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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.4.4a
Chapter 3, Problem 3.4.4a

Motion Along a Coordinate Line


Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.


a. Find the body’s displacement and average velocity for the given time interval.


s = (t⁴/4) − t³ + t², 0 ≤ t ≤ 3

Verified step by step guidance
1
To find the body's displacement over the interval [0, 3], calculate the difference in position at the endpoints of the interval. Evaluate the position function s(t) at t = 3 and t = 0, then subtract: Displacement = s(3) - s(0).
Evaluate s(3) by substituting t = 3 into the position function: s(3) = (3^4/4) - 3^3 + 3^2.
Evaluate s(0) by substituting t = 0 into the position function: s(0) = (0^4/4) - 0^3 + 0^2.
Calculate the displacement using the results from the previous steps: Displacement = s(3) - s(0).
To find the average velocity over the interval [0, 3], use the formula: Average Velocity = Displacement / (3 - 0). Substitute the displacement found in the previous step and simplify.

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Key Concepts

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

Displacement

Displacement refers to the change in position of a body over a specific time interval. It is calculated by finding the difference between the final and initial positions, s(t_final) - s(t_initial). In this context, it helps determine how far the body has moved along the coordinate line from t = 0 to t = 3 seconds.
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Using The Velocity Function

Average Velocity

Average velocity is the total displacement divided by the total time taken. It provides a measure of how fast the body is moving on average over the given time interval. Mathematically, it is expressed as (s(t_final) - s(t_initial)) / (t_final - t_initial), offering insight into the body's overall speed and direction.
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Average Value of a Function

Position Function

The position function, s = f(t), describes the location of a body at any given time t. In this problem, s = (t⁴/4) − t³ + t² represents the body's position in meters as a function of time in seconds. Understanding this function is crucial for calculating displacement and average velocity, as it provides the necessary values for s at different times.
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Relations and Functions
Related Practice
Textbook Question

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.


a. f(x) = (1 − x)⁶

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a. About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value?

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Textbook Question

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Use a CAS to perform the following steps in Exercises 55–62.


a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.


2y² + (xy)¹/³ = x² + 2, P(1,1)

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Textbook Question

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A = (1/2) ab sinθ.


a. How is dA/dt related to dθ/dt if a and b are constant?

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a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).

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Textbook Question

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).

a. Assuming that x, y, and z are differentiable functions of t, how is ds/dt related to dx/dt, dy/dt, and dz/dt?

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