Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 41c

Chapter 3, Problem 41c

Velocity from position The graph of $s = f (t)$ represents the position of an object moving along a line at time $t is greater than or equal to 0$ . <IMAGE>
c. Sketch a graph of the velocity function.

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Understand that velocity is the derivative of the position function with respect to time. This means we need to find f'(t) from the given position function f(t).

Identify key features of the position graph such as intervals where the function is increasing, decreasing, or constant. These features will help determine the behavior of the velocity graph.

For intervals where the position graph is increasing, the velocity will be positive. For intervals where the position graph is decreasing, the velocity will be negative. If the position graph is constant, the velocity will be zero.

Determine the points where the position graph has a maximum or minimum, as these will correspond to points where the velocity graph crosses the t-axis (velocity is zero).

Sketch the velocity graph using the information gathered: positive velocity for increasing intervals, negative velocity for decreasing intervals, and zero velocity for constant intervals. Ensure the graph reflects changes in the slope of the position graph.

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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 line at any given time t. It provides a mathematical representation of how the position changes over time, which is essential for understanding motion. Analyzing this function allows us to determine the object's behavior, such as when it is moving forward or backward.

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 movement. Understanding how to derive the velocity from the position function is crucial for sketching the velocity graph, as it reveals how quickly and in which direction the object is moving at any moment.

Graphing Functions

Graphing functions involves plotting the relationship between variables on a coordinate system, which helps visualize their behavior. For the velocity function, this means representing v(t) against time t. A clear understanding of how to interpret and create graphs is vital for analyzing motion, as it allows one to see trends, such as acceleration or deceleration, in the object's movement.

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Related Practice

Textbook Question

Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.

f(x) = 1/x; a= -5

Textbook Question

Fish length Assume the length L (in centimeters) of a particular species of fish after t years is modeled by the following graph. <IMAGE>

b. What does the derivative tell you about how this species of fish grows?

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Textbook Question

Velocity from position The graph of $s = f (t)$ represents the position of an object moving along a line at time $t is greater than or equal to 0$ . <IMAGE>

a. Assume the velocity of the object is 0 when $t equals 0$ . For what other values of t is the velocity of the object zero?

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Textbook Question

Calculate the derivative of the following functions.

y = ⁴√(2x / (4x - 3))

Textbook Question

Calculate the derivative of the following functions.

y = cos⁴ θ + sin⁴ θ

Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 1/3x-1; a= 2