Q. 88. - Calculus

Question

Suppose Juri drives for two hours and that his distance from home in miles is given by the function $s\left(t\right)$ shown in the following figure.

(a) Find a time interval on which Juri’s acceleration is positive. Is his velocity positive or negative on this interval? Describe what Juri is doing over this time interval. 

(b) The graph $y=s\left(t\right)$ of Juri's position has an inflection point at $t=1$ hour. What does this inflection point say about Juri’s velocity at $t=1?$ About his acceleration at $t=1?$

Step-by-Step Solution

Answer

Part a. $t\in [0,1]$

Part b. At the inflection point $t=1$ the speed and acceleration is zero.

Part a. Step 1. Given information

Distance from home

Part a. Step 2. Explanation

Note the gradient of the curve which represents a changing speed.

In the time interval $[0,1]$speed is increasing and so in this part shows acceleration.

In the time interval  $[1,2]$ speed is decreasing and so in this part shows deceleration.

Thus, Juri is increasing the speed over the time interval $[0,1]$ shows a positive acceleration and positive speed.

Part b. Step 1. Explanation

An inflection point is known as the turning point of a curve.

Around an inflection point the gradient of the curve changes it’s sign from plus to minus or from minus to plus. Therefore, around the inflection point $t=1$ the speed is changing from increasing to decreasing and the acceleration is changing from positive to negative.

$$\begin{gathered} \frac{ds}{dt}=0 \\ \frac{d^{2}s}{d{t}^2}=0 \end{gathered}$$

Thus, at the inflection point $t=1$ the speed and acceleration is zero.