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We are given that \( \mathbf{r}(t) \) is a smooth vector function, and we need to approximate \( \mathbf{r}(0.05) \) using a quadratic polynomial centered at \( t = 0 \).

The Taylor expansion of \( \mathbf{r}(t) \) around \( t = 0 \) is:

\[
\mathbf{r}(t) \approx \mathbf{r}(0) + t\mathbf{r}'(0) + \frac{t^2}{2} \mathbf{r}''(0)
\]

### Given values:
- \( \mathbf{r}(0) = \langle -2, 4, 4 \rangle \)
- \( \mathbf{r}'(0) = \langle 2.5, 2.5, 2 \rangle \)
- \( \mathbf{r}''(0) = \langle -2.5, -2, 1 \rangle \)
- \( t = 0.05 \)

### Step 1: First-order term
The first-order term is \( t\mathbf{r}'(0) \):

\[
t \mathbf{r}'(0) = 0.05 \times \langle 2.5, 2.5, 2 \rangle = \langle 0.125, 0.125, 0.1 \rangle
\]

### Step 2: Second-order term
The second-order term is \( \frac{t^2}{2} \mathbf{r}''(0) \):

\[
\frac{t^2}{2} = \frac{(0.05)^2}{2} = \frac{0.0025}{2} = 0.00125
\]

\[
\frac{t^2}{2} \mathbf{r}''(0) = 0.00125 \times \langle -2.5, -2, 1 \rangle = \langle -0.003125, -0.0025, 0.00125 \rangle
\]

### Step 3: Approximation of \( \mathbf{r}(0.05) \)
Now, sum up the terms:

\[
\mathbf{r}(0.05) \approx \mathbf{r}(0) + t \mathbf{r}'(0) + \frac{t^2}{2} \mathbf{r}''(0)
\]

\[
\mathbf{r}(0.05) \approx \langle -2, 4, 4 \rangle + \langle 0.125, 0.125, 0.1 \rangle + \langle -0.003125, -0.0025, 0.00125 \rangle
\]

\[
\mathbf{r}(0.05) \approx \langle -2 + 0.125 - 0.003125, 4 + 0.125 - 0.0025, 4 + 0.1 + 0.00125 \rangle
\]

\[
\mathbf{r}(0.05) \approx \langle -1.878125, 4.1225, 4.10125 \rangle
\]

### Final Answer:
\[
\mathbf{r}(0.05) \approx \langle -1.878, 4.123, 4.101 \rangle
\]

Would you like further details on this or have any questions?

### Related Questions:
1. How does the Taylor expansion help approximate vector functions?
2. What is the significance of each term in the Taylor expansion?
3. Why do we only use the first and second derivatives for this approximation?
4. How would the approximation change if higher-order derivatives were included?
5. How can the accuracy of this approximation be improved for small values of \( t \)?

**Tip:** The Taylor series provides an excellent approximation for smooth functions when \( t \) is close to the point of expansion (here, \( t = 0 \)).

Suppose that r(t) is a smooth vector function such that r(0) = <−2, 4, 4>, r′(0) = <2.5, 2.5, 2>, and r″(0) = <−2.5, −2, 1>. Use the quadratic polynomial approximation centered at 0 to estimate r(0.05).

Learn how to approximate the value of a smooth vector function r(t) using the quadratic polynomial approximation. This step-by-step guide covers Taylor series expansion with vector functions, using derivatives to estimate r(0.05). Suitable for calculus students.

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