BusCalc 14.2 Partial Derivatives

The paragraph discusses the average electrical power output of a solar panel, which is influenced by the percent of maximum power (p), the latitude (theta), and the fraction of daytime with cloud cover (C). It explains that solar panels are more effective near the equator and less so near the poles due to the angle of sunlight. The power output percent (p) is calculated for a panel at 30 degrees latitude with 25% cloud cover, resulting in approximately 75.5% power output compared to equatorial conditions with no cloud cover.

This section explores the equation for contour lines related to the power output (p) of a solar panel. It explains that when p equals 1, there is only a single point (theta equals 0 and C equals 0) where maximum power is achieved. For p equal to 0, the contour line is at theta equals 81 degrees, indicating no sunlight on the solar panel at or above this latitude. Lastly, for p equal to 0.5, any theta value can be chosen, and C can be solved accordingly, showing a range of possible combinations for half power output.

The paragraph focuses on the trace of the function along the line where theta equals zero, representing the impact of cloud cover (C) on power output (p). It simplifies the power equation for this specific case and shows that as C varies from 0 to 1, p decreases from 1 to 0.5, illustrating the direct relationship between cloud cover and solar panel efficiency. Additionally, it presents a parabola representing the function when C equals 0, indicating the decrease in power output as latitude increases from the equator towards the poles.

The content delves into multi-variable functions, specifically focusing on partial derivatives. It explains that partial derivatives extend the concept of derivatives to functions with more than one independent variable. The paragraph discusses how to take partial derivatives with respect to each variable while treating the others as constants. It also covers evaluating multi-variable functions, understanding contour lines, and tracing the function by setting one variable to a constant.

This section provides an example of evaluating partial derivatives. It demonstrates how to find the partial derivative of a function with respect to x at a specific point (1,2). The process involves differentiating each term of the function with respect to x, treating other variables as constants, and then evaluating the resulting expression at the given point.

The paragraph discusses the concept of tangent lines on surfaces represented by multi-variable functions. It explains that the partial derivative with respect to x represents the slope of the tangent line in the x direction, and similarly for the y direction. It uses a graph to illustrate how the slope of the tangent line changes depending on the direction of the line and the point on the surface.

This part of the script applies the concept of partial derivatives to a real-life scenario involving the effect of caffeine intake on a person's running speed. It uses a function to represent velocity and takes partial derivatives with respect to time (t) and caffeine intake (c) to analyze how these factors influence speed. The example shows how partial derivatives can be used to understand rates of change in different directions.

The paragraph introduces higher order partial derivatives and the associated notation. It explains how to denote first and second partial derivatives using subscripts and how to represent mixed partial derivatives. The content emphasizes the commutative property of partial derivatives, meaning the order in which they are taken does not affect the result.

This section focuses on computing all first and second partial derivatives of a given function. It outlines the process of finding the partial derivatives with respect to x and y, as well as mixed partials. The paragraph demonstrates how to take these derivatives and emphasizes the importance of understanding the process for each term in the function.

The content describes the process of evaluating a specific third derivative, g_xxx_y, at the point where x equals -1 and y equals 1. It explains the steps to find the necessary second derivatives and how to compute them. The paragraph also touches on the concept of a constant function as a result of differentiation.

This paragraph explores the concept of function derivatives in a dynamic system, represented by the variables h, x, and t. It demonstrates how to find mixed partial derivatives and single derivatives at a specific point. The example shows that certain mixed partial derivatives can be constant functions, independent of the variables' values.

The final paragraph deals with finding all first and second derivatives of a function involving natural logarithm and linear terms. It explains the process of taking derivatives with respect to x and y, including the application of the chain rule and power rule. The content also covers the evaluation of mixed partial derivatives and the importance of checking work by taking different paths to the same derivative.