12.4
Imagine a car rounding a mountain curve. At time t, its position is at point P. After a brief interval h, it reaches point Q at time t + h.
The vector from P to Q shows the car's overall change in position during that time and is called a secant vector. It connects two distinct positions along the car's route.
As the time interval h gets smaller, point Q moves closer to point P.
The secant vector starts to reflect the road's direction at P.
Now multiply this secant vector by one over h.
This operation doesn’t change the direction; it only scales the vector’s length.
As h tends to zero, this scaled vector approaches a vector with a well-defined direction and magnitude.
That limiting vector is called the derivative vector, or the tangent vector r′(t).
The vector r′(t) is tangent to the curve at the point P. It represents the instantaneous velocity, both the speed and the direction of the car at any time t.
A vector-valued function describes position as a function of time. For example, in Cartesian coordinates, the position of a car moving along a curved road can be written as
\begin{equation*}\textbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}
Secant Vector and Average Velocity:
This secant vector captures the overall change in position during the interval and provides a crude estimate of the direction of motion.
At time t, the car is at point P, with position r(t). After a short interval h, it reaches point Q, with position r(t+h). The secant vector from P to Q is
\begin{equation*}\textbf{r}(t+h)-\textbf{r}(t)\end{equation*}
\begin{equation*}=\langle x(t+h),y(t+h),z(t+h)\rangle - \langle x(t),y(t),z(t)\rangle\end{equation*}
\begin{equation*}=\langle x(t+h)-x(t),y(t+h)-y(t),z(t+h)-z(t)\rangle\end{equation*}
To get the average velocity, the secant vector is scaled by the reciprocal of the time interval, forming the difference quotient:
\begin{equation*}\jfrac{\textbf{r}(t+h)-\textbf{r}(t)}{h} = \left\langle \jfrac{x(t+h)-x(t)}{h}, \jfrac{y(t+h)-y(t)}{h}, \jfrac{z(t+h)-z(t)}{h}\right\rangle\end{equation*}
While the direction of this vector remains consistent with the secant vector, its magnitude now reflects the average speed over the interval h.
Derivative Vector:
As h approaches zero, point Q converges to point P, and the secant vector becomes infinitesimally short. The limit of the difference quotient as h→0 yields the derivative vector:
\begin{equation*}\textbf{r}'(t)=\lim_{h\to 0}\jfrac{\textbf{r}(t+h)-\textbf{r}(t)}{h}\end{equation*}
The limit can be evaluated component-wise:
\begin{equation*}\textbf{r}'(t)= \left\langle \lim_{h\to 0}\jfrac{x(t+h)-x(t)}{h}, \lim_{h\to 0}\jfrac{y(t+h)-y(t)}{h}, \lim_{h\to 0}\jfrac{z(t+h)-z(t)}{h} \right\rangle\end{equation*}
\begin{equation*}\textbf{r}'(t)=\langle x'(t),y'(t),z'(t)\rangle\end{equation*}
This derivative vector is tangent to the curve at point P and represents the car’s instantaneous velocity, including both speed and direction.
Imagine a car rounding a mountain curve. At time t, its position is at point P. After a brief interval h, it reaches point Q at time t + h.
The vector from P to Q shows the car's overall change in position during that time and is called a secant vector. It connects two distinct positions along the car's route.
As the time interval h gets smaller, point Q moves closer to point P.
The secant vector starts to reflect the road's direction at P.
Now multiply this secant vector by one over h.
This operation doesn’t change the direction; it only scales the vector’s length.
As h tends to zero, this scaled vector approaches a vector with a well-defined direction and magnitude.
That limiting vector is called the derivative vector, or the tangent vector r′(t).
The vector r′(t) is tangent to the curve at the point P. It represents the instantaneous velocity, both the speed and the direction of the car at any time t.
From Chapter 12:
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