Mister Exam
Graphing y = 75*e^(-((1.893*pi*0.219)/(120*10)*x))
The solution
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/1893*pi \
|-------*219|
| 1000 |
|-----------|
\ 1000 /
- -------------*x
1200
f(x) = 75*E
$$f{\left(x \right)} = 75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}}$$
f = 75*E^(-x*(219*(1893*pi/1000)/1000)/1200)
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:
$$75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- \frac{414567 \pi e^{- \frac{138189 \pi x}{400000000}}}{16000000} = 0$$
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
$$- \frac{414567 \pi e^{- \frac{138189 \pi x}{400000000}}}{16000000} = 0$$
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
$$\frac{57288599163 \pi^{2} e^{- \frac{138189 \pi x}{400000000}}}{6400000000000000} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{57288599163 \pi^{2} e^{- \frac{138189 \pi x}{400000000}}}{6400000000000000} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 75*E^(-((1893*pi/1000)*219/1000)/1200*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{75 e^{- \frac{138189 \pi x}{400000000}}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{75 e^{- \frac{138189 \pi x}{400000000}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{75 e^{- \frac{138189 \pi x}{400000000}}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{75 e^{- \frac{138189 \pi x}{400000000}}}{x}\right) = 0$$
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:
$$75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}} = 75 e^{\frac{138189 \pi x}{400000000}}$$
- No
$$75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}} = - 75 e^{\frac{138189 \pi x}{400000000}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}} = 75 e^{\frac{138189 \pi x}{400000000}}$$
- No
$$75 e^{- x \frac{\frac{219}{1000} \frac{1893 \pi}{1000}}{1200}} = - 75 e^{\frac{138189 \pi x}{400000000}}$$
- No
so, the function
not is
neither even, nor odd