-apc.4.2.9 graph of f(x) to g(x)

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Discussion Overview

The discussion revolves around the interpretation of the graphs of functions related to integrals, specifically comparing the graphs of \( f(x) \) and \( g(x) \), where \( g(x) \) is defined as the integral of \( f(t) \) from 2 to \( x \). Participants explore the implications of the relationship between these functions and their graphical representations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that integrals typically introduce curves rather than eliminate them, leading them to choose option D based on the positive slope observed.
  • Another participant identifies significant typos in the problem, clarifying that \( g(x) \) should be defined as \( \int_2^x f(t) \, dt \) and that the question pertains to \( y = g(x) \) rather than \( g(f) \).
  • A participant agrees with the correction regarding the typos and elaborates that since \( f(x) \) is linear, its integral \( g(x) \) will be quadratic, resulting in a parabolic graph that starts negative and becomes positive.
  • There is a mention of the usefulness of graphs in understanding the problem, with one participant emphasizing that the curvature of \( g \) is positive due to the positive slope of \( f \).
  • A participant shares a link to a work-in-progress PDF compiling replies to AP Calculus problems, indicating a goal to make it error-free.
  • Another participant questions the use of abbreviations like "WIP" and "PDF," suggesting that English should be used on the site.

Areas of Agreement / Disagreement

Participants express some agreement on the corrections regarding the problem's formulation and the implications of the graphs, but there are differing views on the appropriateness of language and abbreviations used in the discussion.

Contextual Notes

There are unresolved aspects regarding the specific implications of the graphs and the interpretations of the integral's effect on the function's shape, as well as the clarity of communication within the forum.

karush
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Screenshot 2020-09-30 at 1.04.05 PM.png

image due to graphsok just by observation I chose D since integrals tend to introduce curves not eliminate them and the slope was positive
 
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two significant typos in this problem ...

(1) should be $\displaystyle g(x) = \int_2^x f(t) \, dt$,

(2) which of the following could be $y=g(x)$, not $g(f)$

$g’(x) = f(x) \implies$ the graph of $f(x)$ is the graph of the derivative of $g(x)$

so ... what do you think?
 
good catch
 
karush said:
https://www.physicsforums.com/attachments/10736
image due to graphsok just by observation I chose D since integrals tend to introduce curves not eliminate them and the slope was positive
You should be able to say much more than that! Since the graph of f is a straight line, f is "linear" and its integral is quadratic so its graph is a parabola. Further since the graph of f is below the x-axis the integral starts out negative and becomes positive. Yes, the slope of f is positive so the curvature of g is positive. THAT is why "D" is the correct choice!
 
deriv_analysis2.jpg
 
graphs always help a lot
 
Here is a WIP of a PDF of the MHB replies to the AP Calculus problems given here
A counter has been put into the overleaf document which at this posting is 40,000 views of the replies
My goal is to make the PDF error free and reach 100 problems
Mahalo
https://dl.orangedox.com/6rStfn4eMFHuHvAKuX
 
Last edited:
"WIP', "PDF", "AP"? I thought we were supposed to use English on this site!
 
WIP Work In Progress
 
Last edited:

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