Question

Find the open interval(s) on which the curve given by the vector-valued function is smooth. Enter your answer using interval notation.

Answer 1

To determine the open interval(s) on which the curve given by the vector-valued function is smooth, we need to check the continuity of the function and its derivatives.

Step-by-step explanation:

To determine the open interval(s) on which the curve given by the vector-valued function is smooth, we need to check the continuity of the function and its derivatives. A vector-valued function is smooth if its component functions are continuously differentiable. So, we need to check the continuity of each component function and their derivatives.

Let the vector-valued function be r(t) = <f(t), g(t), h(t)>. To find the open interval(s) on which r(t) is smooth, we need to check the continuity and differentiability of all three component functions f(t), g(t), and h(t) and their derivatives.

An interval is open if its endpoints are not included. So, if all three component functions and their derivative functions are continuous and differentiable on an interval (a, b), where a and b are the endpoints, excluding a and b, then the curve given by the vector-valued function is smooth on that interval.

Answer 2

The open interval on which the curve given by a horizontal vector-valued function f(x) is smooth is (0, 20), as the derivative is continuous and constant over this interval.

Step-by-step explanation:

To find the open interval(s) where the curve of a vector-valued function is smooth, we need to look for intervals where the function, its derivative, and possibly higher-order derivatives are continuous. A function is considered to be smooth on an interval if it has derivatives of all orders throughout that interval and also if there are no sharp corners or cusps in the graph.

Given that the function f(x) is represented by a horizontal line when “0 ≤ x ≤ 20”, the smoothness depends on the first derivative dy(x)/dx being continuous. As the slope of f(x) in this case is zero (since it is a horizontal line), the derivative does not change, and hence the function is smooth over the entire interval [0, 20]. Thus, using interval notation, the open interval on which f(x) is smooth would be (0, 20).