Suppose that the only information we have about a function fis that f(1) and the

Question

Suppose that the only information we have about a function fis that f(1) and the graph of its derivative is as shown. y = f'(x) 0.9) and f(1.1). (a) use a linear approximation to estimate (0.9) (b) Are your estimates in part (a) too large or too small? Explain. O The slopes of the tangent lines are negative, but the tangents are becoming steeper. So the tangent lines lie below the curve f. Thus, the estimates are too large O The slopes of the tangent lines are negative, but the tangents are becoming steeper. So the tangent lines lie below the curve f Thu, the estimates are too small O The slopes of the tangent lines are positive, but the tangents are becoming less steep. So the tangent lines lie above the curve f. Thus, the estimates are too small O The slopes of the tangent lines are positive, but the tangents are becoming less steep. So the tangent lines lie above the curve Thus, the estimates are too large.

Explanation / Answer

Linear approximation

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

Here a=1

f(1) = 0

f'(1) = 2

L(x) =f(1) + f'(1)(x - 1)

L(x) =0 + 2(x -1)

=2(x-1)

f(0.9)

L(0.9) = 2*(0.9-1) = -0.2

f(1.1)

L(1.1) = 2*(1.1- 1) = 0.2

(b)

from the graph f'(x) is positive and decreasing

The tangent of slope is positive (f'(x) > 0) but the tangents are less steeper . so tangent lines lies above the cirve f .thus the estimate are too large

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