how do i find the slope of the curve at the indicated point?
f(x)= absolute value of x, x=2
at x=2, abs(x)= x
f(x)=x
slope= 1
To find the slope of the curve of the function f(x) = |x| at a specific point, you can use the concept of derivatives.
1. Start by finding the derivative of the function f(x). The derivative of the absolute value function |x| is defined differently when x is positive or negative. When x > 0, the derivative is simply 1. When x < 0, the derivative is -1.
2. Determine the sign of x at the given point. In this case, x = 2, which is positive.
3. Since x is positive at x = 2, the derivative of f(x) at x = 2 is 1.
Therefore, the slope of the curve of the function f(x) = |x| at the point x = 2 is 1.
To find the slope of a curve at a specific point, you need to calculate the derivative of the function and then evaluate the derivative at that point.
1. The function given is f(x) = |x| (the absolute value of x).
2. To find the derivative of f(x), you can use the definition of the derivative or the derivative rules. In this case, you can use the derivative rules.
3. The derivative of |x| can be determined by considering the function in two separate cases:
- If x > 0, the function simplifies to f(x) = x. The derivative of x is 1.
- If x < 0, the function simplifies to f(x) = -x. The derivative of -x is -1.
- For x = 0, the function is not differentiable since there is a sharp corner at x = 0.
4. Since the derivative changes depending on whether x is positive or negative, there is no single value for the slope of the curve at x = 2. Instead, you'll need to find the one-sided derivatives.
- To find the slope from the left side, evaluate the derivative for x < 2: f'(x) = -1.
- To find the slope from the right side, evaluate the derivative for x > 2: f'(x) = 1.
5. Therefore, at x = 2, the slope of the curve does not exist because the left and right one-sided derivatives are different.
In summary, the slope of the curve at x = 2 for the function f(x) = |x| does not exist.