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  • How to use it?

  • Graphing y =:
  • -x^2+5x-4
  • |x^2+8x+12|
  • (x^2-4)/(2x+5)
  • -x^2-2x

  • Identical expressions

  • one *(sin(x)-(x)*cos(x))
  • 1 multiply by ( sinus of (x) minus (x) multiply by co sinus of e of (x))
  • one multiply by ( sinus of (x) minus (x) multiply by co sinus of e of (x))
  • 1(sin(x)-(x)cos(x))
  • 1sinx-xcosx

  • Similar expressions

  • 1*(sin(x)+(x)*cos(x))
  • 1*(sinx-(x)*cosx)
Plotting a function graph/ 1*(sin(x)-(x)*cos(x))

Graphing y = 1*(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=20.3713029592876x_{1} = 20.3713029592876
x2=−4.49340945790906x_{2} = -4.49340945790906
x3=−7.72525183693771x_{3} = -7.72525183693771
x4=−42.3879135681319x_{4} = -42.3879135681319
x5=−67.5294347771441x_{5} = -67.5294347771441
x6=−73.8138806006806x_{6} = -73.8138806006806
x7=0x_{7} = 0
x8=70.6716857116195x_{8} = 70.6716857116195
x9=−92.6661922776228x_{9} = -92.6661922776228
x10=−95.8081387868617x_{10} = -95.8081387868617
x11=10.9041216594289x_{11} = 10.9041216594289
x12=26.6660542588127x_{12} = 26.6660542588127
x13=−10.9041216594289x_{13} = -10.9041216594289
x14=102.091966464908x_{14} = 102.091966464908
x15=80.0981286289451x_{15} = 80.0981286289451
x16=−36.1006222443756x_{16} = -36.1006222443756
x17=83.2401924707234x_{17} = 83.2401924707234
x18=48.6741442319544x_{18} = 48.6741442319544
x19=−14.0661939128315x_{19} = -14.0661939128315
x20=14.0661939128315x_{20} = 14.0661939128315
x21=−29.811598790893x_{21} = -29.811598790893
x22=23.519452498689x_{22} = 23.519452498689
x23=76.9560263103312x_{23} = 76.9560263103312
x24=67.5294347771441x_{24} = 67.5294347771441
x25=−17.2207552719308x_{25} = -17.2207552719308
x26=39.2444323611642x_{26} = 39.2444323611642
x27=−23.519452498689x_{27} = -23.519452498689
x28=17.2207552719308x_{28} = 17.2207552719308
x29=29.811598790893x_{29} = 29.811598790893
x30=−48.6741442319544x_{30} = -48.6741442319544
x31=−86.3822220347287x_{31} = -86.3822220347287
x32=89.5242209304172x_{32} = 89.5242209304172
x33=61.2447302603744x_{33} = 61.2447302603744
x34=−39.2444323611642x_{34} = -39.2444323611642
x35=−76.9560263103312x_{35} = -76.9560263103312
x36=−58.1022547544956x_{36} = -58.1022547544956
x37=−54.9596782878889x_{37} = -54.9596782878889
x38=54.9596782878889x_{38} = 54.9596782878889
x39=95.8081387868617x_{39} = 95.8081387868617
x40=92.6661922776228x_{40} = 92.6661922776228
x41=−70.6716857116195x_{41} = -70.6716857116195
x42=86.3822220347287x_{42} = 86.3822220347287
x43=−26.6660542588127x_{43} = -26.6660542588127
x44=−20.3713029592876x_{44} = -20.3713029592876
x45=−89.5242209304172x_{45} = -89.5242209304172
x46=−83.2401924707234x_{46} = -83.2401924707234
x47=45.5311340139913x_{47} = 45.5311340139913
x48=36.1006222443756x_{48} = 36.1006222443756
x49=4.49340945790906x_{49} = 4.49340945790906
x50=−32.9563890398225x_{50} = -32.9563890398225
x51=7.72525183693771x_{51} = 7.72525183693771
x52=98.9500628243319x_{52} = 98.9500628243319
x53=−98.9500628243319x_{53} = -98.9500628243319
x54=−61.2447302603744x_{54} = -61.2447302603744
x55=−0.000109308030426382x_{55} = -0.000109308030426382
x56=42.3879135681319x_{56} = 42.3879135681319
x57=51.8169824872797x_{57} = 51.8169824872797
x58=−51.8169824872797x_{58} = -51.8169824872797
x59=−80.0981286289451x_{59} = -80.0981286289451
x60=58.1022547544956x_{60} = 58.1022547544956
x61=32.9563890398225x_{61} = 32.9563890398225
x62=64.3871195905574x_{62} = 64.3871195905574
x63=−45.5311340139913x_{63} = -45.5311340139913
x64=−64.3871195905574x_{64} = -64.3871195905574
x65=73.8138806006806x_{65} = 73.8138806006806
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=−64.4181717218392x_{1} = -64.4181717218392
x2=−45.57503179559x_{2} = -45.57503179559
x3=70.69997803861x_{3} = 70.69997803861
x4=−51.855560729152x_{4} = -51.855560729152
x5=−48.7152107175577x_{5} = -48.7152107175577
x6=−7.97866571241324x_{6} = -7.97866571241324
x7=89.5465575382492x_{7} = 89.5465575382492
x8=−42.4350618814099x_{8} = -42.4350618814099
x9=64.4181717218392x_{9} = 64.4181717218392
x10=36.1559664195367x_{10} = 36.1559664195367
x11=−29.8785865061074x_{11} = -29.8785865061074
x12=−98.9702722883957x_{12} = -98.9702722883957
x13=45.57503179559x_{13} = 45.57503179559
x14=−73.8409691490209x_{14} = -73.8409691490209
x15=98.9702722883957x_{15} = 98.9702722883957
x16=95.8290108090195x_{16} = 95.8290108090195
x17=14.2074367251912x_{17} = 14.2074367251912
x18=80.1230928148503x_{18} = 80.1230928148503
x19=−20.469167402741x_{19} = -20.469167402741
x20=73.8409691490209x_{20} = 73.8409691490209
x21=−58.1366632448992x_{21} = -58.1366632448992
x22=4.91318043943488x_{22} = 4.91318043943488
x23=−17.3363779239834x_{23} = -17.3363779239834
x24=−61.2773745335697x_{24} = -61.2773745335697
x25=23.6042847729804x_{25} = 23.6042847729804
x26=−39.295350981473x_{26} = -39.295350981473
x27=58.1366632448992x_{27} = 58.1366632448992
x28=−54.9960525574964x_{28} = -54.9960525574964
x29=83.2642147040886x_{29} = 83.2642147040886
x30=39.295350981473x_{30} = 39.295350981473
x31=20.469167402741x_{31} = 20.469167402741
x32=102.111554139654x_{32} = 102.111554139654
x33=51.855560729152x_{33} = 51.855560729152
x34=92.687771772017x_{34} = 92.687771772017
x35=17.3363779239834x_{35} = 17.3363779239834
x36=0x_{36} = 0
x37=−67.5590428388084x_{37} = -67.5590428388084
x38=−11.085538406497x_{38} = -11.085538406497
x39=7.97866571241324x_{39} = 7.97866571241324
x40=−95.8290108090195x_{40} = -95.8290108090195
x41=−14.2074367251912x_{41} = -14.2074367251912
x42=67.5590428388084x_{42} = 67.5590428388084
x43=−70.69997803861x_{43} = -70.69997803861
x44=−23.6042847729804x_{44} = -23.6042847729804
x45=11.085538406497x_{45} = 11.085538406497
x46=−4.91318043943488x_{46} = -4.91318043943488
x47=−76.9820093304187x_{47} = -76.9820093304187
x48=2.02875783811043x_{48} = 2.02875783811043
x49=−26.7409160147873x_{49} = -26.7409160147873
x50=26.7409160147873x_{50} = 26.7409160147873
x51=54.9960525574964x_{51} = 54.9960525574964
x52=−89.5465575382492x_{52} = -89.5465575382492
x53=−36.1559664195367x_{53} = -36.1559664195367
x54=−83.2642147040886x_{54} = -83.2642147040886
x55=86.4053708116885x_{55} = 86.4053708116885
x56=61.2773745335697x_{56} = 61.2773745335697
x57=76.9820093304187x_{57} = 76.9820093304187
x58=−92.687771772017x_{58} = -92.687771772017
x59=42.4350618814099x_{59} = 42.4350618814099
x60=−86.4053708116885x_{60} = -86.4053708116885
x61=48.7152107175577x_{61} = 48.7152107175577
x62=−33.0170010333572x_{62} = -33.0170010333572
x63=33.0170010333572x_{63} = 33.0170010333572
x64=−80.1230928148503x_{64} = -80.1230928148503
x65=−2.02875783811043x_{65} = -2.02875783811043
x66=29.8785865061074x_{66} = 29.8785865061074

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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