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# Lessons 5-9 **Vector-Valued Functions, Calculus & Motions in Space** ## Vector-Valued Functions * **Function:** Understand how to represent space curves and find domains. $\vec{r}(t) = \langle f(t), g(t), h(t)\rangle$ $\vec{r}(t) = \vec{f}(t) + \vec{g}(t) + \vec{h}(t)$ * **Limit:** $\l...

# Lessons 5-9 **Vector-Valued Functions, Calculus & Motions in Space** ## Vector-Valued Functions * **Function:** Understand how to represent space curves and find domains. $\vec{r}(t) = \langle f(t), g(t), h(t)\rangle$ $\vec{r}(t) = \vec{f}(t) + \vec{g}(t) + \vec{h}(t)$ * **Limit:** $\lim_{t\to a} \vec{r}(t) = \lim_{t\to a} \vec{f}(t) + \lim_{t\to a} \vec{g}(t) + \lim_{t\to a} \vec{h}(t)$ ## Calculus of Vector-Valued Functions and Motions in Space * Find the derivatives of vector-valued functions. $\vec{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle$ * Tangent vectors and tangent lines. $\vec{r}'(t_0)$ tangent vector at $t = t_0$ and $\vec{r}(t_0) + {t} {\vec{r}'(t_0)}$ tangent line at point $t_0$. * Evaluate definite integrals. $\int_a^b \vec{r}(t) \, dt = \left< \int_a^b f(t) \, dt, \int_a^b g(t) \, dt, \int_a^b h(t) \, dt \right>$ ## Motion in Space * **Velocity:** $\vec{v}(t) = \vec{r}'(t)$ * **Speed:** $||\vec{v}(t)||$ * **Direction of velocity:** $\vec{T}(t) = \vec{v}(t)/||\vec{v}(t)||$ * **Acceleration:** $\vec{a}(t) = \vec{r}''(t)$ * **Newton's second law of motion:** $\vec{F} = m \vec{a}$ ## Solving Real-World Motion in Space Problems * **2D** * How far the ball went horizontally. * How high the ball went. * When the ball is at a certain height. * Where the ball ended up (horizontally. * **3D** * This case is similar to the 2D motions. ## Length of Curves & Curvature * **Arc length:** $L = \int_a^b ||\vec{r}'(t)|| \, dt$ * **Curvature:** $K = \frac{||\vec{T}'(t)||}{||\vec{r}'(t)||} = \frac{||\vec{r}'(t) \times \vec{r}''(t)||}{||\vec{r}'(t)||^3}$ where $\vec{T}(t) = \frac{\vec{r}'(t)}{||\vec{r}'(t)||} $ ## Functions of Several Variables * **Domains of functions:** Understand restrictions like zeros in the denominator, negative square roots, etc. * **Graphing:** Level curves. Be able to visualize a given curve.