
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:
−xcos⁡(x)+sin⁡(x)=0- x \cos{\left(x \right)} + \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=32.9563890398225x_{1} = 32.9563890398225
x2=80.0981286289451x_{2} = 80.0981286289451
x3=−76.9560263103312x_{3} = -76.9560263103312
x4=−86.3822220347287x_{4} = -86.3822220347287
x5=36.1006222443756x_{5} = 36.1006222443756
x6=−98.9500628243319x_{6} = -98.9500628243319
x7=−39.2444323611642x_{7} = -39.2444323611642
x8=70.6716857116195x_{8} = 70.6716857116195
x9=102.091966464908x_{9} = 102.091966464908
x10=86.3822220347287x_{10} = 86.3822220347287
x11=−89.5242209304172x_{11} = -89.5242209304172
x12=73.8138806006806x_{12} = 73.8138806006806
x13=23.519452498689x_{13} = 23.519452498689
x14=−48.6741442319544x_{14} = -48.6741442319544
x15=−36.1006222443756x_{15} = -36.1006222443756
x16=−45.5311340139913x_{16} = -45.5311340139913
x17=26.6660542588127x_{17} = 26.6660542588127
x18=83.2401924707234x_{18} = 83.2401924707234
x19=10.9041216594289x_{19} = 10.9041216594289
x20=−7.72525183693771x_{20} = -7.72525183693771
x21=42.3879135681319x_{21} = 42.3879135681319
x22=−70.6716857116195x_{22} = -70.6716857116195
x23=−23.519452498689x_{23} = -23.519452498689
x24=45.5311340139913x_{24} = 45.5311340139913
x25=−26.6660542588127x_{25} = -26.6660542588127
x26=64.3871195905574x_{26} = 64.3871195905574
x27=−80.0981286289451x_{27} = -80.0981286289451
x28=98.9500628243319x_{28} = 98.9500628243319
x29=−32.9563890398225x_{29} = -32.9563890398225
x30=4.49340945790906x_{30} = 4.49340945790906
x31=17.2207552719308x_{31} = 17.2207552719308
x32=51.8169824872797x_{32} = 51.8169824872797
x33=76.9560263103312x_{33} = 76.9560263103312
x34=7.72525183693771x_{34} = 7.72525183693771
x35=−0.000109308030426382x_{35} = -0.000109308030426382
x36=89.5242209304172x_{36} = 89.5242209304172
x37=92.6661922776228x_{37} = 92.6661922776228
x38=−92.6661922776228x_{38} = -92.6661922776228
x39=−42.3879135681319x_{39} = -42.3879135681319
x40=20.3713029592876x_{40} = 20.3713029592876
x41=14.0661939128315x_{41} = 14.0661939128315
x42=−83.2401924707234x_{42} = -83.2401924707234
x43=−51.8169824872797x_{43} = -51.8169824872797
x44=−58.1022547544956x_{44} = -58.1022547544956
x45=95.8081387868617x_{45} = 95.8081387868617
x46=39.2444323611642x_{46} = 39.2444323611642
x47=−17.2207552719308x_{47} = -17.2207552719308
x48=−29.811598790893x_{48} = -29.811598790893
x49=−10.9041216594289x_{49} = -10.9041216594289
x50=−73.8138806006806x_{50} = -73.8138806006806
x51=−54.9596782878889x_{51} = -54.9596782878889
x52=54.9596782878889x_{52} = 54.9596782878889
x53=−14.0661939128315x_{53} = -14.0661939128315
x54=29.811598790893x_{54} = 29.811598790893
x55=−61.2447302603744x_{55} = -61.2447302603744
x56=−95.8081387868617x_{56} = -95.8081387868617
x57=−4.49340945790906x_{57} = -4.49340945790906
x58=0x_{58} = 0
x59=67.5294347771441x_{59} = 67.5294347771441
x60=58.1022547544956x_{60} = 58.1022547544956
x61=−67.5294347771441x_{61} = -67.5294347771441
x62=48.6741442319544x_{62} = 48.6741442319544
x63=−64.3871195905574x_{63} = -64.3871195905574
x64=−20.3713029592876x_{64} = -20.3713029592876
x65=61.2447302603744x_{65} = 61.2447302603744
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x) - x*cos(x).
sin⁡(0)−0cos⁡(0)\sin{\left(0 \right)} - 0 \cos{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
xsin⁡(x)=0x \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 0)

(pi, pi)


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:
x2=πx_{2} = \pi
Decreasing at intervals
(−∞,π]\left(-\infty, \pi\right]
Increasing at intervals
[π,∞)\left[\pi, \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
xcos⁡(x)+sin⁡(x)=0x \cos{\left(x \right)} + \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−20.469167402741x_{1} = -20.469167402741
x2=80.1230928148503x_{2} = 80.1230928148503
x3=45.57503179559x_{3} = 45.57503179559
x4=−86.4053708116885x_{4} = -86.4053708116885
x5=−11.085538406497x_{5} = -11.085538406497
x6=95.8290108090195x_{6} = 95.8290108090195
x7=−36.1559664195367x_{7} = -36.1559664195367
x8=−42.4350618814099x_{8} = -42.4350618814099
x9=36.1559664195367x_{9} = 36.1559664195367
x10=7.97866571241324x_{10} = 7.97866571241324
x11=2.02875783811043x_{11} = 2.02875783811043
x12=−80.1230928148503x_{12} = -80.1230928148503
x13=92.687771772017x_{13} = 92.687771772017
x14=86.4053708116885x_{14} = 86.4053708116885
x15=29.8785865061074x_{15} = 29.8785865061074
x16=70.69997803861x_{16} = 70.69997803861
x17=58.1366632448992x_{17} = 58.1366632448992
x18=4.91318043943488x_{18} = 4.91318043943488
x19=73.8409691490209x_{19} = 73.8409691490209
x20=−48.7152107175577x_{20} = -48.7152107175577
x21=54.9960525574964x_{21} = 54.9960525574964
x22=11.085538406497x_{22} = 11.085538406497
x23=−4.91318043943488x_{23} = -4.91318043943488
x24=−76.9820093304187x_{24} = -76.9820093304187
x25=−89.5465575382492x_{25} = -89.5465575382492
x26=−14.2074367251912x_{26} = -14.2074367251912
x27=−33.0170010333572x_{27} = -33.0170010333572
x28=−51.855560729152x_{28} = -51.855560729152
x29=64.4181717218392x_{29} = 64.4181717218392
x30=−67.5590428388084x_{30} = -67.5590428388084
x31=42.4350618814099x_{31} = 42.4350618814099
x32=0x_{32} = 0
x33=−7.97866571241324x_{33} = -7.97866571241324
x34=51.855560729152x_{34} = 51.855560729152
x35=26.7409160147873x_{35} = 26.7409160147873
x36=−26.7409160147873x_{36} = -26.7409160147873
x37=89.5465575382492x_{37} = 89.5465575382492
x38=−83.2642147040886x_{38} = -83.2642147040886
x39=−2.02875783811043x_{39} = -2.02875783811043
x40=83.2642147040886x_{40} = 83.2642147040886
x41=−45.57503179559x_{41} = -45.57503179559
x42=−98.9702722883957x_{42} = -98.9702722883957
x43=67.5590428388084x_{43} = 67.5590428388084
x44=20.469167402741x_{44} = 20.469167402741
x45=−54.9960525574964x_{45} = -54.9960525574964
x46=48.7152107175577x_{46} = 48.7152107175577
x47=39.295350981473x_{47} = 39.295350981473
x48=−17.3363779239834x_{48} = -17.3363779239834
x49=102.111554139654x_{49} = 102.111554139654
x50=−95.8290108090195x_{50} = -95.8290108090195
x51=−64.4181717218392x_{51} = -64.4181717218392
x52=61.2773745335697x_{52} = 61.2773745335697
x53=−29.8785865061074x_{53} = -29.8785865061074
x54=23.6042847729804x_{54} = 23.6042847729804
x55=−39.295350981473x_{55} = -39.295350981473
x56=−58.1366632448992x_{56} = -58.1366632448992
x57=98.9702722883957x_{57} = 98.9702722883957
x58=−23.6042847729804x_{58} = -23.6042847729804
x59=14.2074367251912x_{59} = 14.2074367251912
x60=−73.8409691490209x_{60} = -73.8409691490209
x61=−92.687771772017x_{61} = -92.687771772017
x62=33.0170010333572x_{62} = 33.0170010333572
x63=76.9820093304187x_{63} = 76.9820093304187
x64=−61.2773745335697x_{64} = -61.2773745335697
x65=17.3363779239834x_{65} = 17.3363779239834
x66=−70.69997803861x_{66} = -70.69997803861

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
[98.9702722883957,∞)\left[98.9702722883957, \infty\right)
Convex at the intervals
(−∞,−98.9702722883957]\left(-\infty, -98.9702722883957\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(−xcos⁡(x)+sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(−xcos⁡(x)+sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(- x \cos{\left(x \right)} + \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x) - x*cos(x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(−xcos⁡(x)+sin⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(−xcos⁡(x)+sin⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right)
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:
−xcos⁡(x)+sin⁡(x)=xcos⁡(x)−sin⁡(x)- x \cos{\left(x \right)} + \sin{\left(x \right)} = x \cos{\left(x \right)} - \sin{\left(x \right)}
- No
−xcos⁡(x)+sin⁡(x)=−xcos⁡(x)+sin⁡(x)- x \cos{\left(x \right)} + \sin{\left(x \right)} = - x \cos{\left(x \right)} + \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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