What is the value of f'(x) at x = 4 from the following table of values?x 1 2 3 4 f(x) 20 22 27 35
9.5
Finding the Derivative f'(x) from a Table of Values
When we are given a set of discrete data points in a table, and we need to find the derivative f'(x) at a specific point, we use numerical differentiation methods. These methods approximate the derivative using the available data points.
Understanding the Problem
We are given the following table showing values of x and the corresponding values of f(x):
| x | f(x) |
|---|---|
| 1 | 20 |
| 2 | 22 |
| 3 | 27 |
| 4 | 35 |
We need to find the value of the derivative f'(x) at x = 4.
Numerical Differentiation Methods
Common numerical differentiation formulas include:
- Forward Difference: Approximates f'(x) using $x$ and $x+h$. Formula: $\frac{f(x+h) - f(x)}{h}$
- Backward Difference: Approximates f'(x) using $x$ and $x-h$. Formula: $\frac{f(x) - f(x-h)}{h}$
- Central Difference: Approximates f'(x) using $x-h$ and $x+h$. Formula: $\frac{f(x+h) - f(x-h)}{2h}$
In this table, the points are equally spaced with $h=1$. At x = 4, we only have a point before it (x=3). The standard backward difference formula at x=4 would use points x=4 and x=3:
f'(4) $\approx \frac{f(4) - f(3)}{4 - 3} = \frac{35 - 27}{1} = 8$.
However, 8 is not one of the given options. For approximating the derivative at an endpoint like x=4 when more preceding points are available, a more accurate formula is often used. One such formula is the three-point backward difference formula.
Using the Three-Point Backward Difference Formula
The three-point backward difference formula for approximating f'(x) at x using points x, x-h, and x-2h is:
f'(x) $\approx \frac{3f(x) - 4f(x-h) + f(x-2h)}{2h}$
In our case, we want to find f'(4). We have h=1, so we will use the points:
- x = 4, f(4) = 35
- x - h = 3, f(3) = 27
- x - 2h = 2, f(2) = 22
Let's substitute these values into the three-point backward difference formula for x=4:
f'(4) $\approx \frac{3f(4) - 4f(3) + f(2)}{2(1)}$
f'(4) $\approx \frac{3(35) - 4(27) + 22}{2}$
f'(4) $\approx \frac{105 - 108 + 22}{2}$
f'(4) $\approx \frac{-3 + 22}{2}$
f'(4) $\approx \frac{19}{2}$
f'(4) $\approx 9.5$
Result and Conclusion
Using the three-point backward difference formula, the approximate value of f'(x) at x = 4 is 9.5. This value matches one of the given options.
Revision Table: Key Numerical Differentiation Formulas
| Formula Type | Formula for f'(x) | Points Used | Common Use Case |
|---|---|---|---|
| Forward Difference | $\frac{f(x+h) - f(x)}{h}$ | $x, x+h$ | Beginning of table, general approximation |
| Backward Difference | $\frac{f(x) - f(x-h)}{h}$ | $x-h, x$ | End of table, general approximation |
| Central Difference | $\frac{f(x+h) - f(x-h)}{2h}$ | $x-h, x+h$ | Interior points, generally more accurate than two-point methods |
| Three-Point Backward Difference | $\frac{3f(x) - 4f(x-h) + f(x-2h)}{2h}$ | $x-2h, x-h, x$ | More accurate approximation at the end of the table |
Additional Information: Accuracy of Numerical Derivatives
The accuracy of numerical differentiation depends on the method used and the step size $h$. Smaller values of $h$ generally lead to better approximations, but very small $h$ can introduce errors due to floating-point precision. The formulas used are typically derived from polynomial interpolation. For example, the three-point backward difference formula comes from fitting a quadratic polynomial through the points $(x-2h, f(x-2h))$, $(x-h, f(x-h))$, and $(x, f(x))$, and then differentiating the polynomial at x. Higher-order formulas can provide even greater accuracy if more data points are available.
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