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 =:
-
y=2x⁴-4x²
- y=0.5cosx-1
- exp(-sqrt(x))
- x^4+x^2-10
- Integral of d{x}:
- exp(-sqrt(x))
- Limit of the function:
-
exp(-sqrt(x))
-
Identical expressions
- exp(-sqrt(x))
- exponent of ( minus square root of (x))
- exp(-√(x))
- exp-sqrtx
-
Similar expressions
- exp(sqrt(x))
Graphing y = exp(-sqrt(x))
The solution
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:
$$e^{- \sqrt{x}} = 0$$
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:
$$e^{- \sqrt{x}} = 0$$
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
$$- \frac{e^{- \sqrt{x}}}{2 \sqrt{x}} = 0$$
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
$$- \frac{e^{- \sqrt{x}}}{2 \sqrt{x}} = 0$$
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
$$\frac{\left(\frac{1}{x} + \frac{1}{x^{\frac{3}{2}}}\right) e^{- \sqrt{x}}}{4} = 0$$
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
$$\frac{\left(\frac{1}{x} + \frac{1}{x^{\frac{3}{2}}}\right) e^{- \sqrt{x}}}{4} = 0$$
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
Limit on the left could not be calculated
$$\lim_{x \to -\infty} e^{- \sqrt{x}}$$
$$\lim_{x \to \infty} e^{- \sqrt{x}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Limit on the left could not be calculated
$$\lim_{x \to -\infty} e^{- \sqrt{x}}$$
$$\lim_{x \to \infty} e^{- \sqrt{x}} = 0$$
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 exp(-sqrt(x)), divided by x at x->+oo and x ->-oo
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{x}\right) = 0$$
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:
$$e^{- \sqrt{x}} = e^{- \sqrt{- x}}$$
- No
$$e^{- \sqrt{x}} = - e^{- \sqrt{- x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$e^{- \sqrt{x}} = e^{- \sqrt{- x}}$$
- No
$$e^{- \sqrt{x}} = - e^{- \sqrt{- x}}$$
- No
so, the function
not is
neither even, nor odd