# How do you sketch the graph of #f(θ)=2cosθ+cos2 θ# for #0≤x≤ 2π# using the first and second derivative?

Ok, have a look...my graph is not very nice and please check my maths!

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To sketch the graph of ( f(\theta) = 2\cos(\theta) + \cos(2\theta) ) for ( 0 \leq \theta \leq 2\pi ) using the first and second derivative:

- Find the first derivative ( f'(\theta) ) and the critical points by setting ( f'(\theta) = 0 ).
- Determine the intervals where ( f(\theta) ) is increasing or decreasing using the sign of ( f'(\theta) ).
- Find the second derivative ( f''(\theta) ) and identify the points of inflection by setting ( f''(\theta) = 0 ).
- Determine the intervals where the concavity of ( f(\theta) ) changes using the sign of ( f''(\theta) ).
- Plot the critical points, points of inflection, and intervals of increasing/decreasing and concavity.
- Sketch the graph of ( f(\theta) ) within the specified domain.

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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.

- What are the points of inflection of #f(x)=cos^2x # on the interval #x in [0,2pi]#?
- How do you find intercepts, extrema, points of inflections, asymptotes and graph #y=x/(x^2+1)#?
- How do you determine the concavity for #f(x) = x^4 − 32x^2 + 6#?
- How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3 - 2x^2 - x +1#?
- Is #f(x)=(x-3)(x-2)-x^2# concave or convex at #x=3#?

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