Mister Exam

Other calculators

Plotting a function graph/ cosx/((x-2)(x+10))

Graphing y = cosx/((x-2)(x+10))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            cos(x)     
f(x) = ----------------
       (x - 2)*(x + 10)
f(x)=cos⁡(x)(x−2)(x+10)f{\left(x \right)} = \frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} 
f = cos(x)/(((x - 2)*(x + 10)))
The domain of the function
The points at which the function is not precisely defined:
x1=−10x_{1} = -10
x2=2x_{2} = 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:
cos⁡(x)(x−2)(x+10)=0\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=4.71238898038469x_{1} = 4.71238898038469
x2=17.2787595947439x_{2} = 17.2787595947439
x3=−89.5353906273091x_{3} = -89.5353906273091
x4=64.4026493985908x_{4} = 64.4026493985908
x5=70.6858347057703x_{5} = 70.6858347057703
x6=36.1283155162826x_{6} = 36.1283155162826
x7=−98.9601685880785x_{7} = -98.9601685880785
x8=48.6946861306418x_{8} = 48.6946861306418
x9=−58.1194640914112x_{9} = -58.1194640914112
x10=7.85398163397448x_{10} = 7.85398163397448
x11=39.2699081698724x_{11} = 39.2699081698724
x12=−95.8185759344887x_{12} = -95.8185759344887
x13=−1.5707963267949x_{13} = -1.5707963267949
x14=−92.6769832808989x_{14} = -92.6769832808989
x15=158.650429006285x_{15} = 158.650429006285
x16=−23.5619449019235x_{16} = -23.5619449019235
x17=23.5619449019235x_{17} = 23.5619449019235
x18=61.261056745001x_{18} = 61.261056745001
x19=29.845130209103x_{19} = 29.845130209103
x20=−32.9867228626928x_{20} = -32.9867228626928
x21=−51.8362787842316x_{21} = -51.8362787842316
x22=−80.1106126665397x_{22} = -80.1106126665397
x23=−83.2522053201295x_{23} = -83.2522053201295
x24=67.5442420521806x_{24} = 67.5442420521806
x25=98.9601685880785x_{25} = 98.9601685880785
x26=92.6769832808989x_{26} = 92.6769832808989
x27=−39.2699081698724x_{27} = -39.2699081698724
x28=−717.853921345268x_{28} = -717.853921345268
x29=86.3937979737193x_{29} = 86.3937979737193
x30=45.553093477052x_{30} = 45.553093477052
x31=−67.5442420521806x_{31} = -67.5442420521806
x32=51.8362787842316x_{32} = 51.8362787842316
x33=76.9690200129499x_{33} = 76.9690200129499
x34=−26.7035375555132x_{34} = -26.7035375555132
x35=−4.71238898038469x_{35} = -4.71238898038469
x36=95.8185759344887x_{36} = 95.8185759344887
x37=−86.3937979737193x_{37} = -86.3937979737193
x38=−36.1283155162826x_{38} = -36.1283155162826
x39=83.2522053201295x_{39} = 83.2522053201295
x40=−7.85398163397448x_{40} = -7.85398163397448
x41=−14.1371669411541x_{41} = -14.1371669411541
x42=−17.2787595947439x_{42} = -17.2787595947439
x43=20.4203522483337x_{43} = 20.4203522483337
x44=54.9778714378214x_{44} = 54.9778714378214
x45=−70.6858347057703x_{45} = -70.6858347057703
x46=−48.6946861306418x_{46} = -48.6946861306418
x47=−54.9778714378214x_{47} = -54.9778714378214
x48=−45.553093477052x_{48} = -45.553093477052
x49=−136.659280431156x_{49} = -136.659280431156
x50=14.1371669411541x_{50} = 14.1371669411541
x51=−73.8274273593601x_{51} = -73.8274273593601
x52=26.7035375555132x_{52} = 26.7035375555132
x53=89.5353906273091x_{53} = 89.5353906273091
x54=10.9955742875643x_{54} = 10.9955742875643
x55=174.358392274234x_{55} = 174.358392274234
x56=359.712358836031x_{56} = 359.712358836031
x57=80.1106126665397x_{57} = 80.1106126665397
x58=73.8274273593601x_{58} = 73.8274273593601
x59=58.1194640914112x_{59} = 58.1194640914112
x60=−61.261056745001x_{60} = -61.261056745001
x61=−7858.69402295487x_{61} = -7858.69402295487
x62=−20.4203522483337x_{62} = -20.4203522483337
x63=−42.4115008234622x_{63} = -42.4115008234622
x64=32.9867228626928x_{64} = 32.9867228626928
x65=42.4115008234622x_{65} = 42.4115008234622
x66=−76.9690200129499x_{66} = -76.9690200129499
x67=−64.4026493985908x_{67} = -64.4026493985908
x68=−29.845130209103x_{68} = -29.845130209103
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)/(((x - 2)*(x + 10))).
cos⁡(0)(−1)2⋅10\frac{\cos{\left(0 \right)}}{\left(-1\right) 2 \cdot 10}
The result:
f(0)=−120f{\left(0 \right)} = - \frac{1}{20}
The point:
(0, -1/20)
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
(−2x−8)cos⁡(x)(x−2)2(x+10)2−1(x−2)(x+10)sin⁡(x)=0\frac{\left(- 2 x - 8\right) \cos{\left(x \right)}}{\left(x - 2\right)^{2} \left(x + 10\right)^{2}} - \frac{1}{\left(x - 2\right) \left(x + 10\right)} \sin{\left(x \right)} = 0
Solve this equation
Solutions are not found,
function may have no extrema
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)+4(x+4)sin⁡(x)(x−2)(x+10)+2((x+4)(1x+10+1x−2)+x+4x+10−1+x+4x−2)cos⁡(x)(x−2)(x+10)(x−2)(x+10)=0\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)} = 0
Solve this equation
The roots of this equation
x1=45.4710070173758x_{1} = 45.4710070173758
x2=−86.3449576389006x_{2} = -86.3449576389006
x3=−42.3043930998253x_{3} = -42.3043930998253
x4=−26.5119224835866x_{4} = -26.5119224835866
x5=−83.2014041049884x_{5} = -83.2014041049884
x6=−58.0445093013385x_{6} = -58.0445093013385
x7=−190.044832546368x_{7} = -190.044832546368
x8=−89.4883646207564x_{8} = -89.4883646207564
x9=29.7225546767569x_{9} = 29.7225546767569
x10=−67.4806517839074x_{10} = -67.4806517839074
x11=−39.1526319974029x_{11} = -39.1526319974029
x12=54.9092447837187x_{12} = 54.9092447837187
x13=−76.9137795411141x_{13} = -76.9137795411141
x14=−70.6252971266199x_{14} = -70.6252971266199
x15=42.3236490425477x_{15} = 42.3236490425477
x16=89.4924261335424x_{16} = 89.4924261335424
x17=−92.6316409673177x_{17} = -92.6316409673177
x18=64.3436583352986x_{18} = 64.3436583352986
x19=23.408490381794x_{19} = 23.408490381794
x20=7.35769984295323x_{20} = 7.35769984295323
x21=76.9193095879542x_{21} = 76.9193095879542
x22=−29.6801615396439x_{22} = -29.6801615396439
x23=−32.8416196600561x_{23} = -32.8416196600561
x24=92.6354273858147x_{24} = 92.6354273858147
x25=−13.3916037185683x_{25} = -13.3916037185683
x26=95.7783388455168x_{26} = 95.7783388455168
x27=51.7636909463359x_{27} = 51.7636909463359
x28=−51.7511406381892x_{28} = -51.7511406381892
x29=−64.3356786510842x_{29} = -64.3356786510842
x30=26.5673194144507x_{30} = 26.5673194144507
x31=−16.8781399543188x_{31} = -16.8781399543188
x32=80.062780911349x_{32} = 80.062780911349
x33=17.071740289959x_{33} = 17.071740289959
x34=−20.1312433663008x_{34} = -20.1312433663008
x35=83.2061146839006x_{35} = 83.2061146839006
x36=98.9211688644511x_{36} = 98.9211688644511
x37=−4.80963959200728x_{37} = -4.80963959200728
x38=32.8752374467871x_{38} = 32.8752374467871
x39=−61.1903212799701x_{39} = -61.1903212799701
x40=20.2443394322777x_{40} = 20.2443394322777
x41=86.3493255488836x_{41} = 86.3493255488836
x42=−23.3323622844516x_{42} = -23.3323622844516
x43=−98.9178546297908x_{43} = -98.9178546297908
x44=−54.8981537926459x_{44} = -54.8981537926459
x45=39.1754015123576x_{45} = 39.1754015123576
x46=70.6318829945278x_{46} = 70.6318829945278
x47=61.1991721850654x_{47} = 61.1991721850654
x48=−35.9986561408559x_{48} = -35.9986561408559
x49=−80.0576853892703x_{49} = -80.0576853892703
x50=58.0543844350502x_{50} = 58.0543844350502
x51=73.7756836327888x_{51} = 73.7756836327888
x52=−95.7748003609166x_{52} = -95.7748003609166
x53=1412.14307316361x_{53} = 1412.14307316361
x54=48.6176448144202x_{54} = 48.6176448144202
x55=−48.6033207731632x_{55} = -48.6033207731632
x56=−73.7696605087807x_{56} = -73.7696605087807
x57=10.6659557066639x_{57} = 10.6659557066639
x58=−1.10871099362239x_{58} = -1.10871099362239
x59=36.0260362022415x_{59} = 36.0260362022415
x60=−45.4544954337464x_{60} = -45.4544954337464
x61=67.4878839716098x_{61} = 67.4878839716098
x62=13.884286370083x_{62} = 13.884286370083
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=−10x_{1} = -10
x2=2x_{2} = 2

lim⁡x→−10−(−cos⁡(x)+4(x+4)sin⁡(x)(x−2)(x+10)+2((x+4)(1x+10+1x−2)+x+4x+10−1+x+4x−2)cos⁡(x)(x−2)(x+10)(x−2)(x+10))=−∞\lim_{x \to -10^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = -\infty
lim⁡x→−10+(−cos⁡(x)+4(x+4)sin⁡(x)(x−2)(x+10)+2((x+4)(1x+10+1x−2)+x+4x+10−1+x+4x−2)cos⁡(x)(x−2)(x+10)(x−2)(x+10))=∞\lim_{x \to -10^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = \infty
- the limits are not equal, so
x1=−10x_{1} = -10
- is an inflection point
lim⁡x→2−(−cos⁡(x)+4(x+4)sin⁡(x)(x−2)(x+10)+2((x+4)(1x+10+1x−2)+x+4x+10−1+x+4x−2)cos⁡(x)(x−2)(x+10)(x−2)(x+10))=∞\lim_{x \to 2^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = \infty
lim⁡x→2+(−cos⁡(x)+4(x+4)sin⁡(x)(x−2)(x+10)+2((x+4)(1x+10+1x−2)+x+4x+10−1+x+4x−2)cos⁡(x)(x−2)(x+10)(x−2)(x+10))=−∞\lim_{x \to 2^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = -\infty
- the limits are not equal, so
x2=2x_{2} = 2
- is an inflection point

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
[95.7783388455168,∞)\left[95.7783388455168, \infty\right)
Convex at the intervals
(−∞,−98.9178546297908]\left(-\infty, -98.9178546297908\right]
Vertical asymptotes
Have:
x1=−10x_{1} = -10
x2=2x_{2} = 2
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)(x−2)(x+10))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(cos⁡(x)(x−2)(x+10))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}\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 cos(x)/(((x - 2)*(x + 10))), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(1(x−2)(x+10)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{\left(x - 2\right) \left(x + 10\right)} \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(1(x−2)(x+10)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{\left(x - 2\right) \left(x + 10\right)} \cos{\left(x \right)}}{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)(x−2)(x+10)=cos⁡(x)(10−x)(−x−2)\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = \frac{\cos{\left(x \right)}}{\left(10 - x\right) \left(- x - 2\right)}
- No
cos⁡(x)(x−2)(x+10)=−cos⁡(x)(10−x)(−x−2)\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = - \frac{\cos{\left(x \right)}}{\left(10 - x\right) \left(- x - 2\right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator