Final answer:
The slopes of the curve f(x) = 1/x at x=0.1, x=1, and x=10 are -100, -1, and -0.01 respectively, found by evaluating the derivative of the function at each point.
Step-by-step explanation:
To estimate the slope of the curve f(x) = 1/x at a specific point, we need to determine the derivative of the function, as the derivative gives us the slope of the tangent line at any point along the curve. The derivative of f(x) = 1/x with respect to x is f'(x) = -1/x^2. We can then evaluate this derivative at the points x = 0.1, x = 1, and x = 10 to find the slopes at these x-values.
At x = 0.1:
f'(0.1) = -1/(0.1)^2 = -100
At x = 1:
f'(1) = -1/(1)^2 = -1
At x = 10:
f'(10) = -1/(10)^2 = -0.01
Therefore, the slope of the curve at x = 0.1 is -100, at x = 1 is -1, and at x = 10 is -0.01. These values represent how steep the curve is at each of these points.