Video Example EXAMPLE 1 Find the linearization of the function f(x) = x + 1 at a = 3 and use it to approximate the numbers 3.98
and 4.05 . Are these approximations overestimates or underestimates? SOLUTION The derivative of f(x) = (x + 1)1/2 is f '(x) = Correct: Your answer is correct. . and so we have f(3) = 2 Correct: Your answer is correct. and f '(3) = 1/4 Correct: Your answer is correct. . Putting these values into this equation, we see that the linearization is
1 answer:
Answer:
the linearization is y = 1/4x +5/4
the linearization will produce <em>overestimates</em>
the values computed from this linearization are ...
f(3.98) ≈ 2.245
f(4.05) ≈ 2.2625
Step-by-step explanation:
Apparently, you have ...
from which you have correctly determined that ...
so that f(3) = 2 and f'(3) = 1/4. Putting these values into the point-slope form of the equation of a line, we get the linearization ...
g(x) = (1/4)(x -3) +2
g(x) = (1/4)x +5/4
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The values from this linearization will be overestimates, as the curve f(x) is concave downward everywhere. The tangent (linearization) is necessarily above the curve everywhere.
__
At the given values, we find ...
g(3.98) = 2.245
g(4.05) = 2.2625
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I Hope this helps, greetings ... Dexteright02! =)
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