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


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

Verified step by step guidance

1. Understand the relationship between kinetic energy and velocity. The kinetic energy (KE) of a particle is given by the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>KE</mi> = <mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>m</mi></math> is the mass of the particle and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>v</mi></math> is its velocity.
2. Identify the variable of interest. In this case, we are considering kinetic energy as a function of velocity, so <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>v</mi></math> is the variable.
3. Differentiate the kinetic energy function with respect to velocity to determine the rate of change. The derivative of <math xmlns='http://www.w3.org/1998/Math/MathML'><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math> with respect to <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>v</mi></math> is <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>m</mi><mi>v</mi></math>.
4. Analyze the sign of the derivative. Since mass <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>m</mi></math> is positive and velocity <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>v</mi></math> is also positive for increasing velocity, the derivative <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>m</mi><mi>v</mi></math> is positive.
5. Conclude based on the derivative. Since the derivative is positive, the kinetic energy function is increasing as a function of velocity.

Key Concepts

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.