Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 1 - Functions

Problema 33d

Capitolo 1, Problema 33d

State whether each function is increasing, decreasing, or neither.

d. Kinetic energy as a function of a particle’s velocity

Guida verificata passo dopo passo

Understand the relationship between kinetic energy and velocity. The kinetic energy (KE) of a particle is given by the formula:

KE = \frac{1}{2} m v^{2}

, where

m

is the mass of the particle and

v

is its velocity.

Identify the variable of interest. In this case, we are considering kinetic energy as a function of velocity, so

v

is the variable.

Differentiate the kinetic energy function with respect to velocity to determine the rate of change. The derivative of

\frac{1}{2} m v^{2}

with respect to

v

m v

Analyze the sign of the derivative. Since mass

m

is positive and velocity

v

is also positive for increasing velocity, the derivative

m v

is positive.

Conclude based on the derivative. Since the derivative is positive, the kinetic energy function is increasing as a function of velocity.

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Kinetic Energy Formula

Kinetic energy (KE) is defined by the formula KE = 1/2 mv², where m is the mass of the particle and v is its velocity. This formula indicates that kinetic energy is directly proportional to the square of the velocity, meaning as velocity increases, kinetic energy increases as well.

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Derivatives Applied To Acceleration Example 2

Increasing and Decreasing Functions

A function is considered increasing if, for any two points x1 and x2 where x1 < x2, the function value at x1 is less than the function value at x2 (f(x1) < f(x2)). Conversely, a function is decreasing if f(x1) > f(x2) under the same conditions. Understanding these definitions is crucial for analyzing the behavior of functions.

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Determining Where a Function is Increasing & Decreasing

Derivative and Monotonicity

The derivative of a function provides information about its rate of change. If the derivative of a function is positive over an interval, the function is increasing; if negative, it is decreasing. For kinetic energy as a function of velocity, the derivative with respect to velocity will indicate whether the kinetic energy is increasing or decreasing as velocity changes.

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