Question
Asked Mar 28, 2019
(a) Use an appropriate linearization to approximate 46
(b) Use the same linearization from part (a) to approximate N50
(c) Which of the above do you think is a better approximation? Explain your reasoning.
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(a) Use an appropriate linearization to approximate 46 (b) Use the same linearization from part (a) to approximate N50 (c) Which of the above do you think is a better approximation? Explain your reasoning.

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Expert Answer

Step 1

For a given function f(x), the equation of tangent at a point x = a and y = f(a) is given by:

y - f(a) = f'(a).(x - a)

Hence, y = f(a) + f'(a).(x - a)

Also, since y = f(x), we can rewrite the above equation as:

f(x) = f(a) + f'(a).(x - a)

This is the linearization function for f(x), which is linear in nature with respect to x.

Step 2

Part (a)

We have to find (46)1/2

So, the corresponding function will be f(x) =x1/2 

We now need to choose "a". Let's choose "a" in close vicinity of 46 for which square root is easy to find.

Hence, let's choose a = 49

f(a) = f(49) = 491/2 = 7

f'(x) = 0.5x-1/2 

f'(a) = 0.5 x 49-1/2 = 0.5 / 7 = 0.071428571 = 0.0714

Hence, f(x) = f(a) + f'(a).(x - a)

f(x) = x1/2 = 7 + 0.0714 (x - 49)

Now set x = 46.

Hence,

461/2 = 7 + 0.0714 x (46 – 49) = 6.785714286 = 6.7857

Step 3

Part (b)

Now set x = 50.

Hence,

461/2 = 7 + 0.0714 x...

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Calculus

Derivative

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