Linearization and approximation

Question

Which of the following is the best linear approximation for f(x) = sin(x) near x = π seconds?

Options:

1. y = -x + π - 1
2. y = -1
3. y = -x + π
4. y = -x - π

Answer 1

The time is limited at AP Calculus exam. This multiple-choice question can be quickly answered by the elimination of incorrect options.

The function value at x = π is f(π) = sin(π) = 0. For the options

1. y(π) = -1
2. y(π) = -1
3. y(π) = 0
4. y(π) = -2π

Therefore, only the answer #3 gives the correct value f(π) = 0. Hence the answer #3 is the correct choice.

Answer 2

The linear approximation for the graph of f(x) = sin x is a line with a slope equal to the value of the derivative at the point x = π.

First, find the derivative:

f'(x) = Cos x

At x = π

f'(π) = Cos π = -1

And...

f(π) = Sin π = 0

So now we have a classic Algebra problem. We have a slope = -1 and a point (π , 0). Find the equation:

y = mx + b

y = -(1)x + b

0 = -(1)(π) + b

0 = -π + b

π = b

So the equation is y = -x + π

Choice #3 above is the correct answer.