AP Calculus BC 8.13: Arc Length

Practice AP Calculus BC questions on setting up an arc-length integral for a graph and interpreting it as curve length in context.

[Maximum number: 2]

The function f is twice differentiable for all x with f(0)=0. Values of $f^{\prime}$ , the derivative of f, are given in the table for selected values of x.

What information does $\int_{0}^{\pi} \sqrt{1+\left(f^{\prime}(x)\right)^{2}} d x$ provide about the graph of f ?

$\int_{0}^{\pi} \sqrt{1+\left(f^{\prime}(x)\right)^{2}} d x$ is the arc length of the graph of f on $[0, \pi]$ .}

Arc length of f

1 point

Interval $[0, \pi]$

1 point

Scoring notes:

- A response of "arc length" or "length" earns the first point. Such a response does not need to

reference f. However, if the response references a different function, the response does not earn

the first point and is eligible to earn the second point.

- A response referring to distance explicitly connected to the graph or f (or equivalent) earns the

first point. For example, a response of "distance along the curve" or "distance traveled by a particle

moving along f " earns the first point and is eligible to earn the second point.

- A response referring to distance that is not explicitly connected to the graph of f does not earn the

first point but is eligible to earn the second point. For example, a response of "distance" or

"distance traveled" does not earn the first point but is eligible to earn the second point.

- To earn the second point a response must connect the interval $[0, \pi]$ to arc length, length, or

distance.

Total for part (b) 2 points

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CHA-6.A—Determine the length of a curve in the plane defined by a function, using a definite integral question 1