Question

(a) Use a linear approximation to estimate f(0.9) and f(1.1). f(0.9) ≈ f(1.1) ≈ (b) Are your estimates in part (a) too large or too small? Explain. 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. 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 small. 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 large. 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.

Answer 1

Answer:

(Missing part of the question is attached)

$L(x)=2x+3$

Estimates are too large.

Step-by-step explanation:

Suppose the only information we know about the function is:

$f(1)=5$

where the graph of its derivative is shown in the attachment

(a)

If the function $f$ is differentiable at point $x=1$ , the tangent line to the graph of $f$ at 1 is given by the equation:

$y=f(1) +f'(1)(x-1)$

So we call the linear function:

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

We know the $f(1)=5$ as it is given in the question, and $f'(1)=2$ from the graph attached. Substitute in the equation of $L(x)$.

$L(x)=5+2(x-1)\\L(x)=5+2x-2\\L(x)=2x+3$

(b)

At x=1,  $f'(x)$ is positive but it is decreasing. However. if we draw the tangent lines, we see that the tangent lines are becoming less steeper, so the tangent lines lie above the curve $f$. Thus, The estimates are too large.