Question

Suppose that we don't have a formula for g(x) but we know that g(3) = −1 and g'(x) = x2 + 7 for all x.

Answer 1

Answer:

• g(2.95) ≈ -1.8; g(3.05) ≈ -0.2
• A) tangents are increasing in slope, so the tangent is below the curve, and estimates are too small.

Step-by-step explanation:

(a) The linear approximation of g(x) at x=b will be ...

  g(x) ≈ g'(b)(x -b) +g(b)

Using the given relations, this is ...

  g'(3) = 3² +7 = 16

  g(x) ≈ 16(x -3) -1

Then the points of interest are ...

  g(2.95) ≈ 16(2.95 -3) -1 = -1.8

  g(3.05) ≈ 16(3.05 -3) -1 = -0.2

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(b) At x=3, the slope of the curve is increasing, so the tangent lies below the curve. The estimates are too small. (Matches description A.)