### Home > APCALC > Chapter 3 > Lesson 3.2.3 > Problem3-78

3-78.

The position of a ball as a function of time is given by the function below where $s(t)$ is in meters and $t$ is in seconds. Homework Help ✎

$s ( t ) = \sqrt { t + 1 }$

1. Use your calculator to approximate the instantaneous velocity of the ball at $1, 5, 10,$ and $100$ seconds.

Instantaneous velocity = Instantaneous Rate of Change (IROC) = Derivative

Rewrite $s(t)$ using exponents: $s(t)=(t+1)^{1/2}$

Find $s^\prime (t)$ using the Power rule.

Evaluate $s^\prime (1), s^\prime(5), s^\prime(10)$ and $s^\prime(100)$.

2. What happens to the velocity of the ball after a very long time (i.e. as $t\rightarrow \infty$)?

Note: $t = 100$ seconds is a very long time for a ball to be in motion. (This must be an unearthly situation!) Is $s(t)$ increasing, decreasing or neither as $t\rightarrow \infty$? Is $s(t)$ changing rapidly or not so rapidly? Explain.

3. What happens to the position of the ball after a very long time, (i.e. what is)? Does this make sense given your answer to part (b)?

Does a square root function (such as s(t)) have a horizontal asymptote as t→∞?