Mister Exam
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- The intersection points of functions
-
How to use it?
- Graphing y =:
-
x^3-12x
- x/(x^2+1)
- -x^2+6x-5
- -x^3-27x
-
Identical expressions
- three cos(pi*x/ four)+ two sin(pi*x/3)-tan(pi*x/2)
- 3 co sinus of e of ( Pi multiply by x divide by 4) plus 2 sinus of ( Pi multiply by x divide by 3) minus tangent of ( Pi multiply by x divide by 2)
- three co sinus of e of ( Pi multiply by x divide by four) plus two sinus of ( Pi multiply by x divide by 3) minus tangent of ( Pi multiply by x divide by 2)
- 3cos(pix/4)+2sin(pix/3)-tan(pix/2)
- 3cospix/4+2sinpix/3-tanpix/2
- 3cos(pi*x divide by 4)+2sin(pi*x divide by 3)-tan(pi*x divide by 2)
-
Similar expressions
- 3cos(pi*x/4)-2sin(pi*x/3)-tan(pi*x/2)
- 3cos(pi*x/4)+2sin(pi*x/3)+tan(pi*x/2)
Graphing y = 3cos(pi*x/4)+2sin(pi*x/3)-tan(pi*x/2)
The solution
You have entered
[src]
/pi*x\ /pi*x\ /pi*x\
f(x) = 3*cos|----| + 2*sin|----| - tan|----|
\ 4 / \ 3 / \ 2 /
$$f{\left(x \right)} = \left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}$$
f = 2*sin((pi*x)/3) + 3*cos((pi*x)/4) - tan((pi*x)/2)
The graph of the function
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:
$$\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 88.4398089158009$$
$$x_{2} = -6$$
$$x_{3} = 86.7268001911554$$
$$x_{4} = -9.27319980884459$$
$$x_{5} = 78$$
$$x_{6} = -66$$
$$x_{7} = -18$$
$$x_{8} = -55.5601910841991$$
$$x_{9} = -57.2731998088446$$
$$x_{10} = -79.5601910841991$$
$$x_{11} = -31.5601910841991$$
$$x_{12} = -78$$
$$x_{13} = -81.2731998088446$$
$$x_{14} = -69.6781382075988$$
$$x_{15} = -30$$
$$x_{16} = 64.4398089158009$$
$$x_{17} = 33.6781382075988$$
$$x_{18} = 246$$
$$x_{19} = 90$$
$$x_{20} = 18$$
$$x_{21} = -62.3218617924012$$
$$x_{22} = 43.5601910841991$$
$$x_{23} = -93.6781382075988$$
$$x_{24} = -28.4398089158009$$
$$x_{25} = 98.3218617924012$$
$$x_{26} = 102$$
$$x_{27} = 150$$
$$x_{28} = 126$$
$$x_{29} = 74.3218617924012$$
$$x_{30} = -54$$
$$x_{31} = 2.32186179240115$$
$$x_{32} = 54$$
$$x_{33} = -52.4398089158009$$
$$x_{34} = 9.67813820759885$$
$$x_{35} = -90$$
$$x_{36} = -100.439808915801$$
$$x_{37} = 91.5601910841991$$
$$x_{38} = 62.7268001911554$$
$$x_{39} = 16.4398089158009$$
$$x_{40} = 57.6781382075988$$
$$x_{41} = 67.5601910841991$$
$$x_{42} = 38.7268001911554$$
$$x_{43} = 30$$
$$x_{44} = -7.56019108419914$$
$$x_{45} = 174$$
$$x_{46} = -14.3218617924011$$
$$x_{47} = 14.7268001911554$$
$$x_{48} = 66$$
$$x_{49} = -4.43980891580086$$
$$x_{50} = -76.4398089158009$$
$$x_{51} = -45.6781382075988$$
$$x_{52} = 222$$
$$x_{53} = 114$$
$$x_{54} = -42$$
$$x_{55} = 26.3218617924011$$
$$x_{56} = -38.3218617924012$$
$$x_{57} = 19.5601910841991$$
$$x_{58} = 6$$
$$x_{59} = 40.4398089158009$$
$$x_{60} = -86.3218617924012$$
$$x_{61} = 42$$
$$x_{62} = 50.3218617924012$$
$$x_{63} = 198$$
$$x_{64} = 81.6781382075988$$
$$x_{65} = -33.2731998088446$$
$$x_{66} = -21.6781382075989$$
so we need to solve the equation:
$$\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 88.4398089158009$$
$$x_{2} = -6$$
$$x_{3} = 86.7268001911554$$
$$x_{4} = -9.27319980884459$$
$$x_{5} = 78$$
$$x_{6} = -66$$
$$x_{7} = -18$$
$$x_{8} = -55.5601910841991$$
$$x_{9} = -57.2731998088446$$
$$x_{10} = -79.5601910841991$$
$$x_{11} = -31.5601910841991$$
$$x_{12} = -78$$
$$x_{13} = -81.2731998088446$$
$$x_{14} = -69.6781382075988$$
$$x_{15} = -30$$
$$x_{16} = 64.4398089158009$$
$$x_{17} = 33.6781382075988$$
$$x_{18} = 246$$
$$x_{19} = 90$$
$$x_{20} = 18$$
$$x_{21} = -62.3218617924012$$
$$x_{22} = 43.5601910841991$$
$$x_{23} = -93.6781382075988$$
$$x_{24} = -28.4398089158009$$
$$x_{25} = 98.3218617924012$$
$$x_{26} = 102$$
$$x_{27} = 150$$
$$x_{28} = 126$$
$$x_{29} = 74.3218617924012$$
$$x_{30} = -54$$
$$x_{31} = 2.32186179240115$$
$$x_{32} = 54$$
$$x_{33} = -52.4398089158009$$
$$x_{34} = 9.67813820759885$$
$$x_{35} = -90$$
$$x_{36} = -100.439808915801$$
$$x_{37} = 91.5601910841991$$
$$x_{38} = 62.7268001911554$$
$$x_{39} = 16.4398089158009$$
$$x_{40} = 57.6781382075988$$
$$x_{41} = 67.5601910841991$$
$$x_{42} = 38.7268001911554$$
$$x_{43} = 30$$
$$x_{44} = -7.56019108419914$$
$$x_{45} = 174$$
$$x_{46} = -14.3218617924011$$
$$x_{47} = 14.7268001911554$$
$$x_{48} = 66$$
$$x_{49} = -4.43980891580086$$
$$x_{50} = -76.4398089158009$$
$$x_{51} = -45.6781382075988$$
$$x_{52} = 222$$
$$x_{53} = 114$$
$$x_{54} = -42$$
$$x_{55} = 26.3218617924011$$
$$x_{56} = -38.3218617924012$$
$$x_{57} = 19.5601910841991$$
$$x_{58} = 6$$
$$x_{59} = 40.4398089158009$$
$$x_{60} = -86.3218617924012$$
$$x_{61} = 42$$
$$x_{62} = 50.3218617924012$$
$$x_{63} = 198$$
$$x_{64} = 81.6781382075988$$
$$x_{65} = -33.2731998088446$$
$$x_{66} = -21.6781382075989$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*cos((pi*x)/4) + 2*sin((pi*x)/3) - tan((pi*x)/2), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)}}{x}\right)$$
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:
$$\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)} = - 2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)} + \tan{\left(\frac{\pi x}{2} \right)}$$
- No
$$\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)} = 2 \sin{\left(\frac{\pi x}{3} \right)} - 3 \cos{\left(\frac{\pi x}{4} \right)} - \tan{\left(\frac{\pi x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)} = - 2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)} + \tan{\left(\frac{\pi x}{2} \right)}$$
- No
$$\left(2 \sin{\left(\frac{\pi x}{3} \right)} + 3 \cos{\left(\frac{\pi x}{4} \right)}\right) - \tan{\left(\frac{\pi x}{2} \right)} = 2 \sin{\left(\frac{\pi x}{3} \right)} - 3 \cos{\left(\frac{\pi x}{4} \right)} - \tan{\left(\frac{\pi x}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd