Mister Exam

Other calculators

Plotting a function graph/ cosx/2+1/2

Graphing y = cosx/2+1/2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x)   1
f(x) = ------ + -
         2      2
f(x)=cos⁡(x)2+12f{\left(x \right)} = \frac{\cos{\left(x \right)}}{2} + \frac{1}{2} 
f = cos(x)/2 + 1/2
The graph of the function
02468-8-6-4-2-101002
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:
cos⁡(x)2+12=0\frac{\cos{\left(x \right)}}{2} + \frac{1}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=−78.5398168194507x_{1} = -78.5398168194507
x2=−21.9911490521325x_{2} = -21.9911490521325
x3=−9.42477752082051x_{3} = -9.42477752082051
x4=59.6902606104322x_{4} = 59.6902606104322
x5=40.8407045792514x_{5} = 40.8407045792514
x6=91.1061865667532x_{6} = 91.1061865667532
x7=65.9734452390837x_{7} = 65.9734452390837
x8=15.7079627593774x_{8} = 15.7079627593774
x9=9.42477748794163x_{9} = 9.42477748794163
x10=−59.6902604578012x_{10} = -59.6902604578012
x11=−28.2743337069329x_{11} = -28.2743337069329
x12=78.5398166181283x_{12} = 78.5398166181283
x13=−65.9734449870253x_{13} = -65.9734449870253
x14=−15.707962774825x_{14} = -15.707962774825
x15=91.1061873718352x_{15} = 91.1061873718352
x16=−15.7079632965989x_{16} = -15.7079632965989
x17=65.9734460390947x_{17} = 65.9734460390947
x18=97.3893717959212x_{18} = 97.3893717959212
x19=28.2743338651796x_{19} = 28.2743338651796
x20=−72.2566311847166x_{20} = -72.2566311847166
x21=−40.8407049008781x_{21} = -40.8407049008781
x22=34.5575197055812x_{22} = 34.5575197055812
x23=−34.5575196658297x_{23} = -34.5575196658297
x24=−47.1238901083229x_{24} = -47.1238901083229
x25=−9.4247781365785x_{25} = -9.4247781365785
x26=78.5398168562347x_{26} = 78.5398168562347
x27=78.5398152766482x_{27} = 78.5398152766482
x28=−21.9911485864417x_{28} = -21.9911485864417
x29=84.8230013636028x_{29} = 84.8230013636028
x30=−1127.83176318906x_{30} = -1127.83176318906
x31=47.1238902162437x_{31} = 47.1238902162437
x32=−47.1238893275319x_{32} = -47.1238893275319
x33=−3.14159217367683x_{33} = -3.14159217367683
x34=72.2566315166773x_{34} = 72.2566315166773
x35=78.5398149750205x_{35} = 78.5398149750205
x36=−53.4070745963886x_{36} = -53.4070745963886
x37=34.5575195449229x_{37} = 34.5575195449229
x38=−84.8230012511693x_{38} = -84.8230012511693
x39=21.9911489072506x_{39} = 21.9911489072506
x40=47.123889410773x_{40} = 47.123889410773
x41=15.7079629803241x_{41} = 15.7079629803241
x42=3.1415922548952x_{42} = 3.1415922548952
x43=9.42477826738203x_{43} = 9.42477826738203
x44=−65.9734461969855x_{44} = -65.9734461969855
x45=28.2743343711514x_{45} = 28.2743343711514
x46=3.14159306054457x_{46} = 3.14159306054457
x47=−28.2743343914215x_{47} = -28.2743343914215
x48=72.2566310277176x_{48} = 72.2566310277176
x49=−91.1061864815274x_{49} = -91.1061864815274
x50=−65.9734457649277x_{50} = -65.9734457649277
x51=−28.2743340989896x_{51} = -28.2743340989896
x52=−78.5398160472843x_{52} = -78.5398160472843
x53=53.407075424589x_{53} = 53.407075424589
x54=−72.2566315419804x_{54} = -72.2566315419804
x55=65.9734457529812x_{55} = 65.9734457529812
x56=72.2566306985x_{56} = 72.2566306985
x57=−72.2566308657983x_{57} = -72.2566308657983
x58=28.2743335663982x_{58} = 28.2743335663982
x59=−3.14159295109225x_{59} = -3.14159295109225
x60=21.9911480932338x_{60} = 21.9911480932338
x61=21.9911485852059x_{61} = 21.9911485852059
x62=53.4070746418597x_{62} = 53.4070746418597
x63=40.8407049800347x_{63} = 40.8407049800347
x64=−34.5575188899093x_{64} = -34.5575188899093
x65=−59.6902599212271x_{65} = -59.6902599212271
x66=−59.6902606928653x_{66} = -59.6902606928653
x67=78.5398161804942x_{67} = 78.5398161804942
x68=−53.407075294995x_{68} = -53.407075294995
x69=59.6902599104079x_{69} = 59.6902599104079
x70=−40.8407049290801x_{70} = -40.8407049290801
x71=−91.106187265474x_{71} = -91.106187265474
x72=−15.7079635641079x_{72} = -15.7079635641079
x73=−21.991148226056x_{73} = -21.991148226056
x74=15.707963957033x_{74} = 15.707963957033
x75=−97.3893717476911x_{75} = -97.3893717476911
x76=34.5575190219169x_{76} = 34.5575190219169
x77=−97.3893716284562x_{77} = -97.3893716284562
x78=40.8407045848602x_{78} = 40.8407045848602
x79=−40.8407040952604x_{79} = -40.8407040952604
x80=−9.42477744529557x_{80} = -9.42477744529557
x81=59.6902600526626x_{81} = 59.6902600526626
x82=40.8407042062167x_{82} = 40.8407042062167
x83=−84.8230020565447x_{83} = -84.8230020565447
x84=53.4070766553897x_{84} = 53.4070766553897
x85=−97.3893724533348x_{85} = -97.3893724533348
x86=−65.9734453607004x_{86} = -65.9734453607004
x87=−53.4070745786761x_{87} = -53.4070745786761
x88=15.7079634518075x_{88} = 15.7079634518075
x89=97.389372581711x_{89} = 97.389372581711
x90=84.8230021335997x_{90} = 84.8230021335997
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)/2 + 1/2.
cos⁡(0)2+12\frac{\cos{\left(0 \right)}}{2} + \frac{1}{2}
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)2=0- \frac{\sin{\left(x \right)}}{2} = 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, 1)

(pi, 0)


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=πx_{1} = \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Increasing at intervals
[0,π]\left[0, \pi\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
−cos⁡(x)2=0- \frac{\cos{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}

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
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \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→−∞(cos⁡(x)2+12)=⟨0,1⟩\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{2} + \frac{1}{2}\right) = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞(cos⁡(x)2+12)=⟨0,1⟩\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{2} + \frac{1}{2}\right) = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)/2 + 1/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)2+12x)=0\lim_{x \to -\infty}\left(\frac{\frac{\cos{\left(x \right)}}{2} + \frac{1}{2}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)2+12x)=0\lim_{x \to \infty}\left(\frac{\frac{\cos{\left(x \right)}}{2} + \frac{1}{2}}{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:
cos⁡(x)2+12=cos⁡(x)2+12\frac{\cos{\left(x \right)}}{2} + \frac{1}{2} = \frac{\cos{\left(x \right)}}{2} + \frac{1}{2}
- Yes
cos⁡(x)2+12=−cos⁡(x)2−12\frac{\cos{\left(x \right)}}{2} + \frac{1}{2} = - \frac{\cos{\left(x \right)}}{2} - \frac{1}{2}
- No
so, the function
is
even
    © Mister Exam – Calculator