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Transcript

Constructing a linear approximation: Suppose we have zoomed in on a particular x value or a function, say, x = 3. Then f (3) f (3 h) f (3) h Let’s now use a little bit of algebra to rearrange this to solve for f (3 h) . f (3 h) f (3) f (3)h We let 3 + h = x, where x is close in value to 3, since h is small. So far so good? Then h = x – 3. We now make those substitutions. f ( x) f (3) f (3)( x 3) This is now in the form of a linear equation: y = f(x), the slope m is f (3) , and the constant is “b”. This equation is a linear approximation to the graph near x = 3. Let’s see how this works for the function y x2 . Suppose we want to approximate the equation of the parabola with a straight line near x = 3. We know from our prior work in the previous section that an expression for the derivative is f ( x) 2x So a linear approximation near x = 3 would be: f ( x) f (3) f (3)( x 3) f ( x) 9 6( x 3) y 6x 9 The line y = 6x – 9 and the parabola y x2 are extremely close to each other around x = 3. We graph both equations to verify this: