Question

In each of the following cases, you will be asked to write down a family of parametric curves that have the property that at t = 1, we have x'(t) = y'(t) = 0, but the slope of the curve has the property listed here:

a. Horizontal tangent.
b. Vertical tangent.
c. A slope of 4.2.
d. Slope of the curve is undefined.

Answer 1

Parametric curves with specified tangent slopes at t=1 can be formed by selecting appropriate functions for x(t) and y(t) such that their derivatives with respect to t satisfy the given conditions.

Step-by-step explanation:

The student is asking for a family of parametric curves that satisfy specific conditions at t = 1 in terms of the slope of the tangent to the curve:

Horizontal tangent: This implies that both the first derivatives of x and y with respect to t are zero at t = 1; hence, any pair of functions x(t) and y(t) such that x'(1) = y'(1) = 0 will work. For example, x(t) = (t - 1)^2 and y(t) = (t - 1)^2.Vertical tangent: This suggests that the derivative of y with respect to x is undefined, which happens when x'(t) is zero while y'(t) is not at t = 1. For instance, x(t) = (t - 1)^2 and y(t) = t.A slope of 4.2: We wish for the derivative of y with respect to x to be 4.2 at t = 1. Thus, we could select x(t) = (t - 1)^2 and y(t) = 4.2(t - 1)^2 + k, where k is any constant.Undefined slope: This typically means a vertical tangent, so we can consider the same functions as in the vertical tangent case.