Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- -x^4+4x
- x^x
- (x+1)^2/(x^2+2x)
-
x^2+5x
-
Identical expressions
- sqrt((two *(cos(x)^ two))-sqrt(two))
- square root of ((2 multiply by ( co sinus of e of (x) squared )) minus square root of (2))
- square root of ((two multiply by ( co sinus of e of (x) to the power of two)) minus square root of (two))
- √((2*(cos(x)^2))-√(2))
- sqrt((2*(cos(x)2))-sqrt(2))
- sqrt2*cosx2-sqrt2
- sqrt((2*(cos(x)²))-sqrt(2))
- sqrt((2*(cos(x) to the power of 2))-sqrt(2))
- sqrt((2(cos(x)^2))-sqrt(2))
- sqrt((2(cos(x)2))-sqrt(2))
- sqrt2cosx2-sqrt2
- sqrt2cosx^2-sqrt2
-
Similar expressions
- sqrt((2*(cos(x)^2))+sqrt(2))
- sqrt((2*(cosx^2))-sqrt(2))
Graphing y = sqrt((2*(cos(x)^2))-sqrt(2))
The solution
You have entered
[src]
___________________
/ 2 ___
f(x) = \/ 2*cos (x) - \/ 2
$$f{\left(x \right)} = \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}$$
f = sqrt(2*cos(x)^2 - sqrt(2))
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:
$$\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{acos}{\left(- \frac{2^{\frac{3}{4}}}{2} \right)} + 2 \pi$$
$$x_{2} = - \operatorname{acos}{\left(\frac{2^{\frac{3}{4}}}{2} \right)} + 2 \pi$$
$$x_{3} = \operatorname{acos}{\left(- \frac{2^{\frac{3}{4}}}{2} \right)}$$
$$x_{4} = \operatorname{acos}{\left(\frac{2^{\frac{3}{4}}}{2} \right)}$$
Numerical solution
$$x_{1} = 3.713451523791$$
$$x_{2} = 5.71132643697838$$
$$x_{3} = 2.56973378338858$$
$$x_{4} = 0.57185887020121$$
so we need to solve the equation:
$$\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{acos}{\left(- \frac{2^{\frac{3}{4}}}{2} \right)} + 2 \pi$$
$$x_{2} = - \operatorname{acos}{\left(\frac{2^{\frac{3}{4}}}{2} \right)} + 2 \pi$$
$$x_{3} = \operatorname{acos}{\left(- \frac{2^{\frac{3}{4}}}{2} \right)}$$
$$x_{4} = \operatorname{acos}{\left(\frac{2^{\frac{3}{4}}}{2} \right)}$$
Numerical solution
$$x_{1} = 3.713451523791$$
$$x_{2} = 5.71132643697838$$
$$x_{3} = 2.56973378338858$$
$$x_{4} = 0.57185887020121$$
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
$$- \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
The values of the extrema at the points:
Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
$$x_{3} = 0$$
Decreasing at intervals
$$\left(-\infty, 0\right]$$
Increasing at intervals
$$\left[0, \infty\right)$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
The values of the extrema at the points:
___________
/ ___
(0, \/ 2 - \/ 2 )-pi 4 ___ (----, I*\/ 2 ) 2
pi 4 ___ (--, I*\/ 2 ) 2
Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
$$x_{3} = 0$$
Decreasing at intervals
$$\left(-\infty, 0\right]$$
Increasing at intervals
$$\left[0, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
$$\lim_{x \to \infty} \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
$$\lim_{x \to -\infty} \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
$$\lim_{x \to \infty} \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \sqrt{\left\langle 0, 2\right\rangle - \sqrt{2}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(2*cos(x)^2 - sqrt(2)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}}{x}\right) = 0$$
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:
$$\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}$$
- Yes
$$\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = - \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}$$
- No
so, the function
is
even
So, check:
$$\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}$$
- Yes
$$\sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}} = - \sqrt{2 \cos^{2}{\left(x \right)} - \sqrt{2}}$$
- No
so, the function
is
even