Consider a differentiable function whose graph has been drawn on square grid paper
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider a differentiable function g whose graph has been drawn on square grid paper. The x– and y–axes are scaled so that one grid square corresponds to one unit on each axis. At three particular x–values, the tangent line appears straight enough that its slope can be measured directly against the grid. The following observations are made from the drawing:
• At x=−2, the tangent line rises roughly 3 units while running 5 units to the right.
• At x=0, the tangent line falls roughly 1 unit while running 4 units to the right.
• At x=3, the tangent line rises roughly 2 units while running 1 unit to the right.
Using these visual measurements, answer the questions below.
(a) Compute an estimated numerical value for g′(−2), g′(0), and g′(3) based on the observed rise‐over‐run ratios.
(b) Rank the three estimated derivatives from least to greatest and clearly indicate any equality.
(c) Provide a written explanation linking each numerical estimate to the geometric features of the tangent lines observed on the grid.