Near Earth’s surface

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Near Earth’s surface, a simplified velocity model for a rocket experiencing atmospheric drag is

v(t)=\frac{v_e t}{t-4}, \qquad t>4,

where

v(t)

is the rocket’s upward velocity (in meters per second)

t

seconds after launch and

v_e

is a positive constant representing an adjusted escape velocity. (a) Determine

\displaystyle \lim_{t\to 4^{+}}v(t)

and classify the type of asymptote present at

t=4

. (b) Sketch a qualitative graph of

v

versus

t

for

t>4

, indicating any intercepts, the vertical asymptote, and end behavior as

t\to\infty

. (c) Using the formal limit definition, justify that

v(t)

grows without bound as

t

approaches

4

from the right.