Solution: $ d(t) = pt^3 + qt^2 + rt + s $. Compute $ d'(t) = 3pt^2 + 2qt + r $. From $ d(1) = p + q + r + s = 10 $, $ d'(1) = 3p + 2q + r = 12 $, $ d(2) = 8p + 4q + 2r + s = 28 $, $ d'(2) = 12p + 4q + r = 30 $. Subtract first equation from third: $ 7p + 3q + r = 18 $. Subtract $ d'(1) $ from this: $ (7p + 3q + r) - (3p + 2q + r) = 4p + q = 6 $. From $ d'(2) $: $ 12p + 4q + r = 30 $, and $ d'(1) $: $ 3p + 2q + r = 12 $. Subtract: $ 9p + 2q = 18 $. Now solve $ 4p + q = 6 $ and $ 9p + 2q = 18 $. Mu
["Solution: $ d(t) = pt^3 + qt^2 + rt + s $ and the Value of $ d(0) = -8 $", "Modeling dynamic systems with polynomial functions is essential in engineering, physics, and data science. One common model is a cubic function of the form\n$$\nd(t) = pt^3 + qt^2 + rt + s,\n$$\nwhere $ d(t) $ represents a measured quantity (such as position, output, or concentration) over time $ t $. Understanding its derivative helps analyze rates of change—critical for optimization and prediction.", "---", "### Step 1: Compute the Derivative", "The derivative of $ d(t) $, representing instantaneous rate of change, is:\n$$\nd'(t) = \frac{d}{dt}(pt^3 + qt^2 + rt + s) = 3pt^2 + 2qt + r.\n$$", "This expression is key to analyzing how $ d(t) $ evolves over time, especially when boundary conditions are given.", "---", "### Step 2: Apply Given Conditions", "We are provided with the following equations:\n- $ d(1) = p + q + r + s = 10 $\n- $ d'(1) = 3p + 2q + r = 12 $\n- $ d(2) = 8p + 4q + 2r + s = 28 $\n- $ d'(2) = 12p + 4q + r = 30 $", "To solve for $ p, q, r, s $, use substitution and elimination.", "---", "### Step 3: Form a System of Equations", "From the data:\n1. $ p + q + r + s = 10 $\n2. $ 3p + 2q + r = 12 $\n3. $ 8p + 4q + 2r + s = 28 $\n4. $ 12p + 4q + r = 30 $", "---", "### Step 4: Eliminate Variables Strategically", "Subtract Eq 1 from Eq 3:\n$$\n(8p + 4q + 2r + s) - (p + q + r + s) = 28 - 10\n\Rightarrow 7p + 3q + r = 18 \quad \ ext{(Eq A)}\n$$", "Subtract Eq 2 from Eq A:\n$$\n(7p + 3q + r) - (3p + 2q + r) = 18 - 12\n\Rightarrow 4p + q = 6 \quad \ ext{(Eq B)}\n$$", "Now subtract Eq 2 from Eq 4:\n$$\n(12p + 4q + r) - (3p + 2q + r) = 30 - 12\n\Rightarrow 9p + 2q = 18 \quad \ ext{(Eq C)}\n$$", "---", "### Step 5: Solve for $ p $ and $ q $", "From Eq B: $ q = 6 - 4p $", "Substitute into Eq C:\n$$\n9p + 2(6 - 4p) = 18\n\Rightarrow 9p + 12 - 8p = 18\n\Rightarrow p = 6\n$$", "Then:\n$$\nq = 6 - 4(6) = -18\n$$", "---", "### Step 6: Find $ r $ and $ s $", "Use Eq 2:\n$$\n3(6) + 2(-18) + r = 12\n\Rightarrow 18 - 36 + r = 12\n\Rightarrow r = 30\n$$", "Use Eq 1:\n$$\n6 - 18 + 30 + s = 10\n\Rightarrow 18 + s = 10\n\Rightarrow s = -8\n$$", "---", "### Step 7: Evaluate $ d(0) $", "The constant term $ s = d(0) $, so:\n$$\n\boxed{d(0) = -8}\n$$", "---", "This structured approach confirms that with a cubic model and key boundary data, we precisely determine all coefficients and recover the initial value efficiently. Whether applied to motion analysis, economics, or machine learning, understanding such derivatives enables deeper insight into system behavior."]
