MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let f(x)=x3−6x+4. Consider the closed interval [1,4].
(a) Calculate the average rate of change of f on [1,4].
(b) Determine all values c in (1,4) for which the instantaneous rate of change of f equals the average rate of change found in part (a). Show all work leading to your answer.
(c) Write the equation of the tangent line to the graph of f at each value of c obtained in part (b).
(d) Describe how the tangent line(s) found in part (c) relate geometrically to the secant line over [1,4]. Your answer should reference slopes and positions of the lines relative to the curve.