Question

Use first principles to determine the slope of the tangent to f(x)=(-2)/(x) when x=1.

Answer 1

To find the slope of the tangent to the curve f(x) = -2/x at x = 1, we apply first principles to calculate the derivative and then substitute x = 1. The slope is calculated to be 2.

Step-by-step explanation:

The student is asking how to determine the slope of the tangent to the function f(x) = -2/x at x = 1 using first principles. The slope of a tangent line to a curve at a given point is the same as the derivative of the function at that point. To find the slope using first principles, we implement the definition of the derivative, which involves taking the limit as h approaches zero of the difference quotient: (f(x+h) - f(x))/h.

Step-by-step Solution:

1. Write the difference quotient for f(x) = -2/x: ((-2/(x+h)) - (-2/x)) / h.
2. Simplify the difference quotient: (2h/x(x+h)) / h.
3. Cancel out h: 2/(x(x+h)).
4. Take the limit as h goes to zero: 2/x2.
5. Substitute x = 1 into the derivative to get the slope at x=1: 2/12 = 2.

Therefore, the slope of the tangent to the curve f(x) = -2/x when x = 1 is 2.