Find the curvature of 
r(t) = 

7t, t2, t3

 at the point 
(7, 1, 1).
𝜅 = 


The curvature \(\kappa\) of a vector function \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\) at a given point is given by the formula:

\[
\kappa(t) = \frac{\|\mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}
\]

### Step 1: Find \(\mathbf{r}'(t)\) and \(\mathbf{r}''(t)\)
Given the vector function:
\[
\mathbf{r}(t) = \langle 7t, t^2, t^3 \rangle
\]

#### First derivative \(\mathbf{r}'(t)\):
\[
\mathbf{r}'(t) = \langle \frac{d}{dt}(7t), \frac{d}{dt}(t^2), \frac{d}{dt}(t^3) \rangle = \langle 7, 2t, 3t^2 \rangle
\]

#### Second derivative \(\mathbf{r}''(t)\):
\[
\mathbf{r}''(t) = \langle \frac{d}{dt}(7), \frac{d}{dt}(2t), \frac{d}{dt}(3t^2) \rangle = \langle 0, 2, 6t \rangle
\]

### Step 2: Find \(t\) corresponding to the point \((7, 1, 1)\)
We are given the point \((7, 1, 1)\). Let's use the components of \(\mathbf{r}(t)\) to find \(t\):

\[
\text{From } x(t) = 7t = 7, \quad \Rightarrow t = 1
\]

Now substitute \(t = 1\) into the expressions for \(\mathbf{r}'(t)\) and \(\mathbf{r}''(t)\).

### Step 3: Evaluate \(\mathbf{r}'(1)\) and \(\mathbf{r}''(1)\)
\[
\mathbf{r}'(1) = \langle 7, 2(1), 3(1)^2 \rangle = \langle 7, 2, 3 \rangle
\]
\[
\mathbf{r}''(1) = \langle 0, 2, 6(1) \rangle = \langle 0, 2, 6 \rangle
\]

### Step 4: Calculate the curvature
First, find the magnitudes of \(\mathbf{r}'(1)\) and \(\mathbf{r}''(1)\):

\[
\|\mathbf{r}'(1)\| = \sqrt{7^2 + 2^2 + 3^2} = \sqrt{49 + 4 + 9} = \sqrt{62}
\]

\[
\|\mathbf{r}''(1)\| = \sqrt{0^2 + 2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40}
\]

Now, plug these into the curvature formula:

\[
\kappa(1) = \frac{\|\mathbf{r}''(1)\|}{\|\mathbf{r}'(1)\|^3} = \frac{\sqrt{40}}{(\sqrt{62})^3} = \frac{\sqrt{40}}{62^{3/2}}
\]

### Final Answer:
\[
\kappa(1) = \frac{\sqrt{40}}{62^{3/2}}
\]

Would you like more details on the derivation or have any other questions? Here are a few related questions:

1. How do we calculate the torsion of a space curve?
2. What is the difference between curvature and torsion?
3. How do you interpret curvature geometrically for a curve in space?
4. Can the curvature formula be applied to curves in two dimensions as well?
5. How do higher-order derivatives influence the curvature of a curve?

**Tip:** The curvature at a point provides information about how "curved" the curve is at that specific point. A higher curvature means a sharper bend.

Find the curvature of r(t) = 7t, t^2, t^3 at the point (7, 1, 1).

Learn how to find the curvature of the vector function r(t) = 7t, t^2, t^3 at the point (7, 1, 1) using the curvature formula. This problem explores key concepts in vector calculus, including the calculation of first and second derivatives, magnitudes, and curvature. Suitable for advanced students in Grades 11-12.

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