A twice–differentiable function satisfies for all real
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A twice–differentiable function h satisfies h′(x)=3x2−4x+7 for all real x. A class discussion arises in which one student claims there are infinitely many such functions, whereas another insists the condition h(0)=2 selects exactly one function.
Respond to the discussion in the parts that follow.
(a) Derive an explicit expression for the family of all functions whose derivative is 3x2−4x+7. Clearly indicate any arbitrary constants that appear in your result.
(b) Prove that if two differentiable functions f and g both satisfy f′(x)=g′(x)=3x2−4x+7 for all x and additionally satisfy f(0)=g(0)=2, then f(x)=g(x) for every real number x. Your argument should reference appropriate calculus principles and be written in complete sentences.
(c) Explain, in the context of parts (a) and (b), why specifying a single point (x0,y0) on a tangent curve is sufficient to determine a unique primitive function when its derivative is known and continuous on an interval.