Question: A robotics engineer is calibrating a sensor whose response curve follows the cubic polynomial $ f(x) $ such that $ f(1) = 3, f(2) = -1, f(3) = 5, f(4) = 11 $. Find $ f(0) $.

["Calibrating Precision: Finding the Cubic Polynomial Through Given Data Points", "When engineers calibrate sensors for precise measurements, one key step involves modeling the sensor’s response with a mathematical function. In this case, a robotics engineer faces the challenge of determining a cubic polynomial $ f(x) = ax^3 + bx^2 + cx + d $ that accurately captures sensor behavior, using four known data points:\n$ f(1) = 3 $, $ f(2) = -1 $, $ f(3) = 5 $, and $ f(4) = 11 $.\nUsing these, the engineer reconstructs the full function—and discovers the sensor’s response at $ x = 0 $, a critical input point.", "### The Challenge of a Cubic Fit", "Since $ f(x) $ is cubic, it has four unknown coefficients. With four equations derived from the known point values, we can solve for $ a, b, c, d $ uniquely. Let’s establish the system:", "From $ f(1) = 3 $:\n$$\na(1)^3 + b(1)^2 + c(1) + d = 3 \Rightarrow a + b + c + d = 3 \quad \ ext{(1)}\n$$", "From $ f(2) = -1 $:\n$$\n8a + 4b + 2c + d = -1 \quad \ ext{(2)}\n$$", "From $ f(3) = 5 $:\n$$\n27a + 9b + 3c + d = 5 \quad \ ext{(3)}\n$$", "From $ f(4) = 11 $:\n$$\n64a + 16b + 4c + d = 11 \quad \ ext{(4)}\n$$", "We now solve this system step by step.", "### Step 1: Eliminate $ d $", "Subtract equation (1) from (2):\n$$\n(8a + 4b + 2c + d) - (a + b + c + d) = -1 - 3\n\Rightarrow 7a + 3b + c = -4 \quad \ ext{(5)}\n$$", "Subtract (2) from (3):\n$$\n(27a + 9b + 3c + d) - (8a + 4b + 2c + d) = 5 - (-1)\n\Rightarrow 19a + 5b + c = 6 \quad \ ext{(6)}\n$$", "Subtract (3) from (4):\n$$\n(64a + 16b + 4c + d) - (27a + 9b + 3c + d) = 11 - 5\n\Rightarrow 37a + 7b + c = 6 \quad \ ext{(7)}\n$$", "### Step 2: Eliminate $ c $", "Subtract (5) from (6):\n$$\n(19a + 5b + c) - (7a + 3b + c) = 6 - (-4)\n\Rightarrow 12a + 2b = 10 \Rightarrow 6a + b = 5 \quad \ ext{(8)}\n$$", "Subtract (6) from (7):\n$$\n(37a + 7b + c) - (19a + 5b + c) = 6 - 6\n\Rightarrow 18a + 2b = 0 \Rightarrow 9a + b = 0 \quad \ ext{(9)}\n$$", "### Step 3: Solve for $ a $ and $ b $", "Subtract (8) from (9):\n$$\n(9a + b) - (6a + b) = 0 - 5 \Rightarrow 3a = -5 \Rightarrow a = -\frac{5}{3}\n$$", "Substitute $ a = -\frac{5}{3} $ into (9):\n$$\n9\left(-\frac{5}{3}\right) + b = 0 \Rightarrow -15 + b = 0 \Rightarrow b = 15\n$$", "### Step 4: Solve for $ c $", "Use equation (5):\n$$\n7a + 3b + c = -4\n\Rightarrow 7\left(-\frac{5}{3}\right) + 3(15) + c = -4\n\Rightarrow -\frac{35}{3} + 45 + c = -4\n\Rightarrow \left(\frac{-35 + 135}{3}\right) + c = -4\n\Rightarrow \frac{100}{3} + c = -4\n\Rightarrow c = -4 - \frac{100}{3} = -\frac{112}{3}\n$$", "### Step 5: Solve for $ d $", "Use equation (1):\n$$\na + b + c + d = 3\n\Rightarrow -\frac{5}{3} + 15 - \frac{112}{3} + d = 3\n\Rightarrow \left(-\frac{5 + 112}{3}\right) + 15 + d = 3\n\Rightarrow -\frac{117}{3} + 15 + d = 3\n\Rightarrow -39 + 15 + d = 3\n\Rightarrow -24 + d = 3 \Rightarrow d = 27\n$$", "### Step 6: Evaluate $ f(0) $", "The cubic polynomial is:\n$$\nf(x) = -\frac{5}{3}x^3 + 15x^2 - \frac{112}{3}x + 27\n$$", "At $ x = 0 $:\n$$\nf(0) = d = 27\n$$", "### Conclusion", "By solving the cubic interpolation problem through systematic elimination, the robotics engineer successfully reconstructs the sensor’s response curve. The calibration reveals that the sensor outputs $ f(0) = 27 $, a critical baseline value for system feedback and control. This precision in polynomial fitting exemplifies how mathematical modeling underpins reliable robotic sensor design.", "Key Takeaways:", "- A cubic polynomial is uniquely determined by four distinct data points.\n- Eliminating variables step-by-step efficiently solves systems arising in engineering calibration.\n- The constant term $ d $ corresponds to $ f(0) $, a vital reference point in sensor deployment.", "For engineers and students alike, understanding polynomial interpolation unlocks deeper insights into real-world data fitting and system modeling."]