Mister Exam
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-
How to use it?
- Graphing y =:
- 3x^2+12x+9
-
2x-9x+12x-8
- 2x^3+1
- 2x^2+8x+5
-
Identical expressions
- one -sin(one / two)x
- 1 minus sinus of (1 divide by 2)x
- one minus sinus of (one divide by two)x
- 1-sin1/2x
- 1-sin(1 divide by 2)x
-
Similar expressions
- 1+sin(1/2)x
Graphing y = 1-sin(1/2)x
The solution
You have entered
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f(x) = 1 - sin(1/2)*x
$$f{\left(x \right)} = - x \sin{\left(\frac{1}{2} \right)} + 1$$
f = -x*sin(1/2) + 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:
$$- x \sin{\left(\frac{1}{2} \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{\sin{\left(\frac{1}{2} \right)}}$$
Numerical solution
$$x_{1} = 2.08582964293349$$
so we need to solve the equation:
$$- x \sin{\left(\frac{1}{2} \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{\sin{\left(\frac{1}{2} \right)}}$$
Numerical solution
$$x_{1} = 2.08582964293349$$
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
$$- \sin{\left(\frac{1}{2} \right)} = 0$$
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
$$- \sin{\left(\frac{1}{2} \right)} = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
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
$$\lim_{x \to -\infty}\left(- x \sin{\left(\frac{1}{2} \right)} + 1\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- x \sin{\left(\frac{1}{2} \right)} + 1\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- x \sin{\left(\frac{1}{2} \right)} + 1\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- x \sin{\left(\frac{1}{2} \right)} + 1\right) = -\infty$$
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 1 - sin(1/2)*x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- x \sin{\left(\frac{1}{2} \right)} + 1}{x}\right) = - \sin{\left(\frac{1}{2} \right)}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x \sin{\left(\frac{1}{2} \right)}$$
$$\lim_{x \to \infty}\left(\frac{- x \sin{\left(\frac{1}{2} \right)} + 1}{x}\right) = - \sin{\left(\frac{1}{2} \right)}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x \sin{\left(\frac{1}{2} \right)}$$
$$\lim_{x \to -\infty}\left(\frac{- x \sin{\left(\frac{1}{2} \right)} + 1}{x}\right) = - \sin{\left(\frac{1}{2} \right)}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x \sin{\left(\frac{1}{2} \right)}$$
$$\lim_{x \to \infty}\left(\frac{- x \sin{\left(\frac{1}{2} \right)} + 1}{x}\right) = - \sin{\left(\frac{1}{2} \right)}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x \sin{\left(\frac{1}{2} \right)}$$
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:
$$- x \sin{\left(\frac{1}{2} \right)} + 1 = x \sin{\left(\frac{1}{2} \right)} + 1$$
- No
$$- x \sin{\left(\frac{1}{2} \right)} + 1 = - x \sin{\left(\frac{1}{2} \right)} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- x \sin{\left(\frac{1}{2} \right)} + 1 = x \sin{\left(\frac{1}{2} \right)} + 1$$
- No
$$- x \sin{\left(\frac{1}{2} \right)} + 1 = - x \sin{\left(\frac{1}{2} \right)} - 1$$
- No
so, the function
not is
neither even, nor odd