Practice AP Calculus Unit 5.3 questions on using the sign of the first derivative to justify where a function increases or decreases.
The functions f and g are twice differentiable. The table shown gives values of the functions and their first derivatives at selected values of x.
Is the function m defined in part (c) increasing, decreasing, or neither at x=2 ? Justify your answer.
Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.
m′(x)=15 x^2+f′(x)
Considers m′(x)
1 point
m′(2)=15 · 4+f′(2)=60+(-8)=52
m′(2) with
supporting work
1 point
The graph of m is increasing at x=2 because m′(2)>0.
Answer with justification
1 point
Scoring notes:
- The first point is earned for considering m′(x), m′(2), or m′. This consideration may appear in a justification statement.
- The second point is earned for m′(2)=15 · 2^2+f′(2), m′(2)=60+f′(2), or m′(2)=60-8 but is not earned for an unsupported response of m′(2)=52.
- The third point is earned for an answer and justification consistent with any declared value of m′(2).