Mister Exam
Graphing y = ((e^(x+1))x+e^(x+1))/(x+2)^2
The solution
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x + 1 x + 1
E *x + E
f(x) = -----------------
2
(x + 2)
$$f{\left(x \right)} = \frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}}$$
f = (E^(x + 1)*x + E^(x + 1))/(x + 2)^2
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -2$$
$$x_{1} = -2$$
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:
$$\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
Numerical solution
$$x_{1} = -118.672682588189$$
$$x_{2} = -68.4903272468361$$
$$x_{3} = -114.664789424994$$
$$x_{4} = -96.6197973001996$$
$$x_{5} = -35.9137230448367$$
$$x_{6} = -56.3814476330527$$
$$x_{7} = -1$$
$$x_{8} = -90.6000852288942$$
$$x_{9} = -74.5282683001879$$
$$x_{10} = -60.4241649292603$$
$$x_{11} = -108.651697529499$$
$$x_{12} = -31.6702622256147$$
$$x_{13} = -98.6257441785881$$
$$x_{14} = -50.2989815352984$$
$$x_{15} = -29.4866008532182$$
$$x_{16} = -48.2647379156306$$
$$x_{17} = -120.676407047487$$
$$x_{18} = -64.4599277505068$$
$$x_{19} = -44.1816822874227$$
$$x_{20} = -54.3567986642463$$
$$x_{21} = -76.539275415559$$
$$x_{22} = -70.5038836671442$$
$$x_{23} = -37.9997090392994$$
$$x_{24} = -92.6069918983736$$
$$x_{25} = -110.656242343533$$
$$x_{26} = -27.2218863283251$$
$$x_{27} = -94.6135541594353$$
$$x_{28} = -52.3294686514302$$
$$x_{29} = -102.636830108495$$
$$x_{30} = -42.1305422427734$$
$$x_{31} = -72.5164987518611$$
$$x_{32} = -116.668812945399$$
$$x_{33} = -104.642005111232$$
$$x_{34} = -1$$
$$x_{35} = -86.5851232994608$$
$$x_{36} = -106.646956152332$$
$$x_{37} = -84.5770020080796$$
$$x_{38} = -66.4757184421965$$
$$x_{39} = -62.4428031553755$$
$$x_{40} = -80.5592832130014$$
$$x_{41} = -112.660602621508$$
$$x_{42} = -40.0707362141423$$
$$x_{43} = -78.5495925589064$$
$$x_{44} = -82.5684032417412$$
$$x_{45} = -46.2259710917097$$
$$x_{46} = -88.5928060985669$$
$$x_{47} = -33.8070388825804$$
$$x_{48} = -58.4037989022472$$
$$x_{49} = -100.631415508885$$
so we need to solve the equation:
$$\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
Numerical solution
$$x_{1} = -118.672682588189$$
$$x_{2} = -68.4903272468361$$
$$x_{3} = -114.664789424994$$
$$x_{4} = -96.6197973001996$$
$$x_{5} = -35.9137230448367$$
$$x_{6} = -56.3814476330527$$
$$x_{7} = -1$$
$$x_{8} = -90.6000852288942$$
$$x_{9} = -74.5282683001879$$
$$x_{10} = -60.4241649292603$$
$$x_{11} = -108.651697529499$$
$$x_{12} = -31.6702622256147$$
$$x_{13} = -98.6257441785881$$
$$x_{14} = -50.2989815352984$$
$$x_{15} = -29.4866008532182$$
$$x_{16} = -48.2647379156306$$
$$x_{17} = -120.676407047487$$
$$x_{18} = -64.4599277505068$$
$$x_{19} = -44.1816822874227$$
$$x_{20} = -54.3567986642463$$
$$x_{21} = -76.539275415559$$
$$x_{22} = -70.5038836671442$$
$$x_{23} = -37.9997090392994$$
$$x_{24} = -92.6069918983736$$
$$x_{25} = -110.656242343533$$
$$x_{26} = -27.2218863283251$$
$$x_{27} = -94.6135541594353$$
$$x_{28} = -52.3294686514302$$
$$x_{29} = -102.636830108495$$
$$x_{30} = -42.1305422427734$$
$$x_{31} = -72.5164987518611$$
$$x_{32} = -116.668812945399$$
$$x_{33} = -104.642005111232$$
$$x_{34} = -1$$
$$x_{35} = -86.5851232994608$$
$$x_{36} = -106.646956152332$$
$$x_{37} = -84.5770020080796$$
$$x_{38} = -66.4757184421965$$
$$x_{39} = -62.4428031553755$$
$$x_{40} = -80.5592832130014$$
$$x_{41} = -112.660602621508$$
$$x_{42} = -40.0707362141423$$
$$x_{43} = -78.5495925589064$$
$$x_{44} = -82.5684032417412$$
$$x_{45} = -46.2259710917097$$
$$x_{46} = -88.5928060985669$$
$$x_{47} = -33.8070388825804$$
$$x_{48} = -58.4037989022472$$
$$x_{49} = -100.631415508885$$
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
$$\frac{\left(- 2 x - 4\right) \left(e^{x + 1} x + e^{x + 1}\right)}{\left(x + 2\right)^{4}} + \frac{x e^{x + 1} + 2 e^{x + 1}}{\left(x + 2\right)^{2}} = 0$$
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
$$\frac{\left(- 2 x - 4\right) \left(e^{x + 1} x + e^{x + 1}\right)}{\left(x + 2\right)^{4}} + \frac{x e^{x + 1} + 2 e^{x + 1}}{\left(x + 2\right)^{2}} = 0$$
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
$$\frac{\left(x + \frac{6 \left(x + 1\right)}{\left(x + 2\right)^{2}} - 1\right) e^{x + 1}}{\left(x + 2\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1 - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -2$$
$$\lim_{x \to -2^-}\left(\frac{\left(x + \frac{6 \left(x + 1\right)}{\left(x + 2\right)^{2}} - 1\right) e^{x + 1}}{\left(x + 2\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{\left(x + \frac{6 \left(x + 1\right)}{\left(x + 2\right)^{2}} - 1\right) e^{x + 1}}{\left(x + 2\right)^{2}}\right) = -\infty$$
- 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
$$\left[-1 - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -1 - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\left(x + \frac{6 \left(x + 1\right)}{\left(x + 2\right)^{2}} - 1\right) e^{x + 1}}{\left(x + 2\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1 - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -2$$
$$\lim_{x \to -2^-}\left(\frac{\left(x + \frac{6 \left(x + 1\right)}{\left(x + 2\right)^{2}} - 1\right) e^{x + 1}}{\left(x + 2\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{\left(x + \frac{6 \left(x + 1\right)}{\left(x + 2\right)^{2}} - 1\right) e^{x + 1}}{\left(x + 2\right)^{2}}\right) = -\infty$$
- 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
$$\left[-1 - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -1 - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}\right]$$
Vertical asymptotes
Have:
$$x_{1} = -2$$
$$x_{1} = -2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (E^(x + 1)*x + E^(x + 1))/(x + 2)^2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{x \left(x + 2\right)^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{x \left(x + 2\right)^{2}}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{x \left(x + 2\right)^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{x + 1} x + e^{x + 1}}{x \left(x + 2\right)^{2}}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}} = \frac{- x e^{1 - x} + e^{1 - x}}{\left(2 - x\right)^{2}}$$
- No
$$\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}} = - \frac{- x e^{1 - x} + e^{1 - x}}{\left(2 - x\right)^{2}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}} = \frac{- x e^{1 - x} + e^{1 - x}}{\left(2 - x\right)^{2}}$$
- No
$$\frac{e^{x + 1} x + e^{x + 1}}{\left(x + 2\right)^{2}} = - \frac{- x e^{1 - x} + e^{1 - x}}{\left(2 - x\right)^{2}}$$
- No
so, the function
not is
neither even, nor odd