Question 1181444: An epidemic is spreading through a large western state. Health officials estimate that the number of persons who will be afflicted by the disease is a function of time since the disease was first detected.
Specifically, the function is 𝑛 = 𝑓(𝑡) = 350𝑡 3 − 22𝑡 2 Approximate the instantaneous rate of change at which the disease is expected to be spreading at t=20 days, using,
i. Three Point Central Difference Formula
ii. Three Point Forward Difference Formula
iii. Five Point Central Difference Formula
Also calculate the respective Absolute Relative Error. Let spacing between points equivalent to 0.01
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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