Q.88 - Calculus

Question

Suppose Juri drives for two hours and that his distance from home in miles is given by the function $\mathrm{s}\left(\mathrm{t}\right)$ 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 $\mathrm{y}=\mathrm{s}\left(\mathrm{t}\right)$ of Juri’s position has an inflection point at $\mathrm{t} = 1$hour. What does this inflection point say about Juri’s velocity at $\mathrm{t} = 1?$ About his acceleration at $\mathrm{t} = 1?$

Step-by-Step Solution

Answer

a. The resultant Juri time interval is $\mathrm{t}\in [0,1]$

b. The speed and acceleration are both zero at the inflection point $\mathrm{t}=1$

Part (a) Step 1: Given Information

The given graph is 

Part (a) Step 2: Calculations

Take note of the curve's gradient, which symbolises a change in speed.

The speed is growing in the time interval $[0,1]$, hence this component displays acceleration. The speed decreases in the time range $\left[1,2\right]$ indicating deceleration in this section.

As a result, Juri's speed is growing during the time span $[0,1]$, indicating a positive acceleration and speed.

The resultant Juri time interval is  $\mathrm{t}\in \left[0,1\right]$

Part (b) Step 1: Given Information

The given graph is 

Part (b) Step 2: Calculations

The turning point of a curve is known as an inflection point.

The gradient of the curve changes sign around an inflection point, from plus to minus or negative to plus. As a result, the speed changes from increasing to decreasing near the inflection point $\mathrm{t}=1,$ and the acceleration changes from positive to negative.

$\begin{array}{r}\frac{\mathrm{ds}}{\mathrm{dt}}=0\\ \frac{{\mathrm{d}}^2\mathrm{s}}{{\mathrm{dt}}^2}=0\end{array}$

As a result, at $\mathrm{t}=1$, the speed and acceleration are both zero.