Mister Exam

Other calculators

Plotting a function graph/ (sin(2-x))/(2-x)

Graphing y = (sin(2-x))/(2-x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(2 - x)
f(x) = ----------
         2 - x
f(x)=sin⁡(2−x)2−xf{\left(x \right)} = \frac{\sin{\left(2 - x \right)}}{2 - x} 
f = sin(2 - x)/(2 - x)
The graph of the function
02468-8-6-4-2-10102-1
The domain of the function
The points at which the function is not precisely defined:
x1=2x_{1} = 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:
sin⁡(2−x)2−x=0\frac{\sin{\left(2 - x \right)}}{2 - x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=2+πx_{1} = 2 + \pi
Numerical solution
x1=27.1327412287183x_{1} = 27.1327412287183
x2=89.9645943005142x_{2} = 89.9645943005142
x3=5.14159265358979x_{3} = 5.14159265358979
x4=−29.4159265358979x_{4} = -29.4159265358979
x5=−10.5663706143592x_{5} = -10.5663706143592
x6=−26.2743338823081x_{6} = -26.2743338823081
x7=93.106186954104x_{7} = 93.106186954104
x8=−89.106186954104x_{8} = -89.106186954104
x9=45.9822971502571x_{9} = 45.9822971502571
x10=−57.6902604182061x_{10} = -57.6902604182061
x11=−23.1327412287183x_{11} = -23.1327412287183
x12=−368.707933123596x_{12} = -368.707933123596
x13=−41.9822971502571x_{13} = -41.9822971502571
x14=−32.5575191894877x_{14} = -32.5575191894877
x15=−35.6991118430775x_{15} = -35.6991118430775
x16=−1.14159265358979x_{16} = -1.14159265358979
x17=−51.4070751110265x_{17} = -51.4070751110265
x18=−48.2654824574367x_{18} = -48.2654824574367
x19=14.5663706143592x_{19} = 14.5663706143592
x20=102.530964914873x_{20} = 102.530964914873
x21=17.707963267949x_{21} = 17.707963267949
x22=96.2477796076938x_{22} = 96.2477796076938
x23=67.9734457253857x_{23} = 67.9734457253857
x24=−63.9734457253857x_{24} = -63.9734457253857
x25=39.6991118430775x_{25} = 39.6991118430775
x26=11.4247779607694x_{26} = 11.4247779607694
x27=−76.5398163397448x_{27} = -76.5398163397448
x28=8.28318530717959x_{28} = 8.28318530717959
x29=64.8318530717959x_{29} = 64.8318530717959
x30=−60.8318530717959x_{30} = -60.8318530717959
x31=55.4070751110265x_{31} = 55.4070751110265
x32=23.9911485751286x_{32} = 23.9911485751286
x33=36.5575191894877x_{33} = 36.5575191894877
x34=52.2654824574367x_{34} = 52.2654824574367
x35=−19.9911485751286x_{35} = -19.9911485751286
x36=−98.5309649148734x_{36} = -98.5309649148734
x37=−70.2566310325652x_{37} = -70.2566310325652
x38=−38.8407044966673x_{38} = -38.8407044966673
x39=−45.1238898038469x_{39} = -45.1238898038469
x40=30.2743338823081x_{40} = 30.2743338823081
x41=−85.9645943005142x_{41} = -85.9645943005142
x42=592.619418874881x_{42} = 592.619418874881
x43=86.8230016469244x_{43} = 86.8230016469244
x44=−13.707963267949x_{44} = -13.707963267949
x45=83.6814089933346x_{45} = 83.6814089933346
x46=49.1238898038469x_{46} = 49.1238898038469
x47=77.398223686155x_{47} = 77.398223686155
x48=58.5486677646163x_{48} = 58.5486677646163
x49=−111.097335529233x_{49} = -111.097335529233
x50=155.9380400259x_{50} = 155.9380400259
x51=74.2566310325652x_{51} = 74.2566310325652
x52=−16.8495559215388x_{52} = -16.8495559215388
x53=20.8495559215388x_{53} = 20.8495559215388
x54=−4.28318530717959x_{54} = -4.28318530717959
x55=80.5398163397448x_{55} = 80.5398163397448
x56=33.4159265358979x_{56} = 33.4159265358979
x57=−7.42477796076938x_{57} = -7.42477796076938
x58=61.6902604182061x_{58} = 61.6902604182061
x59=−221.053078404875x_{59} = -221.053078404875
x60=−82.8230016469244x_{60} = -82.8230016469244
x61=−92.2477796076938x_{61} = -92.2477796076938
x62=42.8407044966673x_{62} = 42.8407044966673
x63=−67.1150383789755x_{63} = -67.1150383789755
x64=−54.5486677646163x_{64} = -54.5486677646163
x65=71.1150383789755x_{65} = 71.1150383789755
x66=−95.3893722612836x_{66} = -95.3893722612836
x67=−79.6814089933346x_{67} = -79.6814089933346
x68=−73.398223686155x_{68} = -73.398223686155
x69=99.3893722612836x_{69} = 99.3893722612836
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(2 - x)/(2 - x).
sin⁡(2−0)2−0\frac{\sin{\left(2 - 0 \right)}}{2 - 0}
The result:
f(0)=sin⁡(2)2f{\left(0 \right)} = \frac{\sin{\left(2 \right)}}{2}
The point:
(0, sin(2)/2)
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
−cos⁡(x−2)2−x+sin⁡(2−x)(2−x)2=0- \frac{\cos{\left(x - 2 \right)}}{2 - x} + \frac{\sin{\left(2 - x \right)}}{\left(2 - x\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=66.3871195905574x_{1} = 66.3871195905574
x2=110.375719651675x_{2} = 110.375719651675
x3=−52.9596782878889x_{3} = -52.9596782878889
x4=−2.49340945790906x_{4} = -2.49340945790906
x5=−18.3713029592876x_{5} = -18.3713029592876
x6=60.1022547544956x_{6} = 60.1022547544956
x7=31.811598790893x_{7} = 31.811598790893
x8=25.519452498689x_{8} = 25.519452498689
x9=75.8138806006806x_{9} = 75.8138806006806
x10=19.2207552719308x_{10} = 19.2207552719308
x11=53.8169824872797x_{11} = 53.8169824872797
x12=22.3713029592876x_{12} = 22.3713029592876
x13=−84.3822220347287x_{13} = -84.3822220347287
x14=−90.6661922776228x_{14} = -90.6661922776228
x15=−62.3871195905574x_{15} = -62.3871195905574
x16=−34.1006222443756x_{16} = -34.1006222443756
x17=−37.2444323611642x_{17} = -37.2444323611642
x18=−15.2207552719308x_{18} = -15.2207552719308
x19=91.5242209304172x_{19} = 91.5242209304172
x20=56.9596782878889x_{20} = 56.9596782878889
x21=−5.72525183693771x_{21} = -5.72525183693771
x22=−74.9560263103312x_{22} = -74.9560263103312
x23=−49.8169824872797x_{23} = -49.8169824872797
x24=85.2401924707234x_{24} = 85.2401924707234
x25=−65.5294347771441x_{25} = -65.5294347771441
x26=9.72525183693771x_{26} = 9.72525183693771
x27=−27.811598790893x_{27} = -27.811598790893
x28=−78.0981286289451x_{28} = -78.0981286289451
x29=94.6661922776228x_{29} = 94.6661922776228
x30=−71.8138806006806x_{30} = -71.8138806006806
x31=−56.1022547544956x_{31} = -56.1022547544956
x32=34.9563890398225x_{32} = 34.9563890398225
x33=38.1006222443756x_{33} = 38.1006222443756
x34=41.2444323611642x_{34} = 41.2444323611642
x35=28.6660542588127x_{35} = 28.6660542588127
x36=−4353.81798462425x_{36} = -4353.81798462425
x37=97.8081387868617x_{37} = 97.8081387868617
x38=−87.5242209304172x_{38} = -87.5242209304172
x39=−100.091966464908x_{39} = -100.091966464908
x40=−46.6741442319544x_{40} = -46.6741442319544
x41=−40.3879135681319x_{41} = -40.3879135681319
x42=−96.9500628243319x_{42} = -96.9500628243319
x43=100.950062824332x_{43} = 100.950062824332
x44=−12.0661939128315x_{44} = -12.0661939128315
x45=72.6716857116195x_{45} = 72.6716857116195
x46=78.9560263103312x_{46} = 78.9560263103312
x47=−68.6716857116195x_{47} = -68.6716857116195
x48=12.9041216594289x_{48} = 12.9041216594289
x49=47.5311340139913x_{49} = 47.5311340139913
x50=50.6741442319544x_{50} = 50.6741442319544
x51=−59.2447302603744x_{51} = -59.2447302603744
x52=63.2447302603744x_{52} = 63.2447302603744
x53=69.5294347771441x_{53} = 69.5294347771441
x54=−24.6660542588127x_{54} = -24.6660542588127
x55=88.3822220347287x_{55} = 88.3822220347287
x56=82.0981286289451x_{56} = 82.0981286289451
x57=−30.9563890398225x_{57} = -30.9563890398225
x58=−392.267341680887x_{58} = -392.267341680887
x59=−93.8081387868617x_{59} = -93.8081387868617
x60=−8.9041216594289x_{60} = -8.9041216594289
x61=−43.5311340139913x_{61} = -43.5311340139913
x62=−81.2401924707234x_{62} = -81.2401924707234
x63=16.0661939128315x_{63} = 16.0661939128315
x64=44.3879135681319x_{64} = 44.3879135681319
x65=−21.519452498689x_{65} = -21.519452498689
x66=6.49340945790906x_{66} = 6.49340945790906
The values of the extrema at the points:
(66.38711959055742, 0.0155291838074613)

(110.37571965167469, 0.00922676625078197)

(-52.959678287888934, -0.0181921463218031)

(-2.493409457909064, -0.217233628211222)

(-18.37130295928756, 0.0490296240140742)

(60.10225475449559, 0.0172084874716279)

(31.81159879089296, -0.0335251350213988)

(25.519452498689006, -0.0424796169776126)

(75.81388060068065, -0.01354634434514)

(19.22075527193077, -0.0579718023461539)

(53.81698248727967, 0.019295099487588)

(22.37130295928756, 0.0490296240140742)

(-84.38222203472871, -0.0115756804584678)

(-90.66619227762284, -0.0107907938495342)

(-62.38711959055741, 0.0155291838074613)

(-34.10062224437561, -0.0276897323011492)

(-37.24443236116419, 0.0254730530928808)

(-15.220755271930768, -0.0579718023461539)

(91.52422093041719, 0.0111694646341736)

(56.959678287888934, -0.0181921463218031)

(-5.725251836937707, 0.128374553525899)

(-74.95602631033118, 0.0129933369870427)

(-49.81698248727967, 0.019295099487588)

(85.2401924707234, 0.0120125604820527)

(-65.52943477714412, -0.0148067339465492)

(9.725251836937707, 0.128374553525899)

(-27.81159879089296, -0.0335251350213988)

(-78.09812862894512, -0.012483713321779)

(94.66619227762284, -0.0107907938495342)

(-71.81388060068065, -0.01354634434514)

(-56.10225475449559, 0.0172084874716279)

(34.956389039822476, 0.0303291711863103)

(38.10062224437561, -0.0276897323011492)

(41.24443236116419, 0.0254730530928808)

(28.666054258812675, 0.0374745199939312)

(-4353.817984624248, 0.000229577998248987)

(97.8081387868617, 0.0104369581345658)

(-87.52422093041719, 0.0111694646341736)

(-100.09196646490764, 0.00979462014674114)

(-46.674144231954386, -0.0205404540417537)

(-40.38791356813192, -0.0235850682290164)

(-96.95006282433188, -0.010105591736504)

(100.95006282433188, -0.010105591736504)

(-12.066193912831473, 0.0709134594504622)

(72.6716857116195, 0.0141485220648664)

(78.95602631033118, 0.0129933369870427)

(-68.6716857116195, 0.0141485220648664)

(12.904121659428899, -0.0913252028230577)

(47.53113401399128, 0.0219576982284824)

(50.674144231954386, -0.0205404540417537)

(-59.2447302603744, -0.0163257593209978)

(63.2447302603744, -0.0163257593209978)

(69.52943477714412, -0.0148067339465492)

(-24.666054258812675, 0.0374745199939312)

(88.38222203472871, -0.0115756804584678)

(82.09812862894512, -0.012483713321779)

(-30.956389039822476, 0.0303291711863103)

(-392.26734168088706, -0.00253634191261283)

(-93.8081387868617, 0.0104369581345658)

(-8.904121659428899, -0.0913252028230577)

(-43.53113401399128, 0.0219576982284824)

(-81.2401924707234, 0.0120125604820527)

(16.066193912831473, 0.0709134594504622)

(44.38791356813192, -0.0235850682290164)

(-21.519452498689006, -0.0424796169776126)

(6.493409457909064, -0.217233628211222)


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:
x1=−52.9596782878889x_{1} = -52.9596782878889
x2=−2.49340945790906x_{2} = -2.49340945790906
x3=31.811598790893x_{3} = 31.811598790893
x4=25.519452498689x_{4} = 25.519452498689
x5=75.8138806006806x_{5} = 75.8138806006806
x6=19.2207552719308x_{6} = 19.2207552719308
x7=−84.3822220347287x_{7} = -84.3822220347287
x8=−90.6661922776228x_{8} = -90.6661922776228
x9=−34.1006222443756x_{9} = -34.1006222443756
x10=−15.2207552719308x_{10} = -15.2207552719308
x11=56.9596782878889x_{11} = 56.9596782878889
x12=−65.5294347771441x_{12} = -65.5294347771441
x13=−27.811598790893x_{13} = -27.811598790893
x14=−78.0981286289451x_{14} = -78.0981286289451
x15=94.6661922776228x_{15} = 94.6661922776228
x16=−71.8138806006806x_{16} = -71.8138806006806
x17=38.1006222443756x_{17} = 38.1006222443756
x18=−46.6741442319544x_{18} = -46.6741442319544
x19=−40.3879135681319x_{19} = -40.3879135681319
x20=−96.9500628243319x_{20} = -96.9500628243319
x21=100.950062824332x_{21} = 100.950062824332
x22=12.9041216594289x_{22} = 12.9041216594289
x23=50.6741442319544x_{23} = 50.6741442319544
x24=−59.2447302603744x_{24} = -59.2447302603744
x25=63.2447302603744x_{25} = 63.2447302603744
x26=69.5294347771441x_{26} = 69.5294347771441
x27=88.3822220347287x_{27} = 88.3822220347287
x28=82.0981286289451x_{28} = 82.0981286289451
x29=−392.267341680887x_{29} = -392.267341680887
x30=−8.9041216594289x_{30} = -8.9041216594289
x31=44.3879135681319x_{31} = 44.3879135681319
x32=−21.519452498689x_{32} = -21.519452498689
x33=6.49340945790906x_{33} = 6.49340945790906
Maxima of the function at points:
x33=66.3871195905574x_{33} = 66.3871195905574
x33=110.375719651675x_{33} = 110.375719651675
x33=−18.3713029592876x_{33} = -18.3713029592876
x33=60.1022547544956x_{33} = 60.1022547544956
x33=53.8169824872797x_{33} = 53.8169824872797
x33=22.3713029592876x_{33} = 22.3713029592876
x33=−62.3871195905574x_{33} = -62.3871195905574
x33=−37.2444323611642x_{33} = -37.2444323611642
x33=91.5242209304172x_{33} = 91.5242209304172
x33=−5.72525183693771x_{33} = -5.72525183693771
x33=−74.9560263103312x_{33} = -74.9560263103312
x33=−49.8169824872797x_{33} = -49.8169824872797
x33=85.2401924707234x_{33} = 85.2401924707234
x33=9.72525183693771x_{33} = 9.72525183693771
x33=−56.1022547544956x_{33} = -56.1022547544956
x33=34.9563890398225x_{33} = 34.9563890398225
x33=41.2444323611642x_{33} = 41.2444323611642
x33=28.6660542588127x_{33} = 28.6660542588127
x33=−4353.81798462425x_{33} = -4353.81798462425
x33=97.8081387868617x_{33} = 97.8081387868617
x33=−87.5242209304172x_{33} = -87.5242209304172
x33=−100.091966464908x_{33} = -100.091966464908
x33=−12.0661939128315x_{33} = -12.0661939128315
x33=72.6716857116195x_{33} = 72.6716857116195
x33=78.9560263103312x_{33} = 78.9560263103312
x33=−68.6716857116195x_{33} = -68.6716857116195
x33=47.5311340139913x_{33} = 47.5311340139913
x33=−24.6660542588127x_{33} = -24.6660542588127
x33=−30.9563890398225x_{33} = -30.9563890398225
x33=−93.8081387868617x_{33} = -93.8081387868617
x33=−43.5311340139913x_{33} = -43.5311340139913
x33=−81.2401924707234x_{33} = -81.2401924707234
x33=16.0661939128315x_{33} = 16.0661939128315
Decreasing at intervals
[100.950062824332,∞)\left[100.950062824332, \infty\right)
Increasing at intervals
(−∞,−392.267341680887]\left(-\infty, -392.267341680887\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−2)−2cos⁡(x−2)x−2+2sin⁡(x−2)(x−2)2x−2=0\frac{- \sin{\left(x - 2 \right)} - \frac{2 \cos{\left(x - 2 \right)}}{x - 2} + \frac{2 \sin{\left(x - 2 \right)}}{\left(x - 2\right)^{2}}}{x - 2} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=2x_{1} = 2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(2−x)2−x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 - x \right)}}{2 - x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(sin⁡(2−x)2−x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 - x \right)}}{2 - x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2 - x)/(2 - x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2−x)x(2−x))=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 - x \right)}}{x \left(2 - x\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(2−x)x(2−x))=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 - x \right)}}{x \left(2 - x\right)}\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⁡(2−x)2−x=sin⁡(x+2)x+2\frac{\sin{\left(2 - x \right)}}{2 - x} = \frac{\sin{\left(x + 2 \right)}}{x + 2}
- No
sin⁡(2−x)2−x=−sin⁡(x+2)x+2\frac{\sin{\left(2 - x \right)}}{2 - x} = - \frac{\sin{\left(x + 2 \right)}}{x + 2}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator