Mister Exam

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  • How to use it?

  • Graphing y =:
  • x^2+6x+10
  • x^2+4x+5
  • x^2+2x+4
  • -x^2+2x-3
  • Limit of the function:
  • x/cos(x)^2 x/cos(x)^2
  • Integral of d{x}:
  • x/cos(x)^2 x/cos(x)^2

  • Identical expressions

  • x/cos(x)^ two
  • x divide by co sinus of e of (x) squared
  • x divide by co sinus of e of (x) to the power of two
  • x/cos(x)2
  • x/cosx2
  • x/cos(x)²
  • x/cos(x) to the power of 2
  • x/cosx^2
  • x divide by cos(x)^2

  • Similar expressions

  • x/cosx^2
Plotting a function graph/ x/cos(x)^2

Graphing y = x/cos(x)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          x   
f(x) = -------
          2   
       cos (x)
f(x)=xcos⁡2(x)f{\left(x \right)} = \frac{x}{\cos^{2}{\left(x \right)}} 
f = x/cos(x)^2
The graph of the function
0.000.100.200.300.400.500.600.700.800.901.0005
The domain of the function
The points at which the function is not precisely defined:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
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⁡2(x)=0\frac{x}{\cos^{2}{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x/cos(x)^2.
0cos⁡2(0)\frac{0}{\cos^{2}{\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
2xsin⁡(x)cos⁡3(x)+1cos⁡2(x)=0\frac{2 x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{1}{\cos^{2}{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=−37.6858450405302x_{1} = -37.6858450405302
x2=−50.2555336325565x_{2} = -50.2555336325565
x3=2.97508632168828x_{3} = 2.97508632168828
x4=−6.20274981679304x_{4} = -6.20274981679304
x5=−94.2424741940464x_{5} = -94.2424741940464
x6=−47.1132774827275x_{6} = -47.1132774827275
x7=9.37147510585595x_{7} = 9.37147510585595
x8=−87.9589098892909x_{8} = -87.9589098892909
x9=−25.1128337203766x_{9} = -25.1128337203766
x10=−31.4000043168626x_{10} = -31.4000043168626
x11=37.6858450405302x_{11} = 37.6858450405302
x12=−43.9709264903445x_{12} = -43.9709264903445
x13=18.8229989180076x_{13} = 18.8229989180076
x14=91.1006985770946x_{14} = 91.1006985770946
x15=78.5334497119428x_{15} = 78.5334497119428
x16=65.9658661929102x_{16} = 65.9658661929102
x17=31.4000043168626x_{17} = 31.4000043168626
x18=50.2555336325565x_{18} = 50.2555336325565
x19=97.3842380053013x_{19} = 97.3842380053013
x20=−69.1078034322536x_{20} = -69.1078034322536
x21=84.817106677999x_{21} = 84.817106677999
x22=−2.97508632168828x_{22} = -2.97508632168828
x23=−18.8229989180076x_{23} = -18.8229989180076
x24=−28.2566407733299x_{24} = -28.2566407733299
x25=−21.9683925318703x_{25} = -21.9683925318703
x26=12.5264763376692x_{26} = 12.5264763376692
x27=−75.3915917440781x_{27} = -75.3915917440781
x28=72.2497107001058x_{28} = 72.2497107001058
x29=87.9589098892909x_{29} = 87.9589098892909
x30=40.8284587489214x_{30} = 40.8284587489214
x31=6.20274981679304x_{31} = 6.20274981679304
x32=−53.397711687542x_{32} = -53.397711687542
x33=62.8238944845809x_{33} = 62.8238944845809
x34=−15.6760783451944x_{34} = -15.6760783451944
x35=−9.37147510585595x_{35} = -9.37147510585595
x36=59.6818828624266x_{36} = 59.6818828624266
x37=81.6752872670354x_{37} = 81.6752872670354
x38=75.3915917440781x_{38} = 75.3915917440781
x39=94.2424741940464x_{39} = 94.2424741940464
x40=−34.5430455066495x_{40} = -34.5430455066495
x41=53.397711687542x_{41} = 53.397711687542
x42=47.1132774827275x_{42} = 47.1132774827275
x43=−100.525991117835x_{43} = -100.525991117835
x44=−91.1006985770946x_{44} = -91.1006985770946
x45=−40.8284587489214x_{45} = -40.8284587489214
x46=100.525991117835x_{46} = 100.525991117835
x47=−78.5334497119428x_{47} = -78.5334497119428
x48=15.6760783451944x_{48} = 15.6760783451944
x49=56.5398246709304x_{49} = 56.5398246709304
x50=−12.5264763376692x_{50} = -12.5264763376692
x51=−84.817106677999x_{51} = -84.817106677999
x52=43.9709264903445x_{52} = 43.9709264903445
x53=−81.6752872670354x_{53} = -81.6752872670354
x54=−65.9658661929102x_{54} = -65.9658661929102
x55=34.5430455066495x_{55} = 34.5430455066495
x56=−59.6818828624266x_{56} = -59.6818828624266
x57=−72.2497107001058x_{57} = -72.2497107001058
x58=−62.8238944845809x_{58} = -62.8238944845809
x59=−56.5398246709304x_{59} = -56.5398246709304
x60=28.2566407733299x_{60} = 28.2566407733299
x61=21.9683925318703x_{61} = 21.9683925318703
x62=25.1128337203766x_{62} = 25.1128337203766
x63=−97.3842380053013x_{63} = -97.3842380053013
x64=69.1078034322536x_{64} = 69.1078034322536
The values of the extrema at the points:
(-37.68584504053022, -37.6924788310086)

(-50.255533632556485, -50.2605082091241)

(2.9750863216882792, 3.05911749691083)

(-6.202749816793043, -6.2430545215424)

(-94.24247419404638, -94.2451269257593)

(-47.11327748272753, -47.1185838424919)

(9.371475105855954, 9.3981518026594)

(-87.95890988929088, -87.9617521255159)

(-25.112833720376596, -25.1227887896764)

(-31.400004316862624, -31.4079660992075)

(37.68584504053022, 37.6924788310086)

(-43.97092649034452, -43.9766120653359)

(18.822998918007553, 18.8362805423167)

(91.10069857709462, 91.1034427931534)

(78.53344971194282, 78.5366330688553)

(65.96586619291024, 65.9696560317228)

(31.400004316862624, 31.4079660992075)

(50.255533632556485, 50.2605082091241)

(97.38423800530128, 97.3868051558497)

(-69.10780343225363, -69.1114209687341)

(84.817106677999, 84.8200541966045)

(-2.9750863216882792, -3.05911749691083)

(-18.822998918007553, -18.8362805423167)

(-28.256640773329945, -28.2654882510611)

(-21.968392531870297, -21.9797725178951)

(12.5264763376692, 12.5464340650668)

(-75.39159174407808, -75.3949077637325)

(72.24971070010584, 72.2531709215735)

(87.95890988929088, 87.9617521255159)

(40.8284587489214, 40.8345819288714)

(6.202749816793043, 6.2430545215424)

(-53.39771168754203, -53.40239353611)

(62.82389448458093, 62.8278738622055)

(-15.676078345194368, -15.692026211395)

(-9.371475105855954, -9.3981518026594)

(59.681882862426576, 59.6860717383134)

(81.67528726703536, 81.6783481684214)

(75.39159174407808, 75.3949077637325)

(94.24247419404638, 94.2451269257593)

(-34.54304550664949, -34.5502828534536)

(53.39771168754203, 53.40239353611)

(47.11327748272753, 47.1185838424919)

(-100.52599111783519, -100.528478036862)

(-91.10069857709462, -91.1034427931534)

(-40.8284587489214, -40.8345819288714)

(100.52599111783519, 100.528478036862)

(-78.53344971194282, -78.5366330688553)

(15.676078345194368, 15.692026211395)

(56.53982467093041, 56.5442463330324)

(-12.5264763376692, -12.5464340650668)

(-84.817106677999, -84.8200541966045)

(43.97092649034452, 43.9766120653359)

(-81.67528726703536, -81.6783481684214)

(-65.96586619291024, -65.9696560317228)

(34.54304550664949, 34.5502828534536)

(-59.681882862426576, -59.6860717383134)

(-72.24971070010584, -72.2531709215735)

(-62.82389448458093, -62.8278738622055)

(-56.53982467093041, -56.5442463330324)

(28.256640773329945, 28.2654882510611)

(21.968392531870297, 21.9797725178951)

(25.112833720376596, 25.1227887896764)

(-97.38423800530128, -97.3868051558497)

(69.10780343225363, 69.1114209687341)


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=2.97508632168828x_{1} = 2.97508632168828
x2=9.37147510585595x_{2} = 9.37147510585595
x3=37.6858450405302x_{3} = 37.6858450405302
x4=18.8229989180076x_{4} = 18.8229989180076
x5=91.1006985770946x_{5} = 91.1006985770946
x6=78.5334497119428x_{6} = 78.5334497119428
x7=65.9658661929102x_{7} = 65.9658661929102
x8=31.4000043168626x_{8} = 31.4000043168626
x9=50.2555336325565x_{9} = 50.2555336325565
x10=97.3842380053013x_{10} = 97.3842380053013
x11=84.817106677999x_{11} = 84.817106677999
x12=12.5264763376692x_{12} = 12.5264763376692
x13=72.2497107001058x_{13} = 72.2497107001058
x14=87.9589098892909x_{14} = 87.9589098892909
x15=40.8284587489214x_{15} = 40.8284587489214
x16=6.20274981679304x_{16} = 6.20274981679304
x17=62.8238944845809x_{17} = 62.8238944845809
x18=59.6818828624266x_{18} = 59.6818828624266
x19=81.6752872670354x_{19} = 81.6752872670354
x20=75.3915917440781x_{20} = 75.3915917440781
x21=94.2424741940464x_{21} = 94.2424741940464
x22=53.397711687542x_{22} = 53.397711687542
x23=47.1132774827275x_{23} = 47.1132774827275
x24=100.525991117835x_{24} = 100.525991117835
x25=15.6760783451944x_{25} = 15.6760783451944
x26=56.5398246709304x_{26} = 56.5398246709304
x27=43.9709264903445x_{27} = 43.9709264903445
x28=34.5430455066495x_{28} = 34.5430455066495
x29=28.2566407733299x_{29} = 28.2566407733299
x30=21.9683925318703x_{30} = 21.9683925318703
x31=25.1128337203766x_{31} = 25.1128337203766
x32=69.1078034322536x_{32} = 69.1078034322536
Maxima of the function at points:
x32=−37.6858450405302x_{32} = -37.6858450405302
x32=−50.2555336325565x_{32} = -50.2555336325565
x32=−6.20274981679304x_{32} = -6.20274981679304
x32=−94.2424741940464x_{32} = -94.2424741940464
x32=−47.1132774827275x_{32} = -47.1132774827275
x32=−87.9589098892909x_{32} = -87.9589098892909
x32=−25.1128337203766x_{32} = -25.1128337203766
x32=−31.4000043168626x_{32} = -31.4000043168626
x32=−43.9709264903445x_{32} = -43.9709264903445
x32=−69.1078034322536x_{32} = -69.1078034322536
x32=−2.97508632168828x_{32} = -2.97508632168828
x32=−18.8229989180076x_{32} = -18.8229989180076
x32=−28.2566407733299x_{32} = -28.2566407733299
x32=−21.9683925318703x_{32} = -21.9683925318703
x32=−75.3915917440781x_{32} = -75.3915917440781
x32=−53.397711687542x_{32} = -53.397711687542
x32=−15.6760783451944x_{32} = -15.6760783451944
x32=−9.37147510585595x_{32} = -9.37147510585595
x32=−34.5430455066495x_{32} = -34.5430455066495
x32=−100.525991117835x_{32} = -100.525991117835
x32=−91.1006985770946x_{32} = -91.1006985770946
x32=−40.8284587489214x_{32} = -40.8284587489214
x32=−78.5334497119428x_{32} = -78.5334497119428
x32=−12.5264763376692x_{32} = -12.5264763376692
x32=−84.817106677999x_{32} = -84.817106677999
x32=−81.6752872670354x_{32} = -81.6752872670354
x32=−65.9658661929102x_{32} = -65.9658661929102
x32=−59.6818828624266x_{32} = -59.6818828624266
x32=−72.2497107001058x_{32} = -72.2497107001058
x32=−62.8238944845809x_{32} = -62.8238944845809
x32=−56.5398246709304x_{32} = -56.5398246709304
x32=−97.3842380053013x_{32} = -97.3842380053013
Decreasing at intervals
[100.525991117835,∞)\left[100.525991117835, \infty\right)
Increasing at intervals
[−2.97508632168828,2.97508632168828]\left[-2.97508632168828, 2.97508632168828\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
2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x)=0\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
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=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469

lim⁡x→1.5707963267949−(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=6.70421144615668⋅1065\lim_{x \to 1.5707963267949^-}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 6.70421144615668 \cdot 10^{65}
lim⁡x→1.5707963267949+(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=6.70421144615668⋅1065\lim_{x \to 1.5707963267949^+}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 6.70421144615668 \cdot 10^{65}
- limits are equal, then skip the corresponding point
lim⁡x→4.71238898038469−(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=2.48304127635433⋅1064\lim_{x \to 4.71238898038469^-}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 2.48304127635433 \cdot 10^{64}
lim⁡x→4.71238898038469+(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=2.48304127635433⋅1064\lim_{x \to 4.71238898038469^+}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 2.48304127635433 \cdot 10^{64}
- limits are equal, then skip the corresponding 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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Vertical asymptotes
Have:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(xcos⁡2(x))y = \lim_{x \to -\infty}\left(\frac{x}{\cos^{2}{\left(x \right)}}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(xcos⁡2(x))y = \lim_{x \to \infty}\left(\frac{x}{\cos^{2}{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/cos(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞1cos⁡2(x)=⟨0,∞⟩\lim_{x \to -\infty} \frac{1}{\cos^{2}{\left(x \right)}} = \left\langle 0, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨0,∞⟩xy = \left\langle 0, \infty\right\rangle x
lim⁡x→∞1cos⁡2(x)=⟨0,∞⟩\lim_{x \to \infty} \frac{1}{\cos^{2}{\left(x \right)}} = \left\langle 0, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨0,∞⟩xy = \left\langle 0, \infty\right\rangle x
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⁡2(x)=−xcos⁡2(x)\frac{x}{\cos^{2}{\left(x \right)}} = - \frac{x}{\cos^{2}{\left(x \right)}}
- No
xcos⁡2(x)=xcos⁡2(x)\frac{x}{\cos^{2}{\left(x \right)}} = \frac{x}{\cos^{2}{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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