Calculus of Vector-Valued Functions of One Variable

For each of the following vector-valued functions, state any values of $a$ for which $\displaystyle\lim_{t\to a}\vr(t)$ will not exist and find $\vr\, '(t)$.

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

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

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

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

Consider the two vector-valued functions given by and

1. Determine the point of intersection of the curves generated by $\vr(t)$ and $\vw(s)$. To do so, you will have to find values of $a$ and $b$ that result in $\vr(a)$ and $\vw(b)$ being the same vector.

2. Use the value of $a$ you determined in (a) to find a vector form of the tangent line to $\vr(t)$ at the point where $t = a$.

3. Use the value of $b$ you determined in (a) to find a vector form of the…

For each given function $\vr$, determine $\int \vr(t) \ dt$. In addition, recalling the Fundamental Theorem of Calculus for functions of a single variable, also evaluate $\int_0^1 \vr(t) \ dt$ for each given function $\vr$. For each of your calculations, you should write a sentence that states whether your result is a vector or a scalar and describes what is being measured.

1. $\vr(t) = \left\langle \cos(t), \frac{1}{t+1}, te^t \right\rangle$

2. $\vr(t) = \left\langle \cos(3t), \sin(2t), t \right\rangle$

3. $\vr(t) = \left\langle \frac{t}{1+t^2}, te^{t^2}, \frac{1}{1+t^2} \right\rangle$