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Question
Find a linear approximation for the following functions at the indicated points.
g(x) = `sqrt(x^2 + 9)`, x0 = – 4
Solution
g(x)= `sqrt(x^2 + 9)`, x0 = – 4
g(x)= `1/(2sqrt(x^2 + 9)) (2x) = x/sqrt(x^2 + 9)`
g(x0) = `"g"(- 4)`
= `sqrt(16 + 9)`
= `sqrt(25)`
= 5
g'(x0) = `(- 4)/sqrt(25)`
= `(- 4)/5`
The required linear approximation L(x) = `"g"(x_0) + "g'"(x_0)(x - x_0)`
= `5 - 4/5 (x + 4)`
= `5 - (4x)/5 - 16/5`
= `9/5 - (4x)/5`
= `(9 - 4x)/5`
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