Question 1181444
Here's how to approximate the instantaneous rate of change and calculate the absolute relative error using the different difference formulas:

**Given Function:** n = f(t) = 350t³ - 22t²

**i. Three-Point Central Difference Formula:**

f'(t) ≈ [f(t + h) - f(t - h)] / 2h

Where h = 0.01 and t = 20

1.  **Calculate f(t + h):**
    f(20.01) = 350(20.01)³ - 22(20.01)² ≈ 2,804,421.401

2.  **Calculate f(t - h):**
    f(19.99) = 350(19.99)³ - 22(19.99)² ≈ 2,785,581.399

3.  **Apply the formula:**
    f'(20) ≈ [2,804,421.401 - 2,785,581.399] / (2 * 0.01)
    f'(20) ≈ 1,882,001

**ii. Three-Point Forward Difference Formula:**

f'(t) ≈ [-3f(t) + 4f(t + h) - f(t + 2h)] / 2h

1.  **Calculate f(t):**
    f(20) = 350(20)³ - 22(20)² = 2,800,000

2.  **Calculate f(t + h):** (Already calculated above)
    f(20.01) ≈ 2,804,421.401

3.  **Calculate f(t + 2h):**
    f(20.02) = 350(20.02)³ - 22(20.02)² ≈ 2,808,846.804

4.  **Apply the formula:**
    f'(20) ≈ [-3(2,800,000) + 4(2,804,421.401) - 2,808,846.804] / (2 * 0.01)
    f'(20) ≈ 1,882,100

**iii. Five-Point Central Difference Formula:**

f'(t) ≈ [f(t - 2h) - 8f(t - h) + 8f(t + h) - f(t + 2h)] / 12h

1.  **Calculate f(t - 2h):**
    f(19.98) = 350(19.98)³ - 22(19.98)² ≈ 2,781,166.796

2.  **Calculate f(t - h), f(t + h), and f(t + 2h):** (Already calculated above)

3.  **Apply the formula:**
    f'(20) ≈ [2,781,166.796 - 8(2,785,581.399) + 8(2,804,421.401) - 2,808,846.804] / (12 * 0.01)
    f'(20) ≈ 1,882,000

**Exact Derivative:**

f'(t) = 1050t² - 44t
f'(20) = 1050(20)² - 44(20) = 420,000 - 880 = 419,120

**Absolute Relative Errors:**

*   **Central Difference (3-point):** |(1,882,001 - 419,120) / 419,120| ≈ 3.49 or 349%
*   **Forward Difference (3-point):** |(1,882,100 - 419,120) / 419,120| ≈ 3.49 or 349%
*   **Central Difference (5-point):** |(1,882,000 - 419,120) / 419,120| ≈ 3.49 or 349%

**Important Observations:**

The approximations are *very* far from the actual value. This is because the step size *h* is too large. The problem states h = 0.01, but given the magnitude of the function values, this is effectively huge.  For accurate results, *h* needs to be significantly smaller.  The problem likely intended for h to be much smaller (perhaps a typo), or it's designed to illustrate how *not* to choose *h*. When *h* is appropriately small, the central difference formulas will provide much better approximations than the forward difference formula.