How do you graph #f(x)=sinx-sqrt(3)cos x# for x is between [0, 2pi ]?

Answer 1

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Answer 2

To graph ( f(x) = \sin(x) - \sqrt{3}\cos(x) ) for ( x ) between ( [0, 2\pi] ):

  1. Identify the key points on the graph: the maximum, minimum, and points where the function crosses the x-axis.
  2. Calculate the values of ( f(x) ) at these key points.
  3. Plot the points on the graph.
  4. Sketch the curve connecting the points to form the graph of ( f(x) ).

Here are the steps:

  1. Key points:

    • Maximum: ( f(x) ) reaches its maximum when ( \sin(x) ) is at its maximum value of 1 and ( \cos(x) ) is at its minimum value of -1. So, ( f(x) = 1 - \sqrt{3}(-1) = 1 + \sqrt{3} ) when ( x = \frac{\pi}{2} ).
    • Minimum: ( f(x) ) reaches its minimum when ( \sin(x) ) is at its minimum value of -1 and ( \cos(x) ) is at its maximum value of 1. So, ( f(x) = -1 - \sqrt{3}(1) = -1 - \sqrt{3} ) when ( x = \frac{3\pi}{2} ).
    • X-intercepts: ( f(x) ) crosses the x-axis when ( \sin(x) = \sqrt{3}\cos(x) ). This occurs when ( \tan(x) = \sqrt{3} ), leading to ( x = \frac{\pi}{3} ) and ( x = \frac{4\pi}{3} ).
  2. Calculate ( f(x) ) at the key points:

    • ( f\left(\frac{\pi}{2}\right) = 1 + \sqrt{3} )
    • ( f\left(\frac{3\pi}{2}\right) = -1 - \sqrt{3} )
    • ( f\left(\frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) - \sqrt{3}\cos\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} - \sqrt{3}\left(\frac{1}{2}\right) = 0 )
    • ( f\left(\frac{4\pi}{3}\right) = \sin\left(\frac{4\pi}{3}\right) - \sqrt{3}\cos\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} - \sqrt{3}\left(-\frac{1}{2}\right) = 0 )
  3. Plot the points: ( \left(\frac{\pi}{2}, 1 + \sqrt{3}\right) ), ( \left(\frac{3\pi}{2}, -1 - \sqrt{3}\right) ), ( \left(\frac{\pi}{3}, 0\right) ), and ( \left(\frac{4\pi}{3}, 0\right) ).

  4. Sketch the curve connecting the points to form the graph of ( f(x) ). The curve will oscillate between the maximum and minimum points, crossing the x-axis at the x-intercepts.

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Answer from HIX Tutor

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.

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.

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