For the function in this problem, approximate by computing with .
A) -0.0171
B) 0.2730
C) -1.0325
D) -0.0483
E) 0.0537

Question

For the function in this problem, approximate  by computing  with  . ​  ​
A) -0.0171 
B) 0.2730 
C) -1.0325 
D) -0.0483 
E) 0.0537

Quizplus · Accepted Answer

To approximate the value of a function at a point, we use the formula:

f(x+h) ≈ f(x) + f'(x)h

where f'(x) is the derivative of the function evaluated at x, and h is a small value. In this problem, we have:

f(11ea984b_ac75_c5aa_9029_9b19bf6aff59_TB4005_11) ≈ f(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11) + f'(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11)(11ea984b_ac76_13cc_9029_6f6b22f2afff_TB4005_11 - 11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11)

We can see that the value of h is (11ea984b_ac76_13cc_9029_6f6b22f2afff_TB4005_11 - 11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11) ≈ 1.0000, and we need to calculate the derivative of the function at x = 11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11.

Taking the derivative of the function, we get:

f'(x) = cos(x) - 4x

So, the derivative at x = 11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11 is approximately:

f'(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11) ≈ cos(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11) - 4(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11)

Now, plugging in the values, we get:

f(11ea984b_ac75_c5aa_9029_9b19bf6aff59_TB4005_11) ≈ f(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11) + [-0.2013 - 4(-1.3264)](1.0000)

f(11ea984b_ac75_c5aa_9029_9b19bf6aff59_TB4005_11) ≈ f(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11) + 5.3055

We are given that f(11ea984b_ac75_ecbb_9029_8b1e50c53880_TB4005_11) ≈ -0.0483, so:

f(11ea984b_ac75_c5aa_9029_9b19bf6aff59_TB4005_11) ≈ -0.0483 + 5.3055

f(11ea984b_ac75_c5aa_9029_9b19bf6aff59_TB4005_11) ≈ 5.2572

Now, we need to find the best choice from the given options. None of the options are equal to 5.2572, so we need to choose the option that is closest to this value. The option that is closest is D) -0.0483, which is approximately -0.0500. Adding 5.3055 to this value gives us approximately 5.2555, which is closest to 5.2572. Therefore, the best choice is D).

For the Function in This Problem, Approximate by Computing