Let , defined on the interval . A student claims that the graph of is always…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let h(x)=csc(x), defined on the interval (0,π). A student claims that the graph of h is always decreasing because its derivative is negative for all x in (0,π).
(1) Part 1: (a) Write h(x) in reciprocal notation and explain, in words, why differentiating a reciprocal function should introduce the square of the original denominator into the new denominator.
(2) Part 2: (b) Using the fact that dxdsinx=cosx, argue qualitatively—without completing a detailed algebraic manipulation—why the numerator of h′(x) must contain cosx and why the overall sign of h′(x) is negative on (0,π).
(3) Part 3: (c) Combine your reasoning from parts (a) and (b) to justify the student’s claim that h is decreasing on (0,π). State any trigonometric identities you invoke in your explanation.