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Chapter 2: Q. 49 (page 167)

Think about what you did today and how far north you were from your house or dorm throughout the day. Sketch a graph that represents your distance north from your house or dorm over the course of the day, and explain how the graph reflects what you did today. Then sketch a graph of your velocity.

Short Answer

Expert verified

Ans: The velocity graph for the function can be drawn as follows:

Step by step solution

01

Step 1. Given information.

give,

Think about what you did today and how far north you were from your house or dorm throughout the day.

Sketch a graph according to the question.

02

Step 2. Consider the graph that represents the distance from the house to the north.

In this graph point, A represents the position of the person, and point B represents the place where the person resides. He moves towards A for buying grocery items for the home. He follows the path that has been shown in the graph. On his way, he stops to refill his car, then takes tea in the tea stall, and at last, he came to A for buying the goods.

03

Step 3. The Sketch is:

The velocity graph for the function can be drawn as follows:

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Then $a_{k} = \frac{f^{k} (0)}{k!}$ where $f^{k} (x)$ is the k-th derivative of $f (x) a n d k! = k \cdot (k - 1) \cdot (k - 2) . . . . 2 \cdot 1$

Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park $t$ minutes after she begins her jog is given by the function $s (t)$ shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

(a) What was the average rate of change of Linda’s distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?

(b) On which ten-minute interval was the average rate of change of Linda’s distance from the oak tree the greatest: the first $10$ minutes, the second $10$ minutes, or the last $10$ minutes?

(c) Use the graph of $s (t)$ to estimate Linda’s average velocity during the $5$ -minute interval from $t = 5 t o t = 10$ . What does the sign of this average velocity tell you in real-world terms?

(d) Approximate the times at which Linda’s (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Linda’s jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?

See all solutions

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