# How do you use the first and second derivatives to sketch #f(x) = | (x^2) -1 |#?

# f'(x) \ = { (2x-1,,x lt -1), (1-2x,, -1 lt x lt 1), (2x-1,, x gt 1) :} #

# f''(x) = { (2,,x lt -1), (-2,, -1 lt x lt 1), (2,, x gt 1) :} #

Graphing the function will help to answer the question:

So we can write the function as:

# f(x) = |x^2-1 | #

# " " = { (x^2-1,,x^2-1 gt 0), (-(x^2-1),,x^2-1 lt 0) :} #

# " " = { (x^2-1,,x lt -1), (1-x^2,, -1 lt x lt 1), (x^2-1,, x gt 1) :} #

Note that although

So then we can easily differentiate to get the first derivative:

# f'(x) \ = { (2x-1,,x lt -1), (1-2x,, -1 lt x lt 1), (2x-1,, x gt 1) :} #

And the second derivative is:

# f''(x) = { (2,,x lt -1), (-2,, -1 lt x lt 1), (2,, x gt 1) :} #

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User is interested in learning about calculus, specifically the use of derivatives in sketching functions.To sketch the function ( f(x) = |x^2 - 1| ) using the first and second derivatives, follow these steps:

- Find the critical points by setting the derivative of ( f(x) ) 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 points of inflection by setting the second derivative of ( f(x) ) equal to zero and solving for ( x ).
- Use the second derivative test to determine the concavity of the function in different intervals.
- Sketch the graph using the information obtained from the steps above.

Would you like to see the detailed calculations for these steps?

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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.

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