Vector-Valued Functions: Derivatives, Limits, and Tangent Lines

Vector-Valued Functions and Derivatives

• r(t) is a vector-valued position: \mathbf{r}(t) = x(t)\,\mathbf{i} + y(t)\,\mathbf{j} + z(t)\,\mathbf{k}
• Derivatives are taken component-wise: \mathbf{v}(t) = \mathbf{r}'(t) = x'(t)\,\mathbf{i} + y'(t)\,\mathbf{j} + z'(t)\,\mathbf{k}
  \mathbf{a}(t) = \mathbf{r}''(t) = x''(t)\,\mathbf{i} + y''(t)\,\mathbf{j} + z''(t)\,\mathbf{k}
• Use only smoothly constructed continuous functions for derivatives; refer to textbooks for any missing formulas.
• When evaluating limits of vector-valued expressions, treat each component separately; avoid multi-variable limit mishaps.
• Limits and L’Hôpital's rule:
  • Apply L’Hôpital only to single-variable components (not to expressions with multiple variables like x and y together).
  • For a 0/0 form in a single variable, you may differentiate numerator and denominator until simplification.
  • If a component does not form 0/0, evaluate directly.
• Substitution notes:
  • If a component yields a 0/0, check each component; do not collapse the vector.
  • Be mindful of domain issues (e.g., ln, trig) when evaluating limits component-wise.
• Trigonometric function properties (helpful for limits):
  • Sine and tangent are odd: \sin(-x) = -\sin(x), \tan(-x) = -\tan(x)
  • Cosine is even: \cos(-x) = \cos(x)
• Domain remarks:
  • $\ln$ is defined for positive arguments only; avoid evaluating at negative inputs.
• Number of components depends on the curve:
  • Plane curve: two components (x and y) with i, j.
  • Space curve: three components (x, y, z) with i, j, k.
• From parametric to Cartesian (visualization in the xy-plane):
  • Given x = x(t)\,,\quad y = y(t)\,, eliminate t to obtain a relation between x and y.
  • Example (two components): if x(t) = \frac{t}{t+1}, \; y(t) = \frac{1}{t}
  • Solve t = \frac{1}{y} and substitute into x:
    x = \frac{\frac{1}{y}}{\frac{1}{y}+1} = \frac{1}{1+y}
  • This gives the Cartesian relation: y = \frac{1}{x} - 1 (a shifted rectangular hyperbola; equivalently xy + x - 1 = 0).
  • Visualizing the path in the xy-plane can simplify understanding of the curve.
• Velocity, acceleration, and speed (in 3D):
  • Velocity: \mathbf{v}(t) = \mathbf{r}'(t)
  • Acceleration: \mathbf{a}(t) = \mathbf{r}''(t)
  • Speed (magnitude of velocity): |\mathbf{v}(t)| = \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2}
• Tangent line to a parametric curve at t = t0:
  • Point on curve: \mathbf{r}(t_0)
  • Tangent direction: \mathbf{v}(t0) = \mathbf{r}'(t0)
  • Tangent line (vector form): \mathbf{L}(s) = \mathbf{r}(t0) + s\,\mathbf{r}'(t0)
  • In coordinates: x = x(t0) + s \; x'(t0), \quad y = y(t0) + s \; y'(t0), \quad z = z(t0) + s \; z'(t0)
• Notation reminders:
  • Two common representations are acceptable:
  • \mathbf{r}(t) = \langle x(t), y(t), z(t)\rangle or
  • \mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k}
  • Always keep the I, J, and K components explicit; do not collapse the vector.
• Problem-solving approach (quick guide):
  • Write components and compute derivatives component-wise.
  • If a t-value is given, evaluate each component at that t.
  • To find a Cartesian relationship, eliminate t between x(t) and y(t) (or z(t) if needed).
  • For tangents, compute r(t0) and r'(t0); assemble the line using the tangent form.
• Quick note on integrals (optional reminder): some integrals (e.g., \int \ln t \, dt) use integration by parts; keep this in mind when these arise in related problems.