Draw graph given f'(x), f''(x), domain, etc. Please check?

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SUMMARY

The discussion focuses on sketching a graph based on the first and second derivatives, f'(x) and f''(x), and their implications on concavity and convexity. The user seeks validation on their hand-drawn graph, particularly regarding the interpretation of concavity between specified intervals and the behavior of the graph as it approaches a limit at x = -2. Key points include the necessity of labeling axes correctly and using solid and open dots to indicate points and holes, respectively. The user emphasizes that the graph should reflect the behavior dictated by the second derivative, specifically that f''(x) being positive indicates an upward opening and negative indicates a downward curve.

PREREQUISITES
  • Understanding of first and second derivatives in calculus
  • Knowledge of concavity and convexity concepts
  • Familiarity with graphing techniques and conventions
  • Ability to interpret limits and their graphical implications
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  • Study the relationship between f'(x) and f''(x) in graphing
  • Learn about the implications of concavity and convexity on graph shapes
  • Explore the concept of limits and their effect on graph behavior
  • Review graphing conventions, including the use of solid and open dots
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Students and educators in calculus, mathematicians focusing on graph theory, and anyone involved in visualizing mathematical functions and their derivatives.

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1. I was asked to roughly sketch a graph given the information seen, in the below picture, to the right of the graph.

28rcitw.jpg


2. My actual graph is done by hand and the lines are not as straight, but the general shape is the same.

Does it look approximately right?

The areas I am worried about are:
a) when it specifies being concave of convex between certain intervals: whether it means completely visible between those intervals as concave/convex or just beginning to appear concave/convex

b) where it says "no limit of y x -> -2" exists, whether or not I should have drawn it the same angle as I did, or started the one to the left of -2 from a different angle.
 
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First, label the axis, and adopt the dot convention where a solid dot contains that point and a circle indicates a hole. Everything looks okay until you get to the second derivative/concavity. The points where the second derivative would be zero (in between the domain of concavity given) should be points, and after that it should have a curve. For instance, when it says that from 5 to 10 (for example) f'' is positive, that means the graph should "open upwards" from 5 to 10, and if f'' is negative from 10 to infinity, then it should curve back downwards.
 
Thanks, QuarkCharmer.

To you or anyone who sees, I'm just wondering if this one is more accurate? Again, aside from the straight line, which I know means f"(x) = 0.

2cd9hz.jpg
 

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