Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Motion Analysis

Calculus

4. Applications of Derivatives

Motion Analysis

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2:03 minutes

Problem 4.4.106d

Textbook Question

106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?

Verified step by step guidance

To determine when the acceleration is positive or negative, we need to analyze the concavity of the position function s=f(t). Acceleration is the second derivative of the position function, so we are looking for intervals where the graph is concave up (positive acceleration) and concave down (negative acceleration).

Identify the intervals where the graph of s=f(t) is concave up. This occurs when the graph is curving upwards, resembling a U-shape. In these intervals, the second derivative is positive, indicating positive acceleration.

Identify the intervals where the graph of s=f(t) is concave down. This occurs when the graph is curving downwards, resembling an upside-down U-shape. In these intervals, the second derivative is negative, indicating negative acceleration.

Examine the graph closely to find the points of inflection, where the concavity changes. These are the points where the acceleration changes from positive to negative or vice versa.

Using the graph, estimate the time intervals for positive and negative acceleration by observing the changes in concavity. Note these intervals as approximate times where the acceleration is positive or negative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position Function

The position function, denoted as s = f(t), describes the location of an object along a coordinate line at any given time t. It is a fundamental concept in motion analysis, allowing us to understand how the object's position changes over time. The graph of this function visually represents the object's trajectory, with peaks indicating maximum displacement and troughs indicating minimum displacement.

Velocity

Velocity is the rate of change of the position function with respect to time, mathematically expressed as v(t) = f'(t). It indicates both the speed and direction of the object's motion. When analyzing the graph, the velocity is positive when the graph is increasing and negative when it is decreasing, providing insights into the object's movement along the line.

Acceleration

Acceleration is the rate of change of velocity with respect to time, represented as a(t) = v'(t) = f''(t). It indicates how quickly the object's velocity is changing, which can be positive (speeding up) or negative (slowing down). In the context of the graph, acceleration is positive when the slope of the velocity function is increasing and negative when it is decreasing, helping to determine the object's motion dynamics.

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Related Practice

Textbook Question

Understanding Motion from Graphs

The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.

c. When does the particle move at its greatest speed?

Textbook Question

Understanding Motion from Graphs

The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.

d. When does the particle stand still for more than an instant?

Textbook Question

Finding g on a small airless planet Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of 15 m/sec. Because the acceleration of gravity at the planet’s surface was gₛ m/sec², the explorers expected the ball bearing to reach a height of s = 15t − (1/2)gₛt² m t sec later. The ball bearing reached its maximum height 20 sec after being launched. What was the value of gₛ?

Textbook Question

105. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (c) Acceleration equal to zero?

Textbook Question

106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (b) velocity equal to zero?

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Textbook Question

The accompanying figure shows the velocity v = ds/dt = f(t) (m/sec) of a body moving along a coordinate line.

a. When does the body reverse direction?

Multiple Choice

Given the position equation $s\left(t\right)$ , calculate the average velocity (in meters per second) based on the given interval, and the instantaneous velocity (in meters per second) at the end of the time interval.

$s\left(t\right)=9t-t^2$ , $0\le t\le3$

Multiple Choice

Given the position equation $s\left(t\right)$ , calculate the average velocity (in meters per second) based on the given time interval, and the instantaneous velocity (in meters per second) at the end of the time interval.

$s\left(t\right)=\frac{30}{t+5}$ , $-4\le t\le0$

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