Derivatives are functions that describe the rate of change of another function with respect to one or more of its independent variables. They are a fundamental concept in calculus that allow for the analysis of how a function's output changes as its input changes.
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Derivatives are used to analyze the behavior of parametric curves, which are curves defined by a set of parametric equations.
The derivative of a parametric curve $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ is given by $\frac{d\mathbf{r}}{dt} = \langle \frac{dx}{dt}, \frac{dy}{dt}\rangle$.
Derivatives of parametric curves can be used to find the tangent vector, normal vector, and curvature of the curve at a given point.
Parametric curves are often used to model the motion of objects in physics, as well as to describe the shape of complex geometric objects.
The derivative of a parametric curve can also be used to find the arc length of the curve, which is the distance along the curve between two points.
Review Questions
Explain how the derivative of a parametric curve $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ is calculated.
The derivative of a parametric curve $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ is given by $\frac{d\mathbf{r}}{dt} = \langle \frac{dx}{dt}, \frac{dy}{dt}\rangle$. This means that the derivative of the curve is a vector whose components are the derivatives of the $x$ and $y$ coordinates with respect to the parameter $t$. This allows for the analysis of the rate of change of the curve's position, direction, and curvature.
Describe how the derivative of a parametric curve can be used to find the tangent vector, normal vector, and curvature of the curve.
The derivative of a parametric curve $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ can be used to find the tangent vector, normal vector, and curvature of the curve. The tangent vector is given by $\frac{d\mathbf{r}}{dt}$, the normal vector is given by $\frac{d\mathbf{r}}{dt}$ rotated 90 degrees, and the curvature is given by $\frac{|\frac{d^2\mathbf{r}}{dt^2}|}{|\frac{d\mathbf{r}}{dt}|^3}$. These properties are crucial for understanding the behavior and geometry of the curve.
Explain how the derivative of a parametric curve can be used to find the arc length of the curve between two points.
The arc length of a parametric curve $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ between two points $t_1$ and $t_2$ can be calculated using the derivative of the curve. The arc length is given by the integral $\int_{t_1}^{t_2} |\frac{d\mathbf{r}}{dt}| dt$, where $|\frac{d\mathbf{r}}{dt}|$ is the magnitude of the derivative vector. This allows for the precise measurement of the distance along the curve between any two points, which is useful in many applications.
The line that best approximates the behavior of a function at a given point, with the slope of the tangent line being equal to the derivative of the function at that point.