People's Education Press: Senior High School English, Elective Compulsory Course II (A Edition): Cognitive Transfer – Analogical Reasoning in Sequences and Exploring the Collatz Conjecture

This lesson explores the intrinsic patterns of discrete sequences—such as the iterative process of the Collatz Conjecture and the dual relationship between arithmetic and geometric sequences—to guide students in making a cognitive shift from 'discrete evolution' to 'continuous change.' By using mathematical induction and analogical reasoning as logical scaffolding, the goal is to cultivate students’ ability to identify patterns, naturally leading into the powerful tool of derivatives, which describe instantaneous rates of change for continuous variables.

Detailed Explanation of Core Concepts

Evolution of Patterns and Conjectures:By analyzing the iterative trajectory of the Collatz Conjecture $a_{n+1} = \begin{cases} \frac{a_n}{2}, & a_n \text{ is even} \\ 3a_n+1, & a_n \text{ is odd} \end{cases}$, experience the interplay of uncertainty and determinism within discrete systems, and appreciate how the 'rate of change' exhibits jumps across different states.

Dualism and Transfer in Structured Thinking:Apply the principle of duality (e.g., '+' in arithmetic sequences transforming to '$\times$' in geometric sequences) to understand the isomorphism of mathematical structures. This analogical reasoning provides an essential intuitive foundation for grasping derivative rules (such as the connection between the product rule and the sum rule).

Rigorous Logical Proof:Use the second principle of mathematical induction to verify complex sequence summation formulas (such as $\sum i^2$) or closed-form solutions, thereby building a robust toolkit of proofs for the rigorous derivation of future derivative formulas.

From the 'difference' in sequences to the 'derivative' in functions, we are bridging the logical gap between average trends and local instants. Summary of key formulas:

$$F_n = \frac{1}{\sqrt{5}} \left[ \left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n \right], \quad \sum_{i=1}^n i^2 = \frac{1}{6}n(n+1)(2n+1)$$

QUESTION 1

Find the instantaneous velocity of the diver at $t=2\,\text{s}$ (given the height function $h(t) = -4.9t^2 + 4.8t + 11$).

$-14.8\,\text{m/s}$

$-19.6\,\text{m/s}$

$4.8\,\text{m/s}$

$-10.0\,\text{m/s}$

QUESTION 2

In Example 2, given the crude oil temperature function $y = x^2 - 7x + 15$, calculate the instantaneous rate of change of temperature at $3\text{h}$ and $5\text{h}$, and explain its significance.

$-1$ (cooling) and $3$ (warming)

$-1$ (warming) and $3$ (cooling)

$1$ (warming) and $5$ (warming)

$-4$ (cooling) and $2$ (warming)

QUESTION 3

Based on the description, select the shape of the graph: (1) constant speed driving; (2) accelerating driving; (3) decelerating driving.

(1) straight line; (2) upward curvature; (3) approaching horizontal

(1) straight line; (2) downward curvature; (3) upward curvature

(1) curve; (2) straight line; (3) approaching horizontal

(1) horizontal line; (2) slanted line; (3) curve

QUESTION 4

In the Collatz Conjecture, if the starting number is $a_1 = 7$, what is the third term $a_3$?

$11$

$22$

$34$

$5$

QUESTION 5

According to the 'principle of duality,' if the property of an arithmetic sequence is $a_{n+1} - a_n = d$, what is the corresponding property for a geometric sequence?

$b_{n+1} \div b_n = q$

$b_{n+1} - b_n = q$

$b_{n+1} \cdot b_n = q$

$b_{n+1}^2 = b_n$

QUESTION 6

What is the main difference between the second and first principles of mathematical induction?

Different assumptions: the former assumes all $n \le k$ hold true

Different base steps: the former does not require verifying $n_0$

Different conclusions: the former can only prove finite terms

Different applicability: the former only applies to sequences

QUESTION 7

Given $f(x) = x^2 + 1$. Find the equation of the tangent line to the curve at point $(0, 1)$.

$y = 1$

$y = 0$

$y = x + 1$

$x = 0$

QUESTION 8

For a $3\text{kg}$ object with displacement $y(t) = 1 + t^2$, find the kinetic energy $E_k$ at $t = 5\text{s}$.

$150\,\text{J}$

$75\,\text{J}$

$300\,\text{J}$

$15\,\text{J}$

QUESTION 9

If sequence $b_n$ is geometric and $b_n > 0$, by analogy with the arithmetic mean property of arithmetic sequences $\frac{a_n + a_m}{2} = a_{\frac{n+m}{2}}$, what is the corresponding property for geometric sequences?

$\sqrt{b_n \cdot b_m} = b_{\frac{n+m}{2}}$

$\frac{b_n \cdot b_m}{2} = b_{\frac{n+m}{2}}$

$b_n + b_m = b_{n+m}$

$\sqrt{b_n + b_m} = b_{n+m}$