Mister Exam
Other calculators
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- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- x^3+3x^2-9x
-
x^3-3x+1
- ((x-2)^2)/(x+1)
- x^3-6x^2+9x-3
-
Identical expressions
- cosx−(sqrt(cosx))^ two
- co sinus of e of x−( square root of ( co sinus of e of x)) squared
- co sinus of e of x−( square root of ( co sinus of e of x)) to the power of two
- cosx−(√(cosx))^2
- cosx−(sqrt(cosx))2
- cosx−sqrtcosx2
- cosx−(sqrt(cosx))²
- cosx−(sqrt(cosx)) to the power of 2
- cosx−sqrtcosx^2
Graphing y = cosx−(sqrt(cosx))^2
The solution
You have entered
[src]
2
________
f(x) = cos(x) - \/ cos(x)
$$f{\left(x \right)} = - \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}$$
f = -(sqrt(cos(x)))^2 + cos(x)
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:
$$- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)} = 0$$
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:
$$- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)} = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
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
$$\lim_{x \to -\infty}\left(- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}\right) = 0$$
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 cos(x) - (sqrt(cos(x)))^2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}}{x}\right) = 0$$
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:
$$- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)} = - \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}$$
- Yes
$$- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)} = \left(\sqrt{\cos{\left(x \right)}}\right)^{2} - \cos{\left(x \right)}$$
- No
so, the function
is
even
So, check:
$$- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)} = - \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)}$$
- Yes
$$- \left(\sqrt{\cos{\left(x \right)}}\right)^{2} + \cos{\left(x \right)} = \left(\sqrt{\cos{\left(x \right)}}\right)^{2} - \cos{\left(x \right)}$$
- No
so, the function
is
even