How do you sketch the graph #y=(2+sinx)^2# using the first and second derivatives from #0

Question

How do you sketch the graph #y=(2+sinx)^2# using the first and second derivatives from #0<=x<2pi#?

Paisley Johnson · Accepted Answer

See graph and explanation. x is in radian measure. #pi# measures 3.1 units, nearly. The period of y is #2pi=6.25# units, nearly.

graph{ y= (2+sin x)^2 [-20, 20, -10, 10]} The period of y is #2pi#. As 2 + sin x ranges from 1 to 3, y oscillates between 1 and 9. y' = 2(2+sin x) cos x =0, when cos x = 0 and this x = an odd multiple of #pi/2=1.57#, giving turning points in the waves. y'' #=-2 sin x (2 + sin x)+2 cos^2x# #=-4 sin^2x-4sinx+2=0#, when #sin x=(sqrt3-1)/2# giving #x =kpi+(-1)^k(0.375)# radian,# k = 0, +-1, +-2, +-3, ...#, giving points of inflexion (POI), upon the axial line y = 5. I would like the readers to read my answer once again, looking at the graph after reading every line in my answer. The graph asked for is between x = 0 and x = 6.28, nearly, with ends at (0, 4) and (6.28, 4).

Christopher Agresta · Answer

undefined To sketch the graph of $ y = (2 + \sin x)^2 $ using the first and second derivatives, follow these steps:

1. Find the first derivative of the function $ y = (2 + \sin x)^2 $ with respect to $ x $.
2. Find the critical points by setting the first derivative equal to zero and solving for $ x $.
3. Use the first derivative test to determine the intervals where the function is increasing or decreasing.
4. Find the second derivative of the function $ y = (2 + \sin x)^2 $ with respect to $ x $.
5. Use the second derivative test to determine the concavity and inflection points of the function.
6. Plot the critical points, intervals of increase/decrease, and inflection points on the graph.
7. Sketch the curve of the function, considering the behavior dictated by the first and second derivatives.

Following these steps will allow you to accurately sketch the graph of $ y = (2 + \sin x)^2 $ over the interval $ 0 \leq x < 2\pi $.

How do you sketch the graph #y=(2+sinx)^2# using the first and second derivatives from #0&lt;=x&lt;2pi#?

How do you sketch the graph #y=(2+sinx)^2# using the first and second derivatives from #0<=x<2pi#?