Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경

Hass 15th Edition

Ch. 1 - Functions

문제 33d

1장, 문제 33d

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

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

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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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주요 개념

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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.

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.

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.

State whether each function is increasing, decreasing, or neither.d. Kinetic energy as a function of a particle’s velocity