Mister Exam
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-
How to use it?
- Graphing y =:
- (x-3)2
- x-(1/x)
- cbrt((x+6)x^2)
- 3x-2x
- Limit of the function:
-
((3+n)/(5+n))^(4+n)
-
Identical expressions
- ((three +n)/(five +n))^(four +n)
- ((3 plus n) divide by (5 plus n)) to the power of (4 plus n)
- ((three plus n) divide by (five plus n)) to the power of (four plus n)
- ((3+n)/(5+n))(4+n)
- 3+n/5+n4+n
- 3+n/5+n^4+n
- ((3+n) divide by (5+n))^(4+n)
-
Similar expressions
- ((3+n)/(5-n))^(4+n)
- ((3-n)/(5+n))^(4+n)
- ((3+n)/(5+n))^(4-n)
Graphing y = ((3+n)/(5+n))^(4+n)
The solution
You have entered
[src]
4 + n
/3 + n\
f(n) = |-----|
\5 + n/
$$f{\left(n \right)} = \left(\frac{n + 3}{n + 5}\right)^{n + 4}$$
f = ((n + 3)/(n + 5))^(n + 4)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$n_{1} = -5$$
$$n_{1} = -5$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis N at f = 0
so we need to solve the equation:
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} = 0$$
Solve this equation
The points of intersection with the axis N:
Analytical solution
$$n_{1} = -3$$
Numerical solution
$$n_{1} = -3$$
so we need to solve the equation:
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} = 0$$
Solve this equation
The points of intersection with the axis N:
Analytical solution
$$n_{1} = -3$$
Numerical solution
$$n_{1} = -3$$
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
the second derivative
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} \left(\left(\log{\left(\frac{n + 3}{n + 5} \right)} - \frac{\left(n + 4\right) \left(\frac{n + 3}{n + 5} - 1\right)}{n + 3}\right)^{2} + \frac{\left(\frac{n + 3}{n + 5} - 1\right) \left(\frac{n + 4}{n + 5} - 2 + \frac{n + 4}{n + 3}\right)}{n + 3}\right) = 0$$
Solve this equation
The roots of this equation
$$n_{1} = 8953.45638131695$$
$$n_{2} = 6990.73796438931$$
$$n_{3} = 4373.6596132548$$
$$n_{4} = 11134.2189307864$$
$$n_{5} = -9799.99436668191$$
$$n_{6} = 2846.83074353814$$
$$n_{7} = -1293.39917751243$$
$$n_{8} = 3283.09954620974$$
$$n_{9} = 9607.68801686063$$
$$n_{10} = -5002.17923484701$$
$$n_{11} = 1537.58794963782$$
$$n_{12} = -7619.21149442168$$
$$n_{13} = 5246.04516533381$$
$$n_{14} = -5220.27251953174$$
$$n_{15} = -10018.0707238391$$
$$n_{16} = 2410.51553132971$$
$$n_{17} = 8735.37847256364$$
$$n_{18} = -2384.7302362334$$
$$n_{19} = -8927.68594341854$$
$$n_{20} = -2602.89764731662$$
$$n_{21} = 4591.75972609043$$
$$n_{22} = 8299.221444947$$
$$n_{23} = -8273.45081819186$$
$$n_{24} = -9363.84079584295$$
$$n_{25} = 7863.06258019045$$
$$n_{26} = 5900.31403584993$$
$$n_{27} = 10479.9927977763$$
$$n_{28} = -1074.89269595133$$
$$n_{29} = -4565.98605174309$$
$$n_{30} = 3719.338263495$$
$$n_{31} = -6310.71502417259$$
$$n_{32} = -3693.56228024841$$
$$n_{33} = -2166.54320447881$$
$$n_{34} = 5682.22590017908$$
$$n_{35} = -3911.67420407468$$
$$n_{36} = -9581.91773051752$$
$$n_{37} = 3501.22196302389$$
$$n_{38} = 3937.44946356408$$
$$n_{39} = -2821.05000030132$$
$$n_{40} = -5874.54209509103$$
$$n_{41} = 5464.13633602907$$
$$n_{42} = -1948.32993749252$$
$$n_{43} = -6964.96678817716$$
$$n_{44} = 6554.57129727954$$
$$n_{45} = 10916.1437587139$$
$$n_{46} = -3039.19054446003$$
$$n_{47} = -856.161552216675$$
$$n_{48} = -7837.2918004592$$
$$n_{49} = 6118.40089572796$$
$$n_{50} = 1319.22247608176$$
$$n_{51} = 10043.8409251966$$
$$n_{52} = 2192.33186781124$$
$$n_{53} = -9145.76354129416$$
$$n_{54} = -8491.52961182334$$
$$n_{55} = -10454.2226710028$$
$$n_{56} = -3475.4451159761$$
$$n_{57} = 8081.14226079583$$
$$n_{58} = 1100.73994794893$$
$$n_{59} = 3064.96970801176$$
$$n_{60} = 11352.2939130549$$
$$n_{61} = 882.053205892724$$
$$n_{62} = -5438.36395360976$$
$$n_{63} = 6772.65505256399$$
$$n_{64} = 4809.85713063223$$
$$n_{65} = 4155.55636593842$$
$$n_{66} = 6336.48661148625$$
$$n_{67} = 5027.95217914489$$
$$n_{68} = -8055.37156065333$$
$$n_{69} = -1511.77903829806$$
$$n_{70} = -6528.79986077198$$
$$n_{71} = 9171.53392505234$$
$$n_{72} = -10236.1468198291$$
$$n_{73} = -3257.32165669836$$
$$n_{74} = 10698.0683852336$$
$$n_{75} = -5656.45375132884$$
$$n_{76} = -8709.60797641326$$
$$n_{77} = 2628.68038213321$$
$$n_{78} = -6092.6291411581$$
$$n_{79} = 9389.61112919456$$
$$n_{80} = -4347.88548902282$$
$$n_{81} = -4129.78171888164$$
$$n_{82} = -4784.08384619398$$
$$n_{83} = -7183.04904582307$$
$$n_{84} = 10261.9169827044$$
$$n_{85} = 7208.82010927489$$
$$n_{86} = -7401.1305942593$$
$$n_{87} = 1974.12315685105$$
$$n_{88} = 9825.76460911587$$
$$n_{89} = -6746.88375250113$$
$$n_{90} = 7426.90155475542$$
$$n_{91} = 8517.30017076459$$
$$n_{92} = 7644.98236066206$$
$$n_{93} = 1755.88006355868$$
$$n_{94} = -1730.08046769178$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$n_{1} = -5$$
$$\lim_{n \to -5^-}\left(\left(\frac{n + 3}{n + 5}\right)^{n + 4} \left(\left(\log{\left(\frac{n + 3}{n + 5} \right)} - \frac{\left(n + 4\right) \left(\frac{n + 3}{n + 5} - 1\right)}{n + 3}\right)^{2} + \frac{\left(\frac{n + 3}{n + 5} - 1\right) \left(\frac{n + 4}{n + 5} - 2 + \frac{n + 4}{n + 3}\right)}{n + 3}\right)\right) = 0$$
$$\lim_{n \to -5^+}\left(\left(\frac{n + 3}{n + 5}\right)^{n + 4} \left(\left(\log{\left(\frac{n + 3}{n + 5} \right)} - \frac{\left(n + 4\right) \left(\frac{n + 3}{n + 5} - 1\right)}{n + 3}\right)^{2} + \frac{\left(\frac{n + 3}{n + 5} - 1\right) \left(\frac{n + 4}{n + 5} - 2 + \frac{n + 4}{n + 3}\right)}{n + 3}\right)\right) = 0$$
- 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
$$\left[11352.2939130549, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -9581.91773051752\right]$$
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d n^{2}} f{\left(n \right)} = $$
the second derivative
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} \left(\left(\log{\left(\frac{n + 3}{n + 5} \right)} - \frac{\left(n + 4\right) \left(\frac{n + 3}{n + 5} - 1\right)}{n + 3}\right)^{2} + \frac{\left(\frac{n + 3}{n + 5} - 1\right) \left(\frac{n + 4}{n + 5} - 2 + \frac{n + 4}{n + 3}\right)}{n + 3}\right) = 0$$
Solve this equation
The roots of this equation
$$n_{1} = 8953.45638131695$$
$$n_{2} = 6990.73796438931$$
$$n_{3} = 4373.6596132548$$
$$n_{4} = 11134.2189307864$$
$$n_{5} = -9799.99436668191$$
$$n_{6} = 2846.83074353814$$
$$n_{7} = -1293.39917751243$$
$$n_{8} = 3283.09954620974$$
$$n_{9} = 9607.68801686063$$
$$n_{10} = -5002.17923484701$$
$$n_{11} = 1537.58794963782$$
$$n_{12} = -7619.21149442168$$
$$n_{13} = 5246.04516533381$$
$$n_{14} = -5220.27251953174$$
$$n_{15} = -10018.0707238391$$
$$n_{16} = 2410.51553132971$$
$$n_{17} = 8735.37847256364$$
$$n_{18} = -2384.7302362334$$
$$n_{19} = -8927.68594341854$$
$$n_{20} = -2602.89764731662$$
$$n_{21} = 4591.75972609043$$
$$n_{22} = 8299.221444947$$
$$n_{23} = -8273.45081819186$$
$$n_{24} = -9363.84079584295$$
$$n_{25} = 7863.06258019045$$
$$n_{26} = 5900.31403584993$$
$$n_{27} = 10479.9927977763$$
$$n_{28} = -1074.89269595133$$
$$n_{29} = -4565.98605174309$$
$$n_{30} = 3719.338263495$$
$$n_{31} = -6310.71502417259$$
$$n_{32} = -3693.56228024841$$
$$n_{33} = -2166.54320447881$$
$$n_{34} = 5682.22590017908$$
$$n_{35} = -3911.67420407468$$
$$n_{36} = -9581.91773051752$$
$$n_{37} = 3501.22196302389$$
$$n_{38} = 3937.44946356408$$
$$n_{39} = -2821.05000030132$$
$$n_{40} = -5874.54209509103$$
$$n_{41} = 5464.13633602907$$
$$n_{42} = -1948.32993749252$$
$$n_{43} = -6964.96678817716$$
$$n_{44} = 6554.57129727954$$
$$n_{45} = 10916.1437587139$$
$$n_{46} = -3039.19054446003$$
$$n_{47} = -856.161552216675$$
$$n_{48} = -7837.2918004592$$
$$n_{49} = 6118.40089572796$$
$$n_{50} = 1319.22247608176$$
$$n_{51} = 10043.8409251966$$
$$n_{52} = 2192.33186781124$$
$$n_{53} = -9145.76354129416$$
$$n_{54} = -8491.52961182334$$
$$n_{55} = -10454.2226710028$$
$$n_{56} = -3475.4451159761$$
$$n_{57} = 8081.14226079583$$
$$n_{58} = 1100.73994794893$$
$$n_{59} = 3064.96970801176$$
$$n_{60} = 11352.2939130549$$
$$n_{61} = 882.053205892724$$
$$n_{62} = -5438.36395360976$$
$$n_{63} = 6772.65505256399$$
$$n_{64} = 4809.85713063223$$
$$n_{65} = 4155.55636593842$$
$$n_{66} = 6336.48661148625$$
$$n_{67} = 5027.95217914489$$
$$n_{68} = -8055.37156065333$$
$$n_{69} = -1511.77903829806$$
$$n_{70} = -6528.79986077198$$
$$n_{71} = 9171.53392505234$$
$$n_{72} = -10236.1468198291$$
$$n_{73} = -3257.32165669836$$
$$n_{74} = 10698.0683852336$$
$$n_{75} = -5656.45375132884$$
$$n_{76} = -8709.60797641326$$
$$n_{77} = 2628.68038213321$$
$$n_{78} = -6092.6291411581$$
$$n_{79} = 9389.61112919456$$
$$n_{80} = -4347.88548902282$$
$$n_{81} = -4129.78171888164$$
$$n_{82} = -4784.08384619398$$
$$n_{83} = -7183.04904582307$$
$$n_{84} = 10261.9169827044$$
$$n_{85} = 7208.82010927489$$
$$n_{86} = -7401.1305942593$$
$$n_{87} = 1974.12315685105$$
$$n_{88} = 9825.76460911587$$
$$n_{89} = -6746.88375250113$$
$$n_{90} = 7426.90155475542$$
$$n_{91} = 8517.30017076459$$
$$n_{92} = 7644.98236066206$$
$$n_{93} = 1755.88006355868$$
$$n_{94} = -1730.08046769178$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$n_{1} = -5$$
$$\lim_{n \to -5^-}\left(\left(\frac{n + 3}{n + 5}\right)^{n + 4} \left(\left(\log{\left(\frac{n + 3}{n + 5} \right)} - \frac{\left(n + 4\right) \left(\frac{n + 3}{n + 5} - 1\right)}{n + 3}\right)^{2} + \frac{\left(\frac{n + 3}{n + 5} - 1\right) \left(\frac{n + 4}{n + 5} - 2 + \frac{n + 4}{n + 3}\right)}{n + 3}\right)\right) = 0$$
$$\lim_{n \to -5^+}\left(\left(\frac{n + 3}{n + 5}\right)^{n + 4} \left(\left(\log{\left(\frac{n + 3}{n + 5} \right)} - \frac{\left(n + 4\right) \left(\frac{n + 3}{n + 5} - 1\right)}{n + 3}\right)^{2} + \frac{\left(\frac{n + 3}{n + 5} - 1\right) \left(\frac{n + 4}{n + 5} - 2 + \frac{n + 4}{n + 3}\right)}{n + 3}\right)\right) = 0$$
- 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
$$\left[11352.2939130549, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -9581.91773051752\right]$$
Vertical asymptotes
Have:
$$n_{1} = -5$$
$$n_{1} = -5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at n->+oo and n->-oo
$$\lim_{n \to -\infty} \left(\frac{n + 3}{n + 5}\right)^{n + 4} = e^{-2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = e^{-2}$$
$$\lim_{n \to \infty} \left(\frac{n + 3}{n + 5}\right)^{n + 4} = e^{-2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = e^{-2}$$
$$\lim_{n \to -\infty} \left(\frac{n + 3}{n + 5}\right)^{n + 4} = e^{-2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = e^{-2}$$
$$\lim_{n \to \infty} \left(\frac{n + 3}{n + 5}\right)^{n + 4} = e^{-2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = e^{-2}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((3 + n)/(5 + n))^(4 + n), divided by n at n->+oo and n ->-oo
$$\lim_{n \to -\infty}\left(\frac{\left(\frac{n + 3}{n + 5}\right)^{n + 4}}{n}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{n \to \infty}\left(\frac{\left(\frac{n + 3}{n + 5}\right)^{n + 4}}{n}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{n \to -\infty}\left(\frac{\left(\frac{n + 3}{n + 5}\right)^{n + 4}}{n}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{n \to \infty}\left(\frac{\left(\frac{n + 3}{n + 5}\right)^{n + 4}}{n}\right) = 0$$
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(-n) и f = -f(-n).
So, check:
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} = \left(\frac{3 - n}{5 - n}\right)^{4 - n}$$
- No
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} = - \left(\frac{3 - n}{5 - n}\right)^{4 - n}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} = \left(\frac{3 - n}{5 - n}\right)^{4 - n}$$
- No
$$\left(\frac{n + 3}{n + 5}\right)^{n + 4} = - \left(\frac{3 - n}{5 - n}\right)^{4 - n}$$
- No
so, the function
not is
neither even, nor odd