Mean Values of Function

average_value_derivative(equation_type, coefficients, start, end, precision=4)

Evaluates the average rate of change between two points for a given function

Parameters

• equation_type (str) – Name of the type of function for which the definite integral must be evaluated (e.g., ‘linear’, ‘quadratic’)

• coefficients (list of int or float) – Coefficients of the original function to use for evaluating the average rate of change

• start (int or float) – Value of the x-coordinate of the first point to use for evaluating the rate of change

• end (int or float) – Value of the x-coordinate of the second point to use for evaluating the rate of change

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the result

Raises

• ValueError – First argument must be either ‘linear’, ‘quadratic’, ‘cubic’, ‘hyperbolic’, ‘exponential’, ‘logarithmic’, ‘logistic’, or ‘sinusoidal’

• TypeError – Second argument must be a 1-dimensional list containing elements that are integers or floats

• TypeError – Third and fourth arguments must be integers or floats

• ValueError – Third argument must be less than or equal to fourth argument

• ValueError – Last argument must be a positive integer

Returns

average – Slope of a function between two points; if start and end values are identical, then slope will be zero

Return type

float

See also

mean_values_derivative(), average_value_integral(), mean_values_integral()

Notes

• Slope of a function over an interval: \(m = \frac{f(b) - f(a)}{b - a}\)

• Mean Value Theorem for Derivatives

Examples

Import average_value_derivative function from regressions library

>>> from regressions.analyses.mean_values import average_value_derivative

Evaluate the average rate of change of a cubic function with coefficients 2, 3, 5, and 7 between end points of 10 and 20

>>> average_cubic = average_value_derivative('cubic', [2, 3, 5, 7], 10, 20)
>>> print(average_cubic)
1495.0

Evaluate the average rate of change of a sinusoidal function with coefficients 2, 3, 5, and 7 between end points of 10 and 20

>>> average_sinusoidal = average_value_derivative('sinusoidal', [2, 3, 5, 7], 10, 20)
>>> print(average_sinusoidal)
0.0401

mean_values_derivative(equation_type, coefficients, start, end, precision=4)

Generates a list of all the x-coordinates whose instantaneous rates of change equal the function’s average rate of change between two points

Parameters

• equation_type (str) – Name of the type of function for which an average value must be determined (e.g., ‘linear’, ‘quadratic’)

• coefficients (list of int or float) – Coefficients to use to generate the equation to investigate

• start (int or float) – Value of the x-coordinate of the first point to use for evaluating the rate of change; all results must be greater than this value

• end (int or float) – Value of the x-coordinate of the second point to use for evaluating the rate of change; all results must be less than this value

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the result

Raises

• ValueError – First argument must be either ‘linear’, ‘quadratic’, ‘cubic’, ‘hyperbolic’, ‘exponential’, ‘logarithmic’, ‘logistic’, or ‘sinusoidal’

• TypeError – Second argument must be a 1-dimensional list containing elements that are integers or floats

• TypeError – Third and fourth arguments must be integers or floats

• ValueError – Third argument must be less than or equal to fourth argument

• ValueError – Last argument must be a positive integer

Returns

points – Values of the x-coordinates within the specified interval at which the original function has an instantaneous rate of change equal to its average rate of change over that entire interval; if the function is sinusoidal, then only the initial results within at most a four-period interval within the specified interval will be listed, but general forms will also be included (however, their results may be outside the specified interval); if the algorithm cannot determine any values, then it will return a list of None

Return type

list of float or str

See also

• Roots for key functions: linear_roots(), quadratic_roots(), cubic_roots(), hyperbolic_roots(), exponential_roots(), logarithmic_roots(), logistic_roots(), sinusoidal_roots()

• Mean values: average_value_derivative(), average_value_integral(), mean_values_integral()

Notes

• Mean values for the derivative over an interval: \(f'(c) = \frac{f(b) - f(a)}{b - a}\)

• Mean Value Theorem for Derivatives

Examples

Import mean_values_derivative function from regressions library

>>> from regressions.analyses.mean_values import mean_values_derivative

Generate a list of all the x-coordinates whose instantaneous rates of change equal the function’s average rate of change for a cubic function with coefficients 2, 3, 5, and 7 between end points of 10 and 20

>>> points_cubic = mean_values_derivative('cubic', [2, 3, 5, 7], 10, 20)
>>> print(points_cubic)
[15.2665]

Generate a list of all the x-coordinates whose instantaneous rates of change equal the function’s average rate of change for a cubic function with coefficients 2, 3, 5, and 7 between end points of 10 and 20

>>> points_sinusoidal = mean_values_derivative('sinusoidal', [2, 3, 5, 7], 10, 20)
>>> print(points_sinusoidal)
[10.7618, 11.8046, 12.8562, 13.899, 14.9506, 15.9934, 17.045, 18.0878, 19.1394, '10.7618 + 2.0944k', '11.8046 + 2.0944k']

average_value_integral(equation_type, coefficients, start, end, precision=4)

Evaluates the average value of a given function between two points

Parameters

• equation_type (str) – Name of the type of function for which the definite integral must be evaluated (e.g., ‘linear’, ‘quadratic’)

• coefficients (list of int or float) – Coefficients of the original function to integrate

• start (int or float) – Value of the x-coordinate of the first point to use for evaluating the average value

• end (int or float) – Value of the x-coordinate of the second point to use for evaluating the average value

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the result

Raises

• ValueError – First argument must be either ‘linear’, ‘quadratic’, ‘cubic’, ‘hyperbolic’, ‘exponential’, ‘logarithmic’, ‘logistic’, or ‘sinusoidal’

• TypeError – Second argument must be a 1-dimensional list containing elements that are integers or floats

• TypeError – Third and fourth arguments must be integers or floats

• ValueError – Third argument must be less than or equal to fourth argument

• ValueError – Last argument must be a positive integer

Returns

average – Average value of the function between two points; if start and end values are identical, then average value will be zero

Return type

float

See also

accumulated_area(), mean_values_integral(), average_value_derivative(), mean_values_derivative()

Notes

• Average value of a function over an interval: \(f_{avg} = \frac{1}{b - a}\cdot{\int_{a}^{b} f(x) \,dx}\)

• Mean Value Theorem for Integrals

Examples

Import average_value_integral function from regressions library

>>> from regressions.analyses.mean_values import average_value_integral

Evaluate the average value of a cubic function with coefficients 2, 3, 5, and 7 between end points of 10 and 20

>>> average_cubic = average_value_integral('cubic', [2, 3, 5, 7], 10, 20)
>>> print(average_cubic)
8282.0

Evaluate the average value of a sinusoidal function with coefficients 2, 3, 5, and 7 between end points of 10 and 20

>>> average_sinusoidal = average_value_integral('sinusoidal', [2, 3, 5, 7], 10, 20)
>>> print(average_sinusoidal)
6.9143

mean_values_integral(equation_type, coefficients, start, end, precision=4)

Generates a list of all the x-coordinates between two points at which a function’s value will equal its average value over that interval

Parameters

• equation_type (str) – Name of the type of function for which an average value must be determined (e.g., ‘linear’, ‘quadratic’)

• coefficients (list of int or float) – Coefficients of the origianl function under investigation

• start (int or float) – Value of the x-coordinate of the first point to use for evaluating the average value; all results must be greater than this value

• end (int or float) – Value of the x-coordinate of the second point to use for evaluating the average value; all results must be less than this value

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the result

Raises

• ValueError – First argument must be either ‘linear’, ‘quadratic’, ‘cubic’, ‘hyperbolic’, ‘exponential’, ‘logarithmic’, ‘logistic’, or ‘sinusoidal’

• TypeError – Second argument must be a 1-dimensional list containing elements that are integers or floats

• TypeError – Third and fourth arguments must be integers or floats

• ValueError – Third argument must be less than or equal to fourth argument

• ValueError – Last argument must be a positive integer

Returns

points – Values of the x-coordinates within the specified interval at which the original function has a value equal to its average value over that entire interval; if the function is sinusoidal, then only the initial results within at most a four-period interval within the specified interval will be listed, but general forms will also be included (however, their results may be outside the specified interval); if the algorithm cannot determine any values, then it will return a list of None

Return type

list of float or str

See also

• Roots for key functions: linear_roots(), quadratic_roots(), cubic_roots(), hyperbolic_roots(), exponential_roots(), logarithmic_roots(), logistic_roots(), sinusoidal_roots()

• Mean values: average_value_integral(), average_value_derivative(), mean_values_derivative()

Notes

• Mean values for the integral over an interval: \(f(c) = \frac{1}{b - a}\cdot{\int_{a}^{b} f(x) \,dx}\)

• Mean Value Theorem for Integrals

Examples

Import mean_values_integral function from regressions library

>>> from regressions.analyses.mean_values import mean_values_integral

Generate a list of all the x-coordinates of a cubic function with coefficients 2, 3, 5, and 7 at which the function’s value will equal its average value between 10 and 20

>>> points_cubic = mean_values_integral('cubic', [2, 3, 5, 7], 10, 20)
>>> print(points_cubic)
[15.5188]

Generate a list of all the x-coordinates of a sinusoidal function with coefficients 2, 3, 5, and 7 at which the function’s value will equal its average value between 10 and 20

>>> points_sinusoidal = mean_values_integral('sinusoidal', [2, 3, 5, 7], 10, 20)
>>> print(points_sinusoidal)
[10.2503, 11.2689, 12.3447, 13.3633, 14.4391, 15.4577, 16.5335, 17.5521, 18.6279, 19.6465, '10.2503 + 2.0944k', '11.2689 + 2.0944k']

average_values(equation_type, coefficients, start, end, precision=4)

Calculates the average values for a specific function

Parameters

• equation_type (str) – Name of the type of function for which average values must be determined (e.g., ‘linear’, ‘quadratic’)

• coefficients (list of int or float) – Coefficients of the origianl function under investigation

• start (int or float) – Value of the x-coordinate of the first point to use for evaluating the average values; results within lists must be greater than this value

• end (int or float) – Value of the x-coordinate of the second point to use for evaluating the average values; results within lists must be less than this value

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the result

Raises

• ValueError – First argument must be either ‘linear’, ‘quadratic’, ‘cubic’, ‘hyperbolic’, ‘exponential’, ‘logarithmic’, ‘logistic’, or ‘sinusoidal’

• TypeError – Second argument must be a 1-dimensional list containing elements that are integers or floats

• TypeError – Third and fourth arguments must be integers or floats

• ValueError – Third argument must be less than or equal to fourth argument

• ValueError – Last argument must be a positive integer

Returns

• averages[‘average_value_derivative’] (float) – Slope of a function between two points

• averages[‘mean_values_derivative’] (list of float or str) – Values of the x-coordinates within the specified interval at which the original function has a value equal to its average value over that entire interval; if the function is sinusoidal, then only the initial results within at most a four-period interval within the specified interval will be listed, but general forms will also be included (however, their results may be outside the specified interval); if the algorithm cannot determine any values, then it will return a list of None

• averages[‘average_value_integral’] (float) – Average value of the function between two points

• averages[‘mean_values_integral’] (list of float or str) – Values of the x-coordinates within the specified interval at which the original function has a value equal to its average value over that entire interval; if the function is sinusoidal, then only the initial results within at most a four-period interval within the specified interval will be listed, but general forms will also be included (however, their results may be outside the specified interval); if the algorithm cannot determine any values, then it will return a list of None

See also

average_value_derivative(), mean_values_derivative(), average_value_integral(), mean_values_integral()

Notes

• Mean Value Theorem for Derivatives

• Mean Value Theorem for Integrals

Examples

Import average_values function from regressions library

>>> from regressions.analyses.mean_values import average_values

Calculate the averages of a cubic function with coefficients 2, 3, 5, and 7 between 10 and 20

>>> averages_cubic = average_values('cubic', [2, 3, 5, 7], 10, 20)
>>> print(averages_cubic['average_value_derivative'])
1495.0
>>> print(averages_cubic['mean_values_derivative'])
[15.2665]
>>> print(averages_cubic['average_value_integral'])
8282.0
>>> print(averages_cubic['mean_values_integral'])
[15.5188]

Calculate the averages of a sinusoidal function with coefficients 2, 3, 5, and 7 between 10 and 20

>>> averages_sinusoidal = average_values('sinusoidal', [2, 3, 5, 7], 10, 20)
>>> print(averages_sinusoidal['average_value_derivative'])
0.0401
>>> print(averages_sinusoidal['mean_values_derivative'])
[10.7618, 11.8046, 12.8562, 13.899, 14.9506, 15.9934, 17.045, 18.0878, 19.1394, '10.7618 + 2.0944k', '11.8046 + 2.0944k']
>>> print(averages_sinusoidal['average_value_integral'])
6.9143
>>> print(averages_sinusoidal['mean_values_integral'])
[10.2503, 11.2689, 12.3447, 13.3633, 14.4391, 15.4577, 16.5335, 17.5521, 18.6279, 19.6465, '10.2503 + 2.0944k', '11.2689 + 2.0944k']