Parametric Equations, Polar Coordinates, and Vector–Valued Functions (BC Only)

$\vec{f},'(x) = \langle x^3+1 \rangle$ at $x=1$ gives $\langle 4 \rangle$

$\vec{f},'(x) = \langle 3x^2, 1 \rangle$ at $x=1$ gives $\langle 3,1 \rangle$

No derivative exists for vector-valued functions.

$\vec{f},'(x)=\langle x^2,x+1 \rangle$ at $x=1$ gives $\langle 1,2 \rangle$

Find times when their dot product equals zero.

Set the magnitudes of both vectors equal to each other.

Find times when their cross product equals zero.

Look for times when either vector has a magnitude of zero.

A unit vector in the direction of the original function

The original vector's magnitude

The zero vector

A random vector unrelated to the original function

$<0,-\frac{12}{s^{3}},12s^{2}>$

$<5,-\frac{12}{s},12s>$

$<0,\frac{-6}{s},4>s^{-5}>$

$<5,-\frac{6}{s^{2}},4s^{3}>$

INCORRECT $(\frac{\mathrm{d}}{\mathrm{d}x})[|\dot{\mathbf{x}}| * |\dot{\mathbf{y}}| * \cosθ]$

INCORRECT $(\frac{\mathrm{d}}{\mathrm{d}x})[(\dot{\mathbf{x}}) · (\dot{\mathbf{y}})]$

CORRECT. $(\vec{x}'(t))·(\vec{y}(s))+(\vec{x}(t))·(\frac{\mathrm{d} s}{\mathrm{d} t}⋅(\vec{y})'(s))$

INCORRECT $(\frac{\mathrm{d}}{\mathrm{d}x})[|\dot{\mathbf{x}}| |𝑦| \cosθ]$

There's an error in calculation.

The function is undefined.

The magnitude of the function is increasing.

The function has reached a constant value.

No definitive behavior can be described for infinity.

$\lim (\mathbf{a} \cdot \mathbf{i}) = \infty$

$\lim_{t \to \infty} \mathbf{a}(t) = \mathbf{N}$

$\lim \int (\mathbf{N} \cdot d\mathbf{t})$

$\langle e, \ln(1), 2 \rangle$

$\langle e, \frac{1}{1}, 2 \rangle$

$\langle e^t, t^{-1}, 2t \rangle$

$\langle 0, \frac{1}{e}, 2t \rangle$

The tangent vector $\vec{tangent}(x) = \lim_{h \to 0} \frac{\vec{r}(x+h) - \vec{r}(x)}{h}$

The sum of derivatives for each component $\sum_{component} |tangent|(x) = h \sum_{component} \vec{r}(x)'$

Addition of slopes for tangent $\Sigma tangent = \sum_{component} \Sigma \vec{r}(x) \prime(h)$

The tangent line equation for the magnitude of tangent $|tangent|(x) = h |magnitude| \vec{r}(x')$

Cross product

Component-wise differentiation

Scalar multiplication

Dot product