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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 3.4.5b
Chapter 3, Problem 3.4.5b

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.


b. Find the body’s speed and acceleration at the endpoints of the interval.


s = 25/t² − 5/t, 1 ≤ t ≤ 5

Verified step by step guidance
1
First, find the velocity function v(t) by differentiating the position function s(t) = \(\frac{25}{t^2} - \frac{5}{t}\) with respect to time t. Use the power rule and the chain rule for differentiation.
The velocity function v(t) is the derivative of s(t), so calculate v(t) = \(-\frac{50}{t^3} + \frac{5}{t^2}\).
Next, find the acceleration function a(t) by differentiating the velocity function v(t) with respect to time t. Again, apply the power rule and the chain rule.
The acceleration function a(t) is the derivative of v(t), so calculate a(t) = \(\frac{150}{t^4} - \frac{10}{t^3}\).
Evaluate the velocity v(t) and acceleration a(t) at the endpoints of the interval t = 1 and t = 5 to find the body's speed and acceleration at these points.

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Key Concepts

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

Derivative

The derivative of a function represents the rate of change of the function with respect to a variable. In the context of motion, the derivative of the position function s = f(t) with respect to time t gives the velocity of the body. Calculating the derivative is essential to find the speed and acceleration at specific points.
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Derivatives

Velocity and Speed

Velocity is the derivative of the position function with respect to time, indicating the direction and rate of motion. Speed, however, is the magnitude of velocity, representing how fast an object is moving regardless of direction. To find speed at the endpoints, compute the absolute value of the velocity at those points.
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Derivatives Applied To Velocity

Second Derivative and Acceleration

Acceleration is the rate of change of velocity with respect to time, found by taking the second derivative of the position function. It indicates how quickly the velocity is changing. For this problem, calculate the second derivative of s = f(t) to determine the acceleration at the interval's endpoints.
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The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

A sliding ladder


A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.


b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?


Textbook Question

Quadratic approximations


b. Find the quadratic approximation to f(x) = 1/(1 − x) at x = 0.

Textbook Question

Fruit flies (Continuation of Example 4, Section 2.1.) Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment.


b. During what days does the population seem to be increasing fastest? Slowest?


Textbook Question

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.


b. At what rate is the angle θ changing at this instant (see the figure)?

Textbook Question

By computing the first few derivatives and looking for a pattern, find the following derivatives.


b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

Textbook Question

Temperature The given graph shows the outside temperature T in °F, between 6 a.m. and 6 p.m.

b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times?