Mister Exam

Other calculators

Plotting a function graph/ 10sin(2t)

Graphing y = 10sin(2t)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(t) = 10*sin(2*t)
f(t)=10sin⁡(2t)f{\left(t \right)} = 10 \sin{\left(2 t \right)} 
f = 10*sin(2*t)
The graph of the function
02468-8-6-4-2-1010-2020
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis T at f = 0
so we need to solve the equation:
10sin⁡(2t)=010 \sin{\left(2 t \right)} = 0
Solve this equation
The points of intersection with the axis T:

Analytical solution
t1=0t_{1} = 0
t2=π2t_{2} = \frac{\pi}{2}
Numerical solution
t1=4.71238898038469t_{1} = 4.71238898038469
t2=−59.6902604182061t_{2} = -59.6902604182061
t3=7.85398163397448t_{3} = 7.85398163397448
t4=−21.9911485751286t_{4} = -21.9911485751286
t5=−75.398223686155t_{5} = -75.398223686155
t6=−39.2699081698724t_{6} = -39.2699081698724
t7=50.2654824574367t_{7} = 50.2654824574367
t8=95.8185759344887t_{8} = 95.8185759344887
t9=34.5575191894877t_{9} = 34.5575191894877
t10=−86.3937979737193t_{10} = -86.3937979737193
t11=−48.6946861306418t_{11} = -48.6946861306418
t12=−45.553093477052t_{12} = -45.553093477052
t13=26.7035375555132t_{13} = 26.7035375555132
t14=−20.4203522483337t_{14} = -20.4203522483337
t15=0t_{15} = 0
t16=−64.4026493985908t_{16} = -64.4026493985908
t17=43.9822971502571t_{17} = 43.9822971502571
t18=−43.9822971502571t_{18} = -43.9822971502571
t19=48.6946861306418t_{19} = 48.6946861306418
t20=−58.1194640914112t_{20} = -58.1194640914112
t21=23.5619449019235t_{21} = 23.5619449019235
t22=−80.1106126665397t_{22} = -80.1106126665397
t23=−65.9734457253857t_{23} = -65.9734457253857
t24=−83.2522053201295t_{24} = -83.2522053201295
t25=86.3937979737193t_{25} = 86.3937979737193
t26=45.553093477052t_{26} = 45.553093477052
t27=56.5486677646163t_{27} = 56.5486677646163
t28=51.8362787842316t_{28} = 51.8362787842316
t29=−87.9645943005142t_{29} = -87.9645943005142
t30=−14.1371669411541t_{30} = -14.1371669411541
t31=20.4203522483337t_{31} = 20.4203522483337
t32=87.9645943005142t_{32} = 87.9645943005142
t33=59.6902604182061t_{33} = 59.6902604182061
t34=−73.8274273593601t_{34} = -73.8274273593601
t35=31.4159265358979t_{35} = 31.4159265358979
t36=−81.6814089933346t_{36} = -81.6814089933346
t37=72.2566310325652t_{37} = 72.2566310325652
t38=−9.42477796076938t_{38} = -9.42477796076938
t39=−29.845130209103t_{39} = -29.845130209103
t40=−72.2566310325652t_{40} = -72.2566310325652
t41=81.6814089933346t_{41} = 81.6814089933346
t42=−31.4159265358979t_{42} = -31.4159265358979
t43=−483.805268652828t_{43} = -483.805268652828
t44=−89.5353906273091t_{44} = -89.5353906273091
t45=64.4026493985908t_{45} = 64.4026493985908
t46=−94.2477796076938t_{46} = -94.2477796076938
t47=−50.2654824574367t_{47} = -50.2654824574367
t48=28.2743338823081t_{48} = 28.2743338823081
t49=29.845130209103t_{49} = 29.845130209103
t50=−51.8362787842316t_{50} = -51.8362787842316
t51=92.6769832808989t_{51} = 92.6769832808989
t52=−67.5442420521806t_{52} = -67.5442420521806
t53=100.530964914873t_{53} = 100.530964914873
t54=−53.4070751110265t_{54} = -53.4070751110265
t55=94.2477796076938t_{55} = 94.2477796076938
t56=21.9911485751286t_{56} = 21.9911485751286
t57=−36.1283155162826t_{57} = -36.1283155162826
t58=−15.707963267949t_{58} = -15.707963267949
t59=−7.85398163397448t_{59} = -7.85398163397448
t60=65.9734457253857t_{60} = 65.9734457253857
t61=14.1371669411541t_{61} = 14.1371669411541
t62=−6.28318530717959t_{62} = -6.28318530717959
t63=113.097335529233t_{63} = 113.097335529233
t64=89.5353906273091t_{64} = 89.5353906273091
t65=80.1106126665397t_{65} = 80.1106126665397
t66=78.5398163397448t_{66} = 78.5398163397448
t67=15.707963267949t_{67} = 15.707963267949
t68=37.6991118430775t_{68} = 37.6991118430775
t69=42.4115008234622t_{69} = 42.4115008234622
t70=−97.3893722612836t_{70} = -97.3893722612836
t71=70.6858347057703t_{71} = 70.6858347057703
t72=36.1283155162826t_{72} = 36.1283155162826
t73=−1.5707963267949t_{73} = -1.5707963267949
t74=−95.8185759344887t_{74} = -95.8185759344887
t75=6.28318530717959t_{75} = 6.28318530717959
t76=−23.5619449019235t_{76} = -23.5619449019235
t77=−28.2743338823081t_{77} = -28.2743338823081
t78=−40.8407044966673t_{78} = -40.8407044966673
t79=67.5442420521806t_{79} = 67.5442420521806
t80=12.5663706143592t_{80} = 12.5663706143592
t81=−17.2787595947439t_{81} = -17.2787595947439
t82=−119.380520836412t_{82} = -119.380520836412
t83=−37.6991118430775t_{83} = -37.6991118430775
t84=73.8274273593601t_{84} = 73.8274273593601
t85=58.1194640914112t_{85} = 58.1194640914112
t86=−61.261056745001t_{86} = -61.261056745001
t87=1.5707963267949t_{87} = 1.5707963267949
t88=590.619418874881t_{88} = 590.619418874881
t89=−42.4115008234622t_{89} = -42.4115008234622
The points of intersection with the Y axis coordinate
The graph crosses Y axis when t equals 0:
substitute t = 0 to 10*sin(2*t).
10sin⁡(0⋅2)10 \sin{\left(0 \cdot 2 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddtf(t)=0\frac{d}{d t} f{\left(t \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddtf(t)=\frac{d}{d t} f{\left(t \right)} =
the first derivative
20cos⁡(2t)=020 \cos{\left(2 t \right)} = 0
Solve this equation
The roots of this equation
t1=π4t_{1} = \frac{\pi}{4}
t2=3π4t_{2} = \frac{3 \pi}{4}
The values of the extrema at the points:
 pi     
(--, 10)
 4      

 3*pi      
(----, -10)
  4


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:
Minima of the function at points:
t1=3π4t_{1} = \frac{3 \pi}{4}
Maxima of the function at points:
t1=π4t_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]∪[3π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right)
Increasing at intervals
[π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dt2f(t)=0\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dt2f(t)=\frac{d^{2}}{d t^{2}} f{\left(t \right)} =
the second derivative
−40sin⁡(2t)=0- 40 \sin{\left(2 t \right)} = 0
Solve this equation
The roots of this equation
t1=0t_{1} = 0
t2=π2t_{2} = \frac{\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
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
[0,π2]\left[0, \frac{\pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at t->+oo and t->-oo
lim⁡t→−∞(10sin⁡(2t))=⟨−10,10⟩\lim_{t \to -\infty}\left(10 \sin{\left(2 t \right)}\right) = \left\langle -10, 10\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−10,10⟩y = \left\langle -10, 10\right\rangle
lim⁡t→∞(10sin⁡(2t))=⟨−10,10⟩\lim_{t \to \infty}\left(10 \sin{\left(2 t \right)}\right) = \left\langle -10, 10\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−10,10⟩y = \left\langle -10, 10\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 10*sin(2*t), divided by t at t->+oo and t ->-oo
lim⁡t→−∞(10sin⁡(2t)t)=0\lim_{t \to -\infty}\left(\frac{10 \sin{\left(2 t \right)}}{t}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡t→∞(10sin⁡(2t)t)=0\lim_{t \to \infty}\left(\frac{10 \sin{\left(2 t \right)}}{t}\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(-t) и f = -f(-t).
So, check:
10sin⁡(2t)=−10sin⁡(2t)10 \sin{\left(2 t \right)} = - 10 \sin{\left(2 t \right)}
- No
10sin⁡(2t)=10sin⁡(2t)10 \sin{\left(2 t \right)} = 10 \sin{\left(2 t \right)}
- Yes
so, the function
is
odd
    © Mister Exam – Calculator