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4*(cos(x)+3*x)/10

  • How to use it?

  • Graphing y =:
  • |x^2+8x+12|
  • (x^2-4)/(2x+5)
  • x^2+2x+4
  • x^2+3

  • Identical expressions

  • four *(cos(x)+ three *x)/ ten
  • 4 multiply by ( co sinus of e of (x) plus 3 multiply by x) divide by 10
  • four multiply by ( co sinus of e of (x) plus three multiply by x) divide by ten
  • 4(cos(x)+3x)/10
  • 4cosx+3x/10
  • 4*(cos(x)+3*x) divide by 10

  • Similar expressions

  • 4*(cos(x)-3*x)/10
  • 4*(cosx+3*x)/10
Plotting a function graph/ 4*(cos(x)+3*x)/10

Graphing y = 4*(cos(x)+3*x)/10

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       4*(cos(x) + 3*x)
f(x) = ----------------
              10
f(x)=4⋅(3x+cos⁡(x))10f{\left(x \right)} = \frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} 
f = 4*(3*x + cos(x))/10
The graph of the function
02468-8-6-4-2-1010-2525
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:
4⋅(3x+cos⁡(x))10=0\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−0.316750828771221x_{1} = -0.316750828771221
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 4*(cos(x) + 3*x)/10.
4⋅(3⋅0+cos⁡(0))10\frac{4 \cdot \left(3 \cdot 0 + \cos{\left(0 \right)}\right)}{10}
The result:
f(0)=25f{\left(0 \right)} = \frac{2}{5}
The point:
(0, 2/5)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
65−2sin⁡(x)5=0\frac{6}{5} - \frac{2 \sin{\left(x \right)}}{5} = 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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−2cos⁡(x)5=0- \frac{2 \cos{\left(x \right)}}{5} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}

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:
Concave at the intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \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→−∞(4⋅(3x+cos⁡(x))10)=−∞\lim_{x \to -\infty}\left(\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(4⋅(3x+cos⁡(x))10)=∞\lim_{x \to \infty}\left(\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10}\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 4*(cos(x) + 3*x)/10, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2⋅(3x+cos⁡(x))5x)=65\lim_{x \to -\infty}\left(\frac{2 \cdot \left(3 x + \cos{\left(x \right)}\right)}{5 x}\right) = \frac{6}{5}
Let's take the limit
so,
inclined asymptote equation on the left:
y=6x5y = \frac{6 x}{5}
lim⁡x→∞(2⋅(3x+cos⁡(x))5x)=65\lim_{x \to \infty}\left(\frac{2 \cdot \left(3 x + \cos{\left(x \right)}\right)}{5 x}\right) = \frac{6}{5}
Let's take the limit
so,
inclined asymptote equation on the right:
y=6x5y = \frac{6 x}{5}
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:
4⋅(3x+cos⁡(x))10=−6x5+2cos⁡(x)5\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = - \frac{6 x}{5} + \frac{2 \cos{\left(x \right)}}{5}
- No
4⋅(3x+cos⁡(x))10=6x5−2cos⁡(x)5\frac{4 \cdot \left(3 x + \cos{\left(x \right)}\right)}{10} = \frac{6 x}{5} - \frac{2 \cos{\left(x \right)}}{5}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = 4*(cos(x)+3*x)/10
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