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Numerade Educator

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Problem 44 Hard Difficulty

Suppose that we don't have a formula for $ g(x) $ but we know that $ g(2) = -4 $ and $ g'(x) = \sqrt {x^2 + 5} $ for all $ x. $
(a) Use a linear approximation to estimate $ g(1.95) $ and $ g(2.05). $
(b) Are your estimate in part (a) too large or too small? Explain.

Answer

(a) $g(1.95) \approx-4.15$
$g(2.05) \approx-3.85$
(b) estimates in part (a) are less than actual values

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Video Transcript

Okay. The first thing you know can do is we could substitute G prime of two into two squared plus five, which is a square to four plus five, which is scared of mine, which is equivalent to three. So now we have a G F ax is negative for plus three times X minus two, which is negative for post three acts minus six, which is three X minus 10. Therefore, we have G of 1.95 this three times 1.95 I was 10 negative. 4.15 Three times 2.5 minus tongue as negative. 3.85 Okay, we know that the second derivative is gonna be axe over the square root of ax squared. Plus five. Don't fit the chain role. No, we know the linear approximation is gonna be a line there for all the points away from the middle point to negative four from the center. X equals two are less than the actual values. Remember, G is calm cave up, so the estimates are less. Therefore, we haven't underestimate