1. **State the problem:** We are given a function $G(x)$ with $G(k) = 197$ and $G'(k) = 5 \cdot 8 = 40$. We want to estimate $G(k+2)$ using this information. 2. **Formula used:** We use the linear approximation (or tangent line approximation) formula: $$G(k + h) \approx G(k) + G'(k) \cdot h$$ where $h = 2$ in this case. 3. **Calculate the estimate:** $$G(k + 2) \approx 197 + 40 \cdot 2 = 197 + 80 = 277$$ 4. **Interpretation:** The estimate for $G(k+2)$ is 277 based on the tangent line approximation. --- 5. **Part (ii) reasoning:** Given $G(x) = A e^{bx}$ with $A, b > 0$, the function is exponential and increasing. 6. The derivative is: $$G'(x) = A b e^{bx}$$ which is positive and increasing because $e^{bx}$ grows exponentially. 7. The tangent line approximation uses the slope at $x=k$ to estimate $G(k+2)$, but since $G(x)$ is convex (its second derivative $G''(x) = A b^2 e^{bx} > 0$), the actual value $G(k+2)$ will be **greater** than the linear estimate. 8. **Conclusion:** The estimate is **too small** compared to the actual value. --- **Final answers:** (i) $G(k+2) \approx 277$ (ii) The estimate is **too small**.