Q. 87 - Calculus

Question

Jason’s distance in miles north from the corner of Main Street and High Street t minutes after noon on Tuesday is given by the following function s(t):

(a) Find an interval on which Jason’s velocity is positive and decreasing. Describe what Jason is doing over this time interval.

(b) Find a time interval on which Jason is moving north and his velocity is increasing. Describe what Jason is doing over this time interval.

(c) Find a time interval on which Jason’s acceleration and velocity are both negative. Describe what Jason is doing over this time interval.

(d) At which time is Jason’s velocity at a minimum? What is he doing at that moment?

Step-by-Step Solution

Answer

Part (a) The interval is [0,15] and he is slowing down.

Part (b) The interval is [60,80] and he is speeding up.

Part (c) The interval is [15,40] and he is speeding up.

Part (d) At t = 40 he is starting to slow down.

Step 1. Given Information.

Given a graph of a function which depicts the distance of Jason in miles with respect to time in minutes.

Step 2. Part (a) Velocity is positive and decreasing.

Velocity is positive means the interval in which the slope is positive and decreasing the interval is [0,15] from the graph. He is north from the corner of Main street and High Street walking north and slowing down as the velocity decreases. 

Step 3. Part (b) Velocity is increasing and moving north.

He is moving north means when the graph is moving upwards or when the position is moving upward on the y-axis and the interval in which the velocity is increasing is [60,80]. He is at the south corner now started moving in the north and he is speeding up as the velocity is increasing.

Step 4. Part (c) Acceleration and velocity both negative.

Velocity is negative means the slope is negative and acceleration is negative means slope is decreasing therefore the interval will be [15,40]. He is walking in the south and he is speeding up in the negative direction as the velocity is negative.

Step 5. Part (d) Velocity is minimum.

Velocity is minimum means the slope is minimum and the slope is minimum at t = 40 from the graph. He just finished speeding up in the south direction and is now starting to slow down.