Mathematics
Mathematics
Problem
The function is differentiable for . You want to approximate using a tangent line. Which base point gives the more accurate linear approximation, or ? State the chosen point and one justification based on the information provided.Intuitive explanation
The accuracy of a linear approximation depends on how far you travel from the base point along the curve—the farther you go, the more the tangent line diverges from the actual function. Since is concave down (its second derivative is negative), the tangent line always overestimates the true value, and the error grows with distance. Choosing the base point closest to minimizes this distance and gives the better approximation.
Intuitive explanation
The accuracy of a linear approximation depends on how far you travel from the base point along the curve—the farther you go, the more the tangent line diverges from the actual function. Since is concave down (its second derivative is negative), the tangent line always overestimates the true value, and the error grows with distance. Choosing the base point closest to minimizes this distance and gives the better approximation.