Curve Sketching Using Differential Calculus

Curve sketching utilizes differential calculus to analyze and graph the behavior of function curves. The first derivative, $f'(x)$, identifies critical points-where $f'(x)=0$ or is undefined-that indicate local maxima, minima, or saddle points by revealing intervals where the function increases or decreases. The second derivative, $f''(x)$, reveals concavity: positive values mean the graph curves upward, negative values curve downward, and points where changes sign mark inflection points. The second derivative test classifies critical points by testing : positive value indicates a local minimum, negative a local maximum, and zero leaves the test inconclusive. Additional features like intercepts and asymptotes complement the sketch for more accurate depiction. This method is crucial in engineering mathematics as it aids in optimizing designs, resource allocation, and anticipating system behavior through detailed functional analysis.