AP Calculus BC Fun 4 A Justify Conclusions About the Behavior of a Function Based on the Behavior of Its Derivatives Questions

Practice AP Calculus BC questions on using the sign of the first derivative to justify where a function increases or decreases.

[Maximum number: 2]

Let f be a differentiable function with f(4)=3. On the interval $0 \leq x \leq 7$ , the graph of $f^{\prime}$ , the derivative of f, consists of a semicircle and two line segments, as shown in the figure above.

Let g be the function defined by g(x)=f(x)-x. On what intervals, if any, is g decreasing for $0 \leq x \leq 7$ ? Show the analysis that leads to your answer.

$g^{\prime}(x)=f^{\prime}(x)-1$ $g^{\prime}(x)=f^{\prime}(x)-1$

1 point

$f^{\prime}(x)-1 \leq 0 \Rightarrow f^{\prime}(x) \leq 1$

The graph of g is decreasing on the interval $0 \leq x \leq 5$ because $g^{\prime}(x) \leq 0$ on this interval.

Interval with reason

1 point

Scoring notes:

- The first point can be earned for $f^{\prime}(x) \leq 1$ or the equivalent, in words or symbols.

- Endpoints do not need to be included in the interval to be eligible for the second point.

Total for part (c) 2 points

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AP Calculus BC Fun 4 A Justify Conclusions About the Behavior of a Function Based on the Behavior of Its Derivatives Questions question 1