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

See graph and explanation. x is in radian measure.

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

y''

+-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).

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

- Find the first derivative of the function ( y = (2 + \sin x)^2 ) with respect to ( x ).
- Find the critical points by setting the first derivative equal to zero and solving for ( x ).
- Use the first derivative test to determine the intervals where the function is increasing or decreasing.
- Find the second derivative of the function ( y = (2 + \sin x)^2 ) with respect to ( x ).
- Use the second derivative test to determine the concavity and inflection points of the function.
- Plot the critical points, intervals of increase/decrease, and inflection points on the graph.
- 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 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- If #y = 1 / (1+x^2)#, what are the points of inflection, concavity and critical points?
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- How do you find concavity when #f(x)= x^(7/3) + x^(4/3)#?
- What is the second derivative of #f(x)=x/(x^2+1)#?
- How do you find the inflection points for the function #f(x)=xsqrt(5-x)#?

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