Question

In Exercises 13 and \(14,\) find the slope of the curve at the indicated point. $$f(x)=|x| \quad\( at \)\quad\( (a) \)x=2 \quad\( (b) \)x=-3$$

Short answer

The slope of the function \( f(x) = |x| \) at point \( x = 2 \) is \( 1 \) and at point \( x = -3 \) is \( -1 \).

Step-by-step solution

Understanding \( f(x) = |x| \)

The function \( f(x) = |x| \) can be decomposed into two cases: \( f(x) = x \) when \( x \geq 0 \), and \( f(x) = -x \) when \( x < 0 \) because the absolute value of a number is its non-negative value.

Take derivative of \( f(x) = |x| \)

The derivative of the function, \( f'(x) \), will also have two cases: \( f'(x) = 1 \) when \( x > 0 \) and \( f'(x) = -1 \) when \( x < 0 \). The derivative does not exist when \( x = 0 \) as the function is not differentiable at this point.

Find slope at \( x = 2 \)

To find the slope when \( x = 2 \), substitute \( 2 \) into the equation of \( f'(x) \). Because \( 2 > 0 \), the case of \( f'(x) = 1 \) applies here. So, the slope at \( x = 2 \) is \( 1 \).

Find slope at \( x = -3 \)

To find the slope when \( x = -3 \), substitute \( -3 \) into the equation of \( f'(x) \). Here, \( -3 < 0 \), so the case of \( f'(x) = -1 \) applies. Therefore, the slope at \( x = -3 \) is \( -1 \).