Mister Exam
Graphing y = tan(3*x+5)
The solution
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f(x) = tan(3*x + 5)
$$f{\left(x \right)} = \tan{\left(3 x + 5 \right)}$$
f = tan(3*x + 5)
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:
$$\tan{\left(3 x + 5 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{5}{3}$$
Numerical solution
$$x_{1} = 86.2979276338476$$
$$x_{2} = -17.3746299346156$$
$$x_{3} = -78.1120879040183$$
$$x_{4} = -89.6312609671809$$
$$x_{5} = 77.9203472242748$$
$$x_{6} = -16.327432383419$$
$$x_{7} = -58.2153344312829$$
$$x_{8} = 48.59881579077$$
$$x_{9} = 68.4955692635054$$
$$x_{10} = -1.66666666666667$$
$$x_{11} = -23.6578152417952$$
$$x_{12} = 31.8436549716245$$
$$x_{13} = -67.6401123920523$$
$$x_{14} = 18.2300868060687$$
$$x_{15} = 55.9291986491462$$
$$x_{16} = 58.0235937515394$$
$$x_{17} = -45.6489638169238$$
$$x_{18} = 9.85250639649591$$
$$x_{19} = 60.1179888539326$$
$$x_{20} = 73.7315570194884$$
$$x_{21} = 22.4188770108551$$
$$x_{22} = 53.834803546753$$
$$x_{23} = -82.3008781088047$$
$$x_{24} = -36.2241858561544$$
$$x_{25} = 7.75811129410271$$
$$x_{26} = 99.9114957994033$$
$$x_{27} = 0.427728435726529$$
$$x_{28} = 70.5899643658986$$
$$x_{29} = 84.2035325314543$$
$$x_{30} = 75.8259521218816$$
$$x_{31} = 24.5132721132483$$
$$x_{32} = 26.6076672156415$$
$$x_{33} = 2.52212353811972$$
$$x_{34} = -65.5457172896591$$
$$x_{35} = 20.3244819084619$$
$$x_{36} = 94.6755080434203$$
$$x_{37} = 11.9469014988891$$
$$x_{38} = 66.4011741611122$$
$$x_{39} = -27.8466054465816$$
$$x_{40} = 36.0324451764109$$
$$x_{41} = 38.126840278804$$
$$x_{42} = -14.2330372810258$$
$$x_{43} = -60.3097295336761$$
$$x_{44} = -85.4424707623945$$
$$x_{45} = 95.7227055946169$$
$$x_{46} = -51.9321491241034$$
$$x_{47} = -39.3657785097442$$
$$x_{48} = 44.4100255859836$$
$$x_{49} = 82.1091374290611$$
$$x_{50} = 49.6460133419666$$
$$x_{51} = 46.5044206883768$$
$$x_{52} = -12.1386421786326$$
$$x_{53} = -95.9144462743605$$
$$x_{54} = -7.94985197384625$$
$$x_{55} = 5.66371619170952$$
$$x_{56} = -76.0176928016251$$
$$x_{57} = 33.9380500740177$$
$$x_{58} = -98.0088413767537$$
$$x_{59} = -3.76106176905986$$
$$x_{60} = 40.2212353811972$$
$$x_{61} = 42.3156304835904$$
$$x_{62} = 92.5811129410271$$
$$x_{63} = 4.61651864051292$$
$$x_{64} = 88.3923227362407$$
$$x_{65} = 90.4867178386339$$
$$x_{66} = -29.9410005489748$$
$$x_{67} = 80.014742326668$$
$$x_{68} = -49.8377540217102$$
$$x_{69} = -47.743358919317$$
$$x_{70} = -87.5368658647877$$
$$x_{71} = -25.7522103441884$$
$$x_{72} = -41.4601736121374$$
$$x_{73} = 97.8171006970101$$
$$x_{74} = -10.0442470762394$$
$$x_{75} = -5.85545687145306$$
$$x_{76} = -71.8289025968387$$
$$x_{77} = 71.6371619170952$$
$$x_{78} = -19.4690250370088$$
$$x_{79} = 51.7404084443598$$
$$x_{80} = -83.3480756600013$$
$$x_{81} = -69.7345074944455$$
$$x_{82} = -93.8200511719673$$
$$x_{83} = 29.7492598692313$$
$$x_{84} = -34.1297907537612$$
$$x_{85} = -43.5545687145306$$
$$x_{86} = -100.103236479147$$
$$x_{87} = 27.6548647668381$$
$$x_{88} = -38.3185809585476$$
$$x_{89} = -91.7256560695741$$
$$x_{90} = 62.2123839563258$$
$$x_{91} = -61.3569270848727$$
$$x_{92} = -56.1209393288897$$
$$x_{93} = -54.0265442264966$$
$$x_{94} = -63.4513221872659$$
$$x_{95} = -73.9232976992319$$
$$x_{96} = 16.1356917036755$$
$$x_{97} = 14.0412966012823$$
$$x_{98} = -21.563420139402$$
$$x_{99} = -32.035395651368$$
$$x_{100} = 64.306779058719$$
$$x_{101} = -80.2064830064115$$
so we need to solve the equation:
$$\tan{\left(3 x + 5 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{5}{3}$$
Numerical solution
$$x_{1} = 86.2979276338476$$
$$x_{2} = -17.3746299346156$$
$$x_{3} = -78.1120879040183$$
$$x_{4} = -89.6312609671809$$
$$x_{5} = 77.9203472242748$$
$$x_{6} = -16.327432383419$$
$$x_{7} = -58.2153344312829$$
$$x_{8} = 48.59881579077$$
$$x_{9} = 68.4955692635054$$
$$x_{10} = -1.66666666666667$$
$$x_{11} = -23.6578152417952$$
$$x_{12} = 31.8436549716245$$
$$x_{13} = -67.6401123920523$$
$$x_{14} = 18.2300868060687$$
$$x_{15} = 55.9291986491462$$
$$x_{16} = 58.0235937515394$$
$$x_{17} = -45.6489638169238$$
$$x_{18} = 9.85250639649591$$
$$x_{19} = 60.1179888539326$$
$$x_{20} = 73.7315570194884$$
$$x_{21} = 22.4188770108551$$
$$x_{22} = 53.834803546753$$
$$x_{23} = -82.3008781088047$$
$$x_{24} = -36.2241858561544$$
$$x_{25} = 7.75811129410271$$
$$x_{26} = 99.9114957994033$$
$$x_{27} = 0.427728435726529$$
$$x_{28} = 70.5899643658986$$
$$x_{29} = 84.2035325314543$$
$$x_{30} = 75.8259521218816$$
$$x_{31} = 24.5132721132483$$
$$x_{32} = 26.6076672156415$$
$$x_{33} = 2.52212353811972$$
$$x_{34} = -65.5457172896591$$
$$x_{35} = 20.3244819084619$$
$$x_{36} = 94.6755080434203$$
$$x_{37} = 11.9469014988891$$
$$x_{38} = 66.4011741611122$$
$$x_{39} = -27.8466054465816$$
$$x_{40} = 36.0324451764109$$
$$x_{41} = 38.126840278804$$
$$x_{42} = -14.2330372810258$$
$$x_{43} = -60.3097295336761$$
$$x_{44} = -85.4424707623945$$
$$x_{45} = 95.7227055946169$$
$$x_{46} = -51.9321491241034$$
$$x_{47} = -39.3657785097442$$
$$x_{48} = 44.4100255859836$$
$$x_{49} = 82.1091374290611$$
$$x_{50} = 49.6460133419666$$
$$x_{51} = 46.5044206883768$$
$$x_{52} = -12.1386421786326$$
$$x_{53} = -95.9144462743605$$
$$x_{54} = -7.94985197384625$$
$$x_{55} = 5.66371619170952$$
$$x_{56} = -76.0176928016251$$
$$x_{57} = 33.9380500740177$$
$$x_{58} = -98.0088413767537$$
$$x_{59} = -3.76106176905986$$
$$x_{60} = 40.2212353811972$$
$$x_{61} = 42.3156304835904$$
$$x_{62} = 92.5811129410271$$
$$x_{63} = 4.61651864051292$$
$$x_{64} = 88.3923227362407$$
$$x_{65} = 90.4867178386339$$
$$x_{66} = -29.9410005489748$$
$$x_{67} = 80.014742326668$$
$$x_{68} = -49.8377540217102$$
$$x_{69} = -47.743358919317$$
$$x_{70} = -87.5368658647877$$
$$x_{71} = -25.7522103441884$$
$$x_{72} = -41.4601736121374$$
$$x_{73} = 97.8171006970101$$
$$x_{74} = -10.0442470762394$$
$$x_{75} = -5.85545687145306$$
$$x_{76} = -71.8289025968387$$
$$x_{77} = 71.6371619170952$$
$$x_{78} = -19.4690250370088$$
$$x_{79} = 51.7404084443598$$
$$x_{80} = -83.3480756600013$$
$$x_{81} = -69.7345074944455$$
$$x_{82} = -93.8200511719673$$
$$x_{83} = 29.7492598692313$$
$$x_{84} = -34.1297907537612$$
$$x_{85} = -43.5545687145306$$
$$x_{86} = -100.103236479147$$
$$x_{87} = 27.6548647668381$$
$$x_{88} = -38.3185809585476$$
$$x_{89} = -91.7256560695741$$
$$x_{90} = 62.2123839563258$$
$$x_{91} = -61.3569270848727$$
$$x_{92} = -56.1209393288897$$
$$x_{93} = -54.0265442264966$$
$$x_{94} = -63.4513221872659$$
$$x_{95} = -73.9232976992319$$
$$x_{96} = 16.1356917036755$$
$$x_{97} = 14.0412966012823$$
$$x_{98} = -21.563420139402$$
$$x_{99} = -32.035395651368$$
$$x_{100} = 64.306779058719$$
$$x_{101} = -80.2064830064115$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$3 \tan^{2}{\left(3 x + 5 \right)} + 3 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$3 \tan^{2}{\left(3 x + 5 \right)} + 3 = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$18 \left(\tan^{2}{\left(3 x + 5 \right)} + 1\right) \tan{\left(3 x + 5 \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{5}{3}$$
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
$$\left[- \frac{5}{3}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{5}{3}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$18 \left(\tan^{2}{\left(3 x + 5 \right)} + 1\right) \tan{\left(3 x + 5 \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{5}{3}$$
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
$$\left[- \frac{5}{3}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{5}{3}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \tan{\left(3 x + 5 \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(3 x + 5 \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \tan{\left(3 x + 5 \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(3 x + 5 \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(3*x + 5), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x + 5 \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x + 5 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x + 5 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x + 5 \right)}}{x}\right)$$
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:
$$\tan{\left(3 x + 5 \right)} = - \tan{\left(3 x - 5 \right)}$$
- No
$$\tan{\left(3 x + 5 \right)} = \tan{\left(3 x - 5 \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(3 x + 5 \right)} = - \tan{\left(3 x - 5 \right)}$$
- No
$$\tan{\left(3 x + 5 \right)} = \tan{\left(3 x - 5 \right)}$$
- No
so, the function
not is
neither even, nor odd