Practice AP Calculus 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.
Is the function m defined in part (c) increasing, decreasing, or neither at x=2 ? Justify your answer.
m^(x)=15 x^2+f^(x)
Considers m^(x)
1 point
m^(2)=15. 4+f^(2)=60+(-8)=52m^(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).
Total for part (d)
for question 5
9 points
Part B (AB): Graphing calculator not allowed
Question 6 9 points
General Scoring Notes
The model solution is presented using standard mathematical notation.
Answers (numeric or algebraic) need not be simplified. Answers given as a decimal approximation should be correct to three places after the decimal point. Within each individual free-response question, at most one point is not earned for inappropriate rounding.