Mister Exam
Graphing y = sinh(x)+sin(x)-1
The solution
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f(x) = sinh(x) + sin(x) - 1
$$f{\left(x \right)} = \left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1$$
f = sin(x) + sinh(x) - 1
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:
$$\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.499740253696375$$
so we need to solve the equation:
$$\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.499740253696375$$
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
$$\cos{\left(x \right)} + \cosh{\left(x \right)} = 0$$
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
$$\cos{\left(x \right)} + \cosh{\left(x \right)} = 0$$
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
$$- \sin{\left(x \right)} + \sinh{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \sin{\left(x \right)} + \sinh{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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:
Have no bends at the whole real axis
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1\right) = \infty$$
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 sinh(x) + sin(x) - 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1}{x}\right) = \infty$$
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:
$$\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1 = - \sin{\left(x \right)} - \sinh{\left(x \right)} - 1$$
- No
$$\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1 = \sin{\left(x \right)} + \sinh{\left(x \right)} + 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1 = - \sin{\left(x \right)} - \sinh{\left(x \right)} - 1$$
- No
$$\left(\sin{\left(x \right)} + \sinh{\left(x \right)}\right) - 1 = \sin{\left(x \right)} + \sinh{\left(x \right)} + 1$$
- No
so, the function
not is
neither even, nor odd