Suppose that we don't have a formula for g(x) but we know that g(3) = −5 and g'(x) = x2 + 7 for all x.

(a) use a linear approximation to estimate g(2.9) and g(3.1). g(2.9) ≈ g(3.1) ≈
(b) are your estimates in part (a) too large or too small? explain.

a)

b)

The values are too small since is positive for both values of in. I'm speaking of the values, 2.9 and 3.1.

Step-by-step explanation:

a)

The point-slope of a line is:

where is the slope and is a point on that line.

We want to find the equation of the tangent line of the curve at the point on .

So we know .

To find , we must calculate the derivative of at :

.

So the equation of the tangent line to curve at is:

.

I'm going to solve this for .

Subtract 5 on both sides:

What this means is for values near is that:

.

Let's evaluate this approximation function for .

Let's evaluate this approximation function for .

b) To determine if these are over approximations or under approximations I will require the second derivative.

If is positive, then it leads to underestimation (since the curve is concave up at that number).

If is negative, then it leads to overestimation (since the curve is concave down at that number).

is positive for .

is negative for .

That is, .

So is positive for both values of which means that the values we found in part (a) are underestimations.

its 69.420

step-by-step explanation: