2.3 Calculus of Vector-Valued Functions of One Variable

Calculus of Vector-Valued Functions of One Variable

Introduction

The Derivative

Computing Derivatives

Tangent Lines

Integrating a Vector-Valued Function

As the only employee of Steer Clear, you have decided that you need to understand how the timing of position measurements will change different properties related to your self-driving car. You decide to drive in a figure eight path described by $\vr_8(t)=\langle \cos(t),\sin(2t) \rangle$ for $0 \leq t \leq 2 \pi$. A plot of this path is given in .

In order to understand how often your software should collect location data, you decide to look at your position for a few different times. Calculate the following, rounding the component values to three decimal places. Draw the output vectors of $\vr_8$ on in standard position. -$\vr_8(3)$-$\vr_8(3.1)$-$\vr_8(3.14)$ The tips of these vectors correspond to the locations that would be sampled if you wanted to know the location of your car at $\vr_8(\pi)$ but collected data every second, every tenth of a second, and every hundredth of a second, respectively.

$\vr(t) = \langle \cos(t), t\sin(t), \ln(t) \rangle$.

$\vr(t) = \langle t^2 + 3t, e^{-2t}, \displaystyle\frac{t}{t^2 + 1} \rangle$.

$\vr(t) = \langle \tan(t), \cos(t^2), te^{-t} \rangle$.

$\vr(t) = \left\langle \sqrt{t^4 + 4}, \frac{2}{t^2+t} , e^{2t} \sin(-2t) \right\rangle$.

Let

Sketch the curve using some appropriate tool and make a drawing by hand that labels the point at the terminal point of $\vr(\pi)$.

Recall that we discussed earlier that the vector $\vr\, '(a)$ is tangent to the graph of $\vr(t)$ at the point where $t=a$. Find a direction vector for the line tangent to the graph of $\vr$ at the point where $t=\pi$.

Find the parametric equations of the line tangent to the graph of $\vr$ when $t=\pi$.

On your plot of the curve $\vr(t)$, sketch the tangent line corresponding to $t = \pi$ and highlight the role of $\vr\, '(\pi)$ on your plot.

Work through this exercise and explain your reasoning step by step.

Determine $\va(t)$, the acceleration of the object at time $t$.

Determine $\vr(t)$, position of the object at time $t$.

Compute the position, velocity, and acceleration vectors of the object at time $t=1$ and plot these vectors using Figure.

Give the vector equation for the tangent line, $\vL(t)$, that is tangent to the graph of $\vr(t)$ at $t = 1$.