AP Calculus AB 1.15: Limits at Infinity

Practice AP Calculus questions on evaluating limits at infinity and interpreting end behavior and relative magnitudes of functions.

[Maximum number: 2]

An invasive species of plant appears in a fruit grove at time t=0 and begins to spread. The

function C defined by $C(t)=7.6 \arctan (0.2 t)$ models the number of acres in the fruit grove

affected by the species t weeks after the species appears. It can be shown that $C^{\prime}(t)=\frac{38}{25+t^{2}}$ .

(Note: Your calculator should be in radian mode.)

Assume that the invasive species continues to spread according to the given model for all

times t>0. Write a limit expression that describes the end behavior of the rate of change in

the number of acres affected by the species. Evaluate this limit expression.

C Assume that the invasive species continues to spread according to the given model for all times t>0.

Write a limit expression that describes the end behavior of the rate of change in the number of acres

affected by the species. Evaluate this limit expression.

$\lim _{t \rightarrow \infty} C^{\prime}(t)=\lim _{t \rightarrow \infty} \frac{38}{25+t^{2}}$	Limit expression	Point 5 (P5)
= 0	Value	Point 6 (P6)

Scoring Notes for Part C

- P5 can be earned for either $\lim _{t \rightarrow \infty} C^{\prime}(t)$ or $\lim _{t \rightarrow \infty} C(t)$ .

- A response that includes $\lim _{t \rightarrow \infty} C(t)$ is not eligible to earn P6.

- For P6, arithmetic with infinity, e.g., $\frac{38}{25+\infty^{2}}=0$ , will be considered as scratch work and will not

be considered in scoring.

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LIM-2.D—Interpret the behavior of functions using limits involving infinity—Topic 1.15 question 1