Let’s solve the problems step by step based on the given image.

Problem A:

Position vector:
$\mathbf{r}(t) = \langle \cos(5t), \sin(5t) \rangle$
Find: $\mathbf{r}'\left(\frac{\pi}{4}\right)$.

1. Compute $\mathbf{r}'(t)$:
   Differentiate each component:
   $\mathbf{r}'(t) = \langle -5\sin(5t), 5\cos(5t) \rangle$

2. Substitute $t = \frac{\pi}{4}$:
   $\mathbf{r}'\left(\frac{\pi}{4}\right) = \langle -5\sin\left(\frac{5\pi}{4}\right), 5\cos\left(\frac{5\pi}{4}\right) \rangle$
   Since $\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$ and $\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$, we get:
   $\mathbf{r}'\left(\frac{\pi}{4}\right) = \langle -5\left(-\frac{\sqrt{2}}{2}\right), 5\left(-\frac{\sqrt{2}}{2}\right) \rangle = \langle \frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2} \rangle$

Problem B:

Position vector:
$\mathbf{r}(t) = \langle t^2, t^3 \rangle$
Find: $\mathbf{r}'(7)$.

1. Compute $\mathbf{r}'(t)$:
   Differentiate each component:
   $\mathbf{r}'(t) = \langle 2t, 3t^2 \rangle$

2. Substitute $t = 7$:
   $\mathbf{r}'(7) = \langle 2(7), 3(7^2) \rangle = \langle 14, 147 \rangle$

Problem C:

Position vector:
$\mathbf{r}(t) = e^{5t}\mathbf{i} + e^{-7t}\mathbf{j} + t\mathbf{k}$
Find: $\mathbf{r}'(-1)$.

1. Compute $\mathbf{r}'(t)$:
   Differentiate each component:
   $\mathbf{r}'(t) = \langle 5e^{5t}, -7e^{-7t}, 1 \rangle$

2. Substitute $t = -1$:
   $\mathbf{r}'(-1) = \langle 5e^{-5}, -7e^{7}, 1 \rangle$

Final Answers:

A. $\mathbf{r}'\left(\frac{\pi}{4}\right) = \langle \frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2} \rangle$
B. $\mathbf{r}'(7) = \langle 14, 147 \rangle$
C. $\mathbf{r}'(-1) = \langle 5e^{-5}, -7e^{7}, 1 \rangle$

Would you like more details or explanations?
Here are five related questions to expand your understanding:

1. How do you compute derivatives for vector-valued functions?
2. Why are the tangent vectors useful in physics and geometry?
3. What are the properties of sine and cosine at angles like $\frac{\pi}{4}$?
4. How do exponential functions behave under differentiation?
5. What is the significance of evaluating a derivative at a specific value?

Tip: When differentiating trigonometric or exponential functions, ensure you account for the chain rule!