Final answer:
To determine the values of x where the tangent to the curve has a slope of 1, differentiate the function to find its derivative and solve for f'(x) = 1. For linear functions, the slope is constant and represented by the coefficient b. For nonlinear functions, use the derivative to find where the slope of the tangent is equal to 1.
Step-by-step explanation:
To find the values of x within the interval (a, b) where the tangent to the curve has a slope of 1, we use the concept that the slope of a curve at a particular point corresponds to the slope of the tangent at that point. If we have a function, say f(x), then the slope of the tangent at a given point is calculated by finding the derivative f'(x) and evaluating it at that point.
In case of a linear equation of the form y = a + bx, for example, the slope is represented by the coefficient b, which remains constant irrespective of the point on the line. However, for non-linear functions, we need to differentiate the function to find the derivative and then solve the equation f'(x) = 1 to find the specific values of x where the slope of the tangent is equal to 1.
If the specific equation of the curve or function is not given, we generally use the endpoints of the tangent line to calculate the slope. The tangent's slope can be found as the change in y coordinates over the change in x