A continuous function g is shown on a coordinate grid. The curve is increasing on the interval 2≤x≤4 and passes through the point P(3,5). Two additional labeled points on the curve are A(2.5,4.0) and B(3.5,6.2). Near x=3 the graph is concave up.
Throughout this problem, assume g is differentiable on the open interval that contains the three points described.
(a) Describe a procedure that uses the information given in the graph to obtain a numerical approximation for g′(3). Your answer should identify the quantities that must be computed and how they are combined.
(b) Without performing any calculations, explain why an approximation that relies on points closer to x=3 is generally more accurate than an approximation that uses points farther from x=3. Reference specific geometric features of the graph in your explanation.
(c) Decide whether the slope of the line through A and B is an overestimate or an underestimate of g′(3). Justify your answer using the concavity information provided.