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  • How to use it?

  • Graphing y =:
  • x^2+6x+10
  • x^2+4x+5
  • 16x^3+12x^2-5
  • x^4+2x^2+5

  • Identical expressions

  • one /2sin(x)+ one .7cos(x)
  • 1 divide by 2 sinus of (x) plus 1.7 co sinus of e of (x)
  • one divide by 2 sinus of (x) plus one .7 co sinus of e of (x)
  • 1/2sinx+1.7cosx
  • 1 divide by 2sin(x)+1.7cos(x)

  • Similar expressions

  • 1/2sin(x)-1.7cos(x)
  • 1/2sinx+1.7cosx
Plotting a function graph/ 1/2sin(x)+1.7cos(x)

Graphing y = 1/2sin(x)+1.7cos(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       sin(x)   17*cos(x)
f(x) = ------ + ---------
         2          10
f(x)=sin⁡(x)2+17cos⁡(x)10f{\left(x \right)} = \frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10} 
f = sin(x)/2 + 17*cos(x)/10
The graph of the function
02468-8-6-4-2-10105-5
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:
sin⁡(x)2+17cos⁡(x)10=0\frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−atan⁡(175)x_{1} = - \operatorname{atan}{\left(\frac{17}{5} \right)}
Numerical solution
x1=−23.2758934602061x_{1} = -23.2758934602061
x2=−26.4174861137959x_{2} = -26.4174861137959
x3=55.2639228795387x_{3} = 55.2639228795387
x4=−45.2670420353347x_{4} = -45.2670420353347
x5=−86.107746532002x_{5} = -86.107746532002
x6=−51.5502273425143x_{6} = -51.5502273425143
x7=74.1134788010775x_{7} = 74.1134788010775
x8=−67.2581906104632x_{8} = -67.2581906104632
x9=−38.9838567281551x_{9} = -38.9838567281551
x10=86.6798494154366x_{10} = 86.6798494154366
x11=26.9895889972306x_{11} = 26.9895889972306
x12=1.85684776851221x_{12} = 1.85684776851221
x13=23.8479963436408x_{13} = 23.8479963436408
x14=83.5382567618468x_{14} = 83.5382567618468
x15=−16.9927081530265x_{15} = -16.9927081530265
x16=−57.8334126496939x_{16} = -57.8334126496939
x17=58.4055155331285x_{17} = 58.4055155331285
x18=−64.1165979568734x_{18} = -64.1165979568734
x19=−35.8422640745653x_{19} = -35.8422640745653
x20=−13.8511154994368x_{20} = -13.8511154994368
x21=64.6887008403081x_{21} = 64.6887008403081
x22=33.2727743044101x_{22} = 33.2727743044101
x23=11.2816257292816x_{23} = 11.2816257292816
x24=14.4232183828714x_{24} = 14.4232183828714
x25=42.6975522651795x_{25} = 42.6975522651795
x26=99.2462200297958x_{26} = 99.2462200297958
x27=77.2550714546673x_{27} = 77.2550714546673
x28=−10.709522845847x_{28} = -10.709522845847
x29=8.1400330756918x_{29} = 8.1400330756918
x30=92.9630347226162x_{30} = 92.9630347226162
x31=−95.5325244927714x_{31} = -95.5325244927714
x32=67.8302934938979x_{32} = 67.8302934938979
x33=−32.7006714209755x_{33} = -32.7006714209755
x34=4.99844042210201x_{34} = 4.99844042210201
x35=−79.8245612248224x_{35} = -79.8245612248224
x36=−7.56793019225716x_{36} = -7.56793019225716
x37=−42.1254493817449x_{37} = -42.1254493817449
x38=48.9807375723591x_{38} = 48.9807375723591
x39=−48.4086346889245x_{39} = -48.4086346889245
x40=−76.6829685712326x_{40} = -76.6829685712326
x41=30.1311816508204x_{41} = 30.1311816508204
x42=89.8214420690264x_{42} = 89.8214420690264
x43=52.1223302259489x_{43} = 52.1223302259489
x44=−29.5590787673857x_{44} = -29.5590787673857
x45=96.104627376206x_{45} = 96.104627376206
x46=−89.2493391855918x_{46} = -89.2493391855918
x47=−98.6741171463612x_{47} = -98.6741171463612
x48=165899.940105885x_{48} = 165899.940105885
x49=−92.3909318391816x_{49} = -92.3909318391816
x50=70.9718861474877x_{50} = 70.9718861474877
x51=−60.9750053032837x_{51} = -60.9750053032837
x52=45.8391449187693x_{52} = 45.8391449187693
x53=−4.42633753866737x_{53} = -4.42633753866737
x54=−1.28474488507758x_{54} = -1.28474488507758
x55=39.5559596115897x_{55} = 39.5559596115897
x56=−82.9661538784122x_{56} = -82.9661538784122
x57=−70.399783264053x_{57} = -70.399783264053
x58=−54.6918199961041x_{58} = -54.6918199961041
x59=−73.5413759176428x_{59} = -73.5413759176428
x60=17.5648110364612x_{60} = 17.5648110364612
x61=20.706403690051x_{61} = 20.706403690051
x62=36.4143669579999x_{62} = 36.4143669579999
x63=61.5471081867183x_{63} = 61.5471081867183
x64=80.396664108257x_{64} = 80.396664108257
x65=−20.1343008066163x_{65} = -20.1343008066163
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)/2 + 17*cos(x)/10.
sin⁡(0)2+17cos⁡(0)10\frac{\sin{\left(0 \right)}}{2} + \frac{17 \cos{\left(0 \right)}}{10}
The result:
f(0)=1710f{\left(0 \right)} = \frac{17}{10}
The point:
(0, 17/10)
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
−17sin⁡(x)10+cos⁡(x)2=0- \frac{17 \sin{\left(x \right)}}{10} + \frac{\cos{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=atan⁡(517)x_{1} = \operatorname{atan}{\left(\frac{5}{17} \right)}
The values of the extrema at the points:
               _____ 
             \/ 314  
(atan(5/17), -------)
                10


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:
x1=atan⁡(517)x_{1} = \operatorname{atan}{\left(\frac{5}{17} \right)}
Decreasing at intervals
(−∞,atan⁡(517)]\left(-\infty, \operatorname{atan}{\left(\frac{5}{17} \right)}\right]
Increasing at intervals
[atan⁡(517),∞)\left[\operatorname{atan}{\left(\frac{5}{17} \right)}, \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
−5sin⁡(x)+17cos⁡(x)10=0- \frac{5 \sin{\left(x \right)} + 17 \cos{\left(x \right)}}{10} = 0
Solve this equation
The roots of this equation
x1=−atan⁡(175)x_{1} = - \operatorname{atan}{\left(\frac{17}{5} \right)}

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
(−∞,−atan⁡(175)]\left(-\infty, - \operatorname{atan}{\left(\frac{17}{5} \right)}\right]
Convex at the intervals
[−atan⁡(175),∞)\left[- \operatorname{atan}{\left(\frac{17}{5} \right)}, \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→−∞(sin⁡(x)2+17cos⁡(x)10)=⟨−115,115⟩\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10}\right) = \left\langle - \frac{11}{5}, \frac{11}{5}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−115,115⟩y = \left\langle - \frac{11}{5}, \frac{11}{5}\right\rangle
lim⁡x→∞(sin⁡(x)2+17cos⁡(x)10)=⟨−115,115⟩\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10}\right) = \left\langle - \frac{11}{5}, \frac{11}{5}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−115,115⟩y = \left\langle - \frac{11}{5}, \frac{11}{5}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)/2 + 17*cos(x)/10, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)2+17cos⁡(x)10x)=0\lim_{x \to -\infty}\left(\frac{\frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)2+17cos⁡(x)10x)=0\lim_{x \to \infty}\left(\frac{\frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10}}{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:
sin⁡(x)2+17cos⁡(x)10=−sin⁡(x)2+17cos⁡(x)10\frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10} = - \frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10}
- No
sin⁡(x)2+17cos⁡(x)10=sin⁡(x)2−17cos⁡(x)10\frac{\sin{\left(x \right)}}{2} + \frac{17 \cos{\left(x \right)}}{10} = \frac{\sin{\left(x \right)}}{2} - \frac{17 \cos{\left(x \right)}}{10}
- No
so, the function
not is
neither even, nor odd
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