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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 41a
Chapter 3, Problem 41a

Velocity from position The graph of s=f(t)s = f(t) represents the position of an object moving along a line at time t≥0 t≥0 . <IMAGE>
a. Assume the velocity of the object is 0 when t=0 t=0 . For what other values of t is the velocity of the object zero?

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Step 1: Understand that the velocity of an object is the derivative of its position function with respect to time, i.e., v(t) = f'(t).
Step 2: Since the velocity is zero at t = 0, we need to find other values of t where the derivative f'(t) = 0.
Step 3: Analyze the graph of s = f(t) to identify points where the tangent to the curve is horizontal, as these correspond to points where f'(t) = 0.
Step 4: Look for critical points on the graph where the slope of the tangent line is zero, indicating potential values of t where the velocity is zero.
Step 5: Verify these points by considering the behavior of the graph around these critical points to ensure they are indeed points where the velocity is zero.

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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. Understanding this function is crucial because it provides the basis for analyzing the object's motion. The graph of this function visually represents how the position changes over time, allowing us to infer other properties like velocity and acceleration.
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Relations and Functions

Velocity

Velocity is defined as the rate of change of position with respect to time, mathematically expressed as v(t) = f'(t), where f'(t) is the derivative of the position function. It indicates how fast and in what direction the object is moving. When the velocity is zero, it signifies that the object is momentarily at rest, which is essential for determining points of interest in motion analysis.
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Derivatives Applied To Velocity

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. In the context of velocity, finding critical points helps identify when the object's velocity is zero, which corresponds to moments when the object stops or changes direction. Analyzing these points is vital for understanding the overall motion of the object.
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Related Practice
Textbook Question

Find the derivative function f' for the following functions f.

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

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

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

Velocity from position The graph of s=f(t)s = f(t) represents the position of an object moving along a line at time t≥0t≥0 . <IMAGE>

c. Sketch a graph of the velocity function.

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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