Let f be the function defined by . Find the approximate change in y if x changes from 2 to 2.05. Find the actual change in y if x changes from 2 to 2.05.

A) ,
B) ,
C) ,
D) ,

Question

Let f be the function defined by  . Find the approximate change in y if x changes from 2 to 2.05. Find the actual change in y if x changes from 2 to 2.05.
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A) , 
B) , 
C) ​ , 
D) ​ ,

Quizplus · Accepted Answer

To find the approximate change in y, we can use the formula:

Δy ≈ f'(x)Δx

where f'(x) is the derivative of f with respect to x evaluated at x=2, and Δx = 2.05 - 2 = 0.05.

Using the power rule for differentiation, we have:

f'(x) = 3x^2 - 8x + 2

f'(2) = 3(2)^2 - 8(2) + 2 = -10

Substituting into the formula, we get:

Δy ≈ (-10)(0.05) = -0.5

Therefore, the approximate change in y is -0.5.

To find the actual change in y, we can compute:

Δy = f(2.05) - f(2)

= (2.05)^3 - 4(2.05)^2 + 2(2.05) - [(2)^3 - 4(2)^2 + 2(2)]

= 0.040125 - 0.08 + 0.1 - 2

= -1.839875

Therefore, the actual change in y is approximately -1.84.

The only choice that includes both -0.5 and -1.84 as options is B.

Let F Be the Function Defined by