Derivative of f(x), g(x) and h(x) from Calculus Problem A.1: Solve Urgent"

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Discussion Overview

The discussion revolves around finding the derivative of a function λ(x) defined in terms of three other functions f(x), g(x), and h(x) from a previous problem (Problem A.1). Participants explore the application of calculus rules such as the product rule and sum rule in this context.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Debate/contested

Main Points Raised

  • One participant requests the derivative λ0(x) of the function λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x).
  • Another participant reiterates the request for the derivative and asks about the progress made on the problem.
  • A participant provides a solution approach using the sum rule and product rule, suggesting that the derivative is f'g + fg' + f'h + fh' - g'h - gh'.
  • One participant identifies specific functions: f(x) = x - 2, h(x) = x - 6, and describes g(x) as a semicircle with endpoints (4, 2) and (8, 2), prompting questions about the properties of these functions.
  • Another participant questions the classification of a semicircle as a line segment.
  • A later reply humorously defends the classification of the semicircle as an "honorary line segment."

Areas of Agreement / Disagreement

There is no consensus on the classification of g(x) as a semicircle versus a line segment, and the discussion includes differing interpretations of the functions involved.

Contextual Notes

Participants have not fully defined the functions f(x), g(x), and h(x) in terms of their mathematical properties, leading to some ambiguity in their discussions. The context of Problem A.1 is assumed but not elaborated upon.

Who May Find This Useful

This discussion may be of interest to students or individuals working on calculus problems involving derivatives, particularly those exploring the application of product and sum rules in differentiation.

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Please also solve this :

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ0(x) of the following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)Note: here problem A.1 is the upper problem.Please solve both the problem.
 
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sfvdsc said:
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Please also solve this :

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ0(x) of the following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)Note: here problem A.1 is the upper problem.Please solve both the problem.
What have you done so far?
 
That's a pretty straight forward problem. Using the "sum rule", that (f+ g)'= f'+ g', and the "product rule", that (fg)'= f'g+ fg' the derivative of fg+ fh- gh is f'g+ fg'+ f'h+ fh'- g'h- gh'.

You should see immediately that f'= h'= 1. What is g'?
 
f(x)= x- 2. That line segment starts at (2, 0) and ends at (4, 2). What is the length of that line segment?

h(x)= x- 6. That line segment starts at (8, 2) and ends at (10, 6). What is the length of that line segment?

g(x) is a semi-circle with end points (4, 2) and (8, 2). What is its radius? What is its circumference?
 
Since when is a semicircle a line segment?
 
For this problem it was an honorary line segment!
 

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