Mister Exam

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  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • 6x-2x^2
  • (6x^2-x^4)/9
  • -3x+5

  • Identical expressions

  • ln(one +(sqrt(x*sin(x))))
  • ln(1 plus ( square root of (x multiply by sinus of (x))))
  • ln(one plus ( square root of (x multiply by sinus of (x))))
  • ln(1+(√(x*sin(x))))
  • ln(1+(sqrt(xsin(x))))
  • ln1+sqrtxsinx

  • Similar expressions

  • ln(1-(sqrt(x*sin(x))))
  • ln(1+(sqrt(x*sinx)))
Plotting a function graph/ ln(1+(sqrt(x*sin(x))))

Graphing y = ln(1+(sqrt(x*sin(x))))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /      __________\
f(x) = log\1 + \/ x*sin(x) /
f(x)=log⁡(xsin⁡(x)+1)f{\left(x \right)} = \log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)} 
f = log(sqrt(x*sin(x)) + 1)
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:
log⁡(xsin⁡(x)+1)=0\log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(1 + sqrt(x*sin(x))).
log⁡(0sin⁡(0)+1)\log{\left(\sqrt{0 \sin{\left(0 \right)}} + 1 \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
xsin⁡(x)(xcos⁡(x)2+sin⁡(x)2)x(xsin⁡(x)+1)sin⁡(x)=0\frac{\sqrt{x \sin{\left(x \right)}} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{x \left(\sqrt{x \sin{\left(x \right)}} + 1\right) \sin{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=−26.7409160147873x_{1} = -26.7409160147873
x2=−89.5465575382492x_{2} = -89.5465575382492
x3=73.8409691490209x_{3} = 73.8409691490209
x4=−39.295350981473x_{4} = -39.295350981473
x5=70.69997803861x_{5} = 70.69997803861
x6=−2.02875783811043x_{6} = -2.02875783811043
x7=−64.4181717218392x_{7} = -64.4181717218392
x8=−20.469167402741x_{8} = -20.469167402741
x9=−61.2773745335697x_{9} = -61.2773745335697
x10=26.7409160147873x_{10} = 26.7409160147873
x11=61.2773745335697x_{11} = 61.2773745335697
x12=−98.9702722883957x_{12} = -98.9702722883957
x13=−83.2642147040886x_{13} = -83.2642147040886
x14=−51.855560729152x_{14} = -51.855560729152
x15=−86.4053708116885x_{15} = -86.4053708116885
x16=54.9960525574964x_{16} = 54.9960525574964
x17=−92.687771772017x_{17} = -92.687771772017
x18=−58.1366632448992x_{18} = -58.1366632448992
x19=−4.91318043943488x_{19} = -4.91318043943488
x20=23.6042847729804x_{20} = 23.6042847729804
x21=48.7152107175577x_{21} = 48.7152107175577
x22=17.3363779239834x_{22} = 17.3363779239834
x23=95.8290108090195x_{23} = 95.8290108090195
x24=−70.69997803861x_{24} = -70.69997803861
x25=33.0170010333572x_{25} = 33.0170010333572
x26=−36.1559664195367x_{26} = -36.1559664195367
x27=−54.9960525574964x_{27} = -54.9960525574964
x28=76.9820093304187x_{28} = 76.9820093304187
x29=80.1230928148503x_{29} = 80.1230928148503
x30=−76.9820093304187x_{30} = -76.9820093304187
x31=64.4181717218392x_{31} = 64.4181717218392
x32=−23.6042847729804x_{32} = -23.6042847729804
x33=−80.1230928148503x_{33} = -80.1230928148503
x34=98.9702722883957x_{34} = 98.9702722883957
x35=14.2074367251912x_{35} = 14.2074367251912
x36=42.4350618814099x_{36} = 42.4350618814099
x37=51.855560729152x_{37} = 51.855560729152
x38=7.97866571241324x_{38} = 7.97866571241324
x39=−33.0170010333572x_{39} = -33.0170010333572
x40=−67.5590428388084x_{40} = -67.5590428388084
x41=58.1366632448992x_{41} = 58.1366632448992
x42=11.085538406497x_{42} = 11.085538406497
x43=29.8785865061074x_{43} = 29.8785865061074
x44=92.687771772017x_{44} = 92.687771772017
x45=67.5590428388084x_{45} = 67.5590428388084
x46=−45.57503179559x_{46} = -45.57503179559
x47=−73.8409691490209x_{47} = -73.8409691490209
x48=−11.085538406497x_{48} = -11.085538406497
x49=−14.2074367251912x_{49} = -14.2074367251912
x50=−48.7152107175577x_{50} = -48.7152107175577
x51=86.4053708116885x_{51} = 86.4053708116885
x52=−17.3363779239834x_{52} = -17.3363779239834
x53=20.469167402741x_{53} = 20.469167402741
x54=45.57503179559x_{54} = 45.57503179559
x55=−29.8785865061074x_{55} = -29.8785865061074
x56=4.91318043943488x_{56} = 4.91318043943488
x57=−7.97866571241324x_{57} = -7.97866571241324
x58=−42.4350618814099x_{58} = -42.4350618814099
x59=2.02875783811043x_{59} = 2.02875783811043
x60=83.2642147040886x_{60} = 83.2642147040886
x61=89.5465575382492x_{61} = 89.5465575382492
x62=−95.8290108090195x_{62} = -95.8290108090195
x63=39.295350981473x_{63} = 39.295350981473
The values of the extrema at the points:
(-26.74091601478731, 1.81959439703613)

(-89.54655753824919, 2.34780788061745)

(73.8409691490209, 2.15763749166158 + 1.45493941316353*I)

(-39.295350981472986, 1.98342417304979)

(70.69997803861, 2.24155054914526)

(-2.028757838110434, 0.853974674628296)

(-64.41817172183916, 2.20006606067476)

(-20.46916740274095, 1.70866602110739)

(-61.277374533569656, 2.0657335936083 + 1.44372937374428*I)

(26.74091601478731, 1.81959439703613)

(61.277374533569656, 2.0657335936083 + 1.44372937374428*I)

(-98.9702722883957, 2.30241116601255 + 1.47061142377839*I)

(-83.26421470408864, 2.31496748478214)

(-51.85556072915197, 2.10418429767025)

(-86.40537081168854, 2.23524526555628 + 1.46362539911914*I)

(54.99605255749639, 2.01255943226034 + 1.43674906731317*I)

(-92.687771772017, 2.26995505049594 + 1.46729480728072*I)

(-58.13666324489916, 2.15456946051663)

(-4.913180439434884, 0.88017480962742 + 1.14317179398954*I)

(23.604284772980407, 1.60103023087858 + 1.36771465614255*I)

(48.715210717557724, 1.95305226513135 + 1.42847588502425*I)

(17.33637792398336, 1.45365837217991 + 1.3349008968913*I)

(95.82901080901948, 2.37852365290896)

(-70.69997803861, 2.24155054914526)

(33.017001033357246, 1.90876074892496)

(-36.15596641953672, 1.80737615936079 + 1.40596686862376*I)

(-54.99605255749639, 2.01255943226034 + 1.43674906731317*I)

(76.98200933041872, 2.2796817137158)

(80.12309281485025, 2.19794537325366 + 1.45953587048814*I)

(-76.98200933041872, 2.2796817137158)

(64.41817172183916, 2.20006606067476)

(-23.604284772980407, 1.60103023087858 + 1.36771465614255*I)

(-80.12309281485025, 2.19794537325366 + 1.45953587048814*I)

(98.9702722883957, 2.30241116601255 + 1.47061142377839*I)

(14.207436725191188, 1.56121807481243)

(42.43506188140989, 1.88549789835899 + 1.41845426567079*I)

(51.85556072915197, 2.10418429767025)

(7.978665712413241, 1.3385914443644)

(-33.017001033357246, 1.90876074892496)

(-67.5590428388084, 2.11379368769621 + 1.4497218389579*I)

(58.13666324489916, 2.15456946051663)

(11.085538406497022, 1.24414662173076 + 1.27846363206011*I)

(29.878586506107393, 1.71476065844424 + 1.3898029639899*I)

(92.687771772017, 2.26995505049594 + 1.46729480728072*I)

(67.5590428388084, 2.11379368769621 + 1.4497218389579*I)

(-45.57503179559002, 2.04770779974173)

(-73.8409691490209, 2.15763749166158 + 1.45493941316353*I)

(-11.085538406497022, 1.24414662173076 + 1.27846363206011*I)

(-14.207436725191188, 1.56121807481243)

(-48.715210717557724, 1.95305226513135 + 1.42847588502425*I)

(86.40537081168854, 2.23524526555628 + 1.46362539911914*I)

(-17.33637792398336, 1.45365837217991 + 1.3349008968913*I)

(20.46916740274095, 1.70866602110739)

(45.57503179559002, 2.04770779974173)

(-29.878586506107393, 1.71476065844424 + 1.3898029639899*I)

(4.913180439434884, 0.88017480962742 + 1.14317179398954*I)

(-7.978665712413241, 1.3385914443644)

(-42.43506188140989, 1.88549789835899 + 1.41845426567079*I)

(2.028757838110434, 0.853974674628296)

(83.26421470408864, 2.31496748478214)

(89.54655753824919, 2.34780788061745)

(-95.82901080901948, 2.37852365290896)

(39.295350981472986, 1.98342417304979)


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:
The function has no minima
Maxima of the function at points:
x63=−26.7409160147873x_{63} = -26.7409160147873
x63=−89.5465575382492x_{63} = -89.5465575382492
x63=−39.295350981473x_{63} = -39.295350981473
x63=70.69997803861x_{63} = 70.69997803861
x63=−2.02875783811043x_{63} = -2.02875783811043
x63=−64.4181717218392x_{63} = -64.4181717218392
x63=−20.469167402741x_{63} = -20.469167402741
x63=26.7409160147873x_{63} = 26.7409160147873
x63=−83.2642147040886x_{63} = -83.2642147040886
x63=−51.855560729152x_{63} = -51.855560729152
x63=−58.1366632448992x_{63} = -58.1366632448992
x63=95.8290108090195x_{63} = 95.8290108090195
x63=−70.69997803861x_{63} = -70.69997803861
x63=33.0170010333572x_{63} = 33.0170010333572
x63=76.9820093304187x_{63} = 76.9820093304187
x63=−76.9820093304187x_{63} = -76.9820093304187
x63=64.4181717218392x_{63} = 64.4181717218392
x63=14.2074367251912x_{63} = 14.2074367251912
x63=51.855560729152x_{63} = 51.855560729152
x63=7.97866571241324x_{63} = 7.97866571241324
x63=−33.0170010333572x_{63} = -33.0170010333572
x63=58.1366632448992x_{63} = 58.1366632448992
x63=−45.57503179559x_{63} = -45.57503179559
x63=−14.2074367251912x_{63} = -14.2074367251912
x63=20.469167402741x_{63} = 20.469167402741
x63=45.57503179559x_{63} = 45.57503179559
x63=−7.97866571241324x_{63} = -7.97866571241324
x63=2.02875783811043x_{63} = 2.02875783811043
x63=83.2642147040886x_{63} = 83.2642147040886
x63=89.5465575382492x_{63} = 89.5465575382492
x63=−95.8290108090195x_{63} = -95.8290108090195
x63=39.295350981473x_{63} = 39.295350981473
Decreasing at intervals
(−∞,−95.8290108090195]\left(-\infty, -95.8290108090195\right]
Increasing at intervals
[95.8290108090195,∞)\left[95.8290108090195, \infty\right)
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→−∞log⁡(xsin⁡(x)+1)y = \lim_{x \to -\infty} \log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞log⁡(xsin⁡(x)+1)y = \lim_{x \to \infty} \log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(1 + sqrt(x*sin(x))), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(log⁡(xsin⁡(x)+1)x)y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(log⁡(xsin⁡(x)+1)x)y = x \lim_{x \to \infty}\left(\frac{\log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)}}{x}\right)
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:
log⁡(xsin⁡(x)+1)=log⁡(xsin⁡(x)+1)\log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)} = \log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)}
- Yes
log⁡(xsin⁡(x)+1)=−log⁡(xsin⁡(x)+1)\log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)} = - \log{\left(\sqrt{x \sin{\left(x \right)}} + 1 \right)}
- No
so, the function
is
even
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