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(0,5-x)cosx+sinx
Plotting a function graph/ (0,5-x)cosx+sinx

Graphing y = (0,5-x)cosx+sinx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = (1/2 - x)*cos(x) + sin(x)
f(x)=(−x+12)cos⁡(x)+sin⁡(x)f{\left(x \right)} = \left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)} 
f = (1/2 - x)*cos(x) + sin(x)
The graph of the function
0-80-70-60-50-40-30-20-10102002
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:
(−x+12)cos⁡(x)+sin⁡(x)=0\left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−76.956110192893x_{1} = -76.956110192893
x2=23.5185289387051x_{2} = 23.5185289387051
x3=−80.0982060790942x_{3} = -80.0982060790942
x4=14.0635732069164x_{4} = 14.0635732069164
x5=−14.0686338222896x_{5} = -14.0686338222896
x6=95.8080840300867x_{6} = 95.8080840300867
x7=−36.1010006541423x_{7} = -36.1010006541423
x8=−64.387239267837x_{8} = -64.387239267837
x9=36.1002331970341x_{9} = 36.1002331970341
x10=−98.9501136342677x_{10} = -98.9501136342677
x11=−4.51559129296106x_{11} = -4.51559129296106
x12=80.0980502056287x_{12} = 80.0980502056287
x13=92.6661337343071x_{13} = 92.6661337343071
x14=−17.2223935780088x_{14} = -17.2223935780088
x15=−89.5242829701305x_{15} = -89.5242829701305
x16=76.9559413304481x_{16} = 76.9559413304481
x17=64.386998039561x_{17} = 64.386998039561
x18=20.3700677184872x_{18} = 20.3700677184872
x19=48.6739309964161x_{19} = 48.6739309964161
x20=73.8137882061413x_{20} = 73.8137882061413
x21=17.2190186447695x_{21} = 17.2190186447695
x22=−39.2447529232359x_{22} = -39.2447529232359
x23=−73.8139717516921x_{23} = -73.8139717516921
x24=−48.6743531293462x_{24} = -48.6743531293462
x25=−23.5203375400701x_{25} = -23.5203375400701
x26=−54.9598423269172x_{26} = -54.9598423269172
x27=−7.73311295281087x_{27} = -7.73311295281087
x28=86.3821546373973x_{28} = 86.3821546373973
x29=29.8110265832429x_{29} = 29.8110265832429
x30=−45.5313725794722x_{30} = -45.5313725794722
x31=−61.2448624813776x_{31} = -61.2448624813776
x32=32.9559215884855x_{32} = 32.9559215884855
x33=−51.8171669266635x_{33} = -51.8171669266635
x34=45.5308901481363x_{34} = 45.5308901481363
x35=42.3876319619764x_{35} = 42.3876319619764
x36=54.9595112357352x_{36} = 54.9595112357352
x37=−83.2402642010158x_{37} = -83.2402642010158
x38=70.6715848877719x_{38} = 70.6715848877719
x39=−20.3724788770916x_{39} = -20.3724788770916
x40=−67.5295436148378x_{40} = -67.5295436148378
x41=39.2441035198914x_{41} = 39.2441035198914
x42=83.2401198733711x_{42} = 83.2401198733711
x43=−86.3822886562246x_{43} = -86.3822886562246
x44=10.8997125066948x_{44} = 10.8997125066948
x45=−26.6667444566263x_{45} = -26.6667444566263
x46=7.71628308643291x_{46} = 7.71628308643291
x47=89.5241581937306x_{47} = 89.5241581937306
x48=−42.388188604579x_{48} = -42.388188604579
x49=−95.8081929750169x_{49} = -95.8081929750169
x50=−70.6717851185568x_{50} = -70.6717851185568
x51=51.8167944524242x_{51} = 51.8167944524242
x52=−10.9081410672717x_{52} = -10.9081410672717
x53=−58.1024016003196x_{53} = -58.1024016003196
x54=58.1021053586029x_{54} = 58.1021053586029
x55=−92.6662501924907x_{55} = -92.6662501924907
x56=67.5293243153618x_{56} = 67.5293243153618
x57=−0.975017193264127x_{57} = -0.975017193264127
x58=26.6653376455723x_{58} = 26.6653376455723
x59=98.9500114982446x_{59} = 98.9500114982446
x60=4.46535566966087x_{60} = 4.46535566966087
x61=−32.9568425065237x_{61} = -32.9568425065237
x62=−29.8121521001303x_{62} = -29.8121521001303
x63=61.2445958621199x_{63} = 61.2445958621199
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (1/2 - x)*cos(x) + sin(x).
sin⁡(0)+((−1)0+12)cos⁡(0)\sin{\left(0 \right)} + \left(\left(-1\right) 0 + \frac{1}{2}\right) \cos{\left(0 \right)}
The result:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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
−(−x+12)sin⁡(x)=0- \left(- x + \frac{1}{2}\right) \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=12x_{2} = \frac{1}{2}
x3=πx_{3} = \pi
The values of the extrema at the points:
(0, 1/2)

(1/2, sin(1/2))

(pi, -1/2 + 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:
Minima of the function at points:
x1=12x_{1} = \frac{1}{2}
Maxima of the function at points:
x1=0x_{1} = 0
x1=πx_{1} = \pi
Decreasing at intervals
(−∞,0]∪[12,∞)\left(-\infty, 0\right] \cup \left[\frac{1}{2}, \infty\right)
Increasing at intervals
(−∞,12]∪[π,∞)\left(-\infty, \frac{1}{2}\right] \cup \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
(2x−1)cos⁡(x)2+sin⁡(x)=0\frac{\left(2 x - 1\right) \cos{\left(x \right)}}{2} + \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=95.829065529839x_{1} = 95.829065529839
x2=102.1116023198x_{2} = 102.1116023198
x3=61.2775087154266x_{3} = 61.2775087154266
x4=89.5466202277414x_{4} = 89.5466202277414
x5=33.0174658775265x_{5} = 33.0174658775265
x6=4.93419822854993x_{6} = 4.93419822854993
x7=86.4054381545562x_{7} = 86.4054381545562
x8=−20.4680078422429x_{8} = -20.4680078422429
x9=−58.1365166573738x_{9} = -58.1365166573738
x10=−98.9702215094204x_{10} = -98.9702215094204
x11=14.2099775813926x_{11} = 14.2099775813926
x12=51.8557483406994x_{12} = 51.8557483406994
x13=−83.2641430354848x_{13} = -83.2641430354848
x14=36.1563536592178x_{14} = 36.1563536592178
x15=17.3380791158534x_{15} = 17.3380791158534
x16=70.700078740623x_{16} = 70.700078740623
x17=−86.4053042434102x_{17} = -86.4053042434102
x18=7.98676475119172x_{18} = 7.98676475119172
x19=20.4703846071522x_{19} = 20.4703846071522
x20=92.6878302742345x_{20} = 92.6878302742345
x21=−14.2050661771509x_{21} = -14.2050661771509
x22=−80.123015436615x_{22} = -80.123015436615
x23=−67.5589341430727x_{23} = -67.5589341430727
x24=80.1231711644351x_{24} = 80.1231711644351
x25=−23.6034090301611x_{25} = -23.6034090301611
x26=−95.8289566560771x_{26} = -95.8289566560771
x27=−42.4347877496486x_{27} = -42.4347877496486
x28=64.4182930958041x_{28} = 64.4182930958041
x29=0.247412484885142x_{29} = 0.247412484885142
x30=98.9703235828905x_{30} = 98.9703235828905
x31=−4.89564432915531x_{31} = -4.89564432915531
x32=58.1368123734526x_{32} = 58.1368123734526
x33=45.5752749499286x_{33} = 45.5752749499286
x34=83.2642872382528x_{34} = 83.2642872382528
x35=−73.8408780976001x_{35} = -73.8408780976001
x36=−7.97148100902349x_{36} = -7.97148100902349
x37=−51.8553766970605x_{37} = -51.8553766970605
x38=−26.7402314854239x_{38} = -26.7402314854239
x39=−76.9819255322054x_{39} = -76.9819255322054
x40=−33.0165500205799x_{40} = -33.0165500205799
x41=54.9962192754584x_{41} = 54.9962192754584
x42=−64.4180522161792x_{42} = -64.4180522161792
x43=−48.7150023424838x_{43} = -48.7150023424838
x44=−54.9958888407247x_{44} = -54.9958888407247
x45=11.0897262388501x_{45} = 11.0897262388501
x46=26.7416265193495x_{46} = 26.7416265193495
x47=42.4353425392198x_{47} = 42.4353425392198
x48=−92.6877138973701x_{48} = -92.6877138973701
x49=48.715423408888x_{49} = 48.715423408888
x50=23.6051982121417x_{50} = 23.6051982121417
x51=29.8791548121049x_{51} = 29.8791548121049
x52=−61.2772425220152x_{52} = -61.2772425220152
x53=240.336007491163x_{53} = 240.336007491163
x54=67.5591531543674x_{54} = 67.5591531543674
x55=−45.5747939110765x_{55} = -45.5747939110765
x56=−11.0817037582484x_{56} = -11.0817037582484
x57=−36.1555897201517x_{57} = -36.1555897201517
x58=−89.5464955446878x_{58} = -89.5464955446878
x59=76.9820942237331x_{59} = 76.9820942237331
x60=−70.699878750109x_{60} = -70.699878750109
x61=−17.3347711916489x_{61} = -17.3347711916489
x62=2.12300090681457x_{62} = 2.12300090681457
x63=39.2956785303244x_{63} = 39.2956785303244
x64=−1.95728275422062x_{64} = -1.95728275422062
x65=−39.2950316476879x_{65} = -39.2950316476879
x66=−29.8780368458978x_{66} = -29.8780368458978
x67=73.8410614412353x_{67} = 73.8410614412353

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
[98.9703235828905,∞)\left[98.9703235828905, \infty\right)
Convex at the intervals
(−∞,−98.9702215094204]\left(-\infty, -98.9702215094204\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((−x+12)cos⁡(x)+sin⁡(x))=−∞sign⁡(⟨−1,1⟩)\lim_{x \to -\infty}\left(\left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)}\right) = - \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−∞sign⁡(⟨−1,1⟩)y = - \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→∞((−x+12)cos⁡(x)+sin⁡(x))=∞sign⁡(⟨−1,1⟩)\lim_{x \to \infty}\left(\left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)}\right) = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∞sign⁡(⟨−1,1⟩)y = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1/2 - x)*cos(x) + sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((−x+12)cos⁡(x)+sin⁡(x)x)=⟨−1,1⟩\lim_{x \to -\infty}\left(\frac{\left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=x⟨−1,1⟩y = x \left\langle -1, 1\right\rangle
lim⁡x→∞((−x+12)cos⁡(x)+sin⁡(x)x)=⟨−1,1⟩\lim_{x \to \infty}\left(\frac{\left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)}}{x}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=x⟨−1,1⟩y = x \left\langle -1, 1\right\rangle
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:
(−x+12)cos⁡(x)+sin⁡(x)=(x+12)cos⁡(x)−sin⁡(x)\left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)} = \left(x + \frac{1}{2}\right) \cos{\left(x \right)} - \sin{\left(x \right)}
- No
(−x+12)cos⁡(x)+sin⁡(x)=−(x+12)cos⁡(x)+sin⁡(x)\left(- x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)} = - \left(x + \frac{1}{2}\right) \cos{\left(x \right)} + \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = (0,5-x)cosx+sinx
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