We are given a table of values for a function $f(x)$ and we want to estimate the value of $f(2)$. The table provides the following data: $x = -5$, $f(x) = -2$ $x = 3$, $f(x) = 2$ $x = 5$, $f(x) = 3$

Applied MathematicsNumerical AnalysisInterpolationLinear InterpolationFunction Approximation
2025/3/6

1. Problem Description

We are given a table of values for a function f(x)f(x) and we want to estimate the value of f(2)f(2). The table provides the following data:
x=5x = -5, f(x)=2f(x) = -2
x=3x = 3, f(x)=2f(x) = 2
x=5x = 5, f(x)=3f(x) = 3

2. Solution Steps

We can use linear interpolation to estimate the value of f(2)f(2). Since x=2x=2 is between x=5x=-5 and x=3x=3, we can consider the points (5,2)(-5, -2) and (3,2)(3, 2).
The equation of the line passing through the points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:
yy1=y2y1x2x1(xx1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)
Using the points (5,2)(-5, -2) and (3,2)(3, 2), we have:
y(2)=2(2)3(5)(x(5))y - (-2) = \frac{2 - (-2)}{3 - (-5)} (x - (-5))
y+2=48(x+5)y + 2 = \frac{4}{8} (x + 5)
y+2=12(x+5)y + 2 = \frac{1}{2} (x + 5)
y=12x+522y = \frac{1}{2}x + \frac{5}{2} - 2
y=12x+5242y = \frac{1}{2}x + \frac{5}{2} - \frac{4}{2}
y=12x+12y = \frac{1}{2}x + \frac{1}{2}
Now, we can estimate f(2)f(2) by plugging in x=2x = 2:
f(2)=12(2)+12f(2) = \frac{1}{2}(2) + \frac{1}{2}
f(2)=1+12f(2) = 1 + \frac{1}{2}
f(2)=32f(2) = \frac{3}{2}

3. Final Answer

f(2)=32f(2) = \frac{3}{2}

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