goo.gl/photos/2RPU1dAWoKzUn4Pf8
1. Which of the above functions are not continuous where they are defined? Why?
2. Which of the above functions are not differentiable where they are defined? Why?
For the first question, I said that graph F is not continuous because of a jump, but I'm missing another one from those 6 graphs.
For the second question, I said b is not differentiable because of corners, but I appear to be missing 3 more. I initially answered C, D, and F but those aren't correct.
(All 6 graphs are part of the questions and A and a are two different graphs fyi)
Any help is greatly appreciated!
Any ideas?
To determine which functions in the given graphs are not continuous or differentiable, let's analyze each graph step by step:
1. Graph A:
- This graph appears to be continuous since there are no abrupt jumps or breaks in the curve.
2. Graph a:
- Similar to Graph A, Graph a is continuous with no sudden interruptions.
3. Graph b:
- Graph b is not continuous since there is a jump/discontinuity at the point where the curve changes abruptly. This could be due to a sudden change in the function's definition or behavior at that specific point.
4. Graph c:
- Graph c seems to be continuous without any visible breaks or jumps.
5. Graph d:
- Similar to Graph c, Graph d appears to be continuous without any noticeable interruptions.
6. Graph e:
- Graph e is continuous since the curve is smooth without any apparent jumps or breaks.
7. Graph F:
- As you correctly mentioned, Graph F is not continuous due to the presence of a jump. However, there is one more graph that is not continuous.
To identify the additional graph that is not continuous, we can see that Graph B is also not continuous. It exhibits a discontinuity where a vertical asymptote appears, causing a break in the curve.
Now let's move on to the differentiability of the functions:
1. Graph A:
- Graph A appears to be differentiable since the curve is smooth without any sharp corners or cusps.
2. Graph a:
- Graph a also seems to be differentiable without any visible abrupt changes or corners in the curve.
3. Graph b:
- As you correctly identified, Graph b is not differentiable due to the presence of corners or cusps.
4. Graph c:
- Contrary to your initial answer, Graph c is actually differentiable. It lacks any sharp corners or cusps, and the curve seems to be smooth throughout.
5. Graph d:
- Similar to Graph c, Graph d is differentiable. There are no sharp corners or cusps in the curve.
6. Graph e:
- Graph e appears to be differentiable without any visible breaks or corners.
7. Graph F:
- Just like before, Graph F is not differentiable due to the presence of corners.
So, to summarize the answers to your questions:
1. The graphs that are not continuous where defined are: Graph b, Graph F, and Graph B (vertical asymptote).
2. The graphs that are not differentiable where defined are: Graph b, Graph F.
I hope this clarifies the analysis of the given graphs. Let me know if you have any further questions!
To determine which of the given functions are not continuous where they are defined, you need to examine the graphs and identify any points where there are sudden jumps, gaps, or vertical asymptotes.
Let's go through each of the functions:
1. Function A: Looking at the graph, there are no sudden jumps, gaps, or vertical asymptotes. It appears to be continuous where it is defined.
2. Function B: There is a sudden jump or discontinuity at x = 0, where the graph changes abruptly. Therefore, Function B is not continuous at x = 0.
3. Function C: From the graph, it is continuous without any jumps or gaps. Therefore, Function C is continuous where it is defined.
4. Function D: There is a vertical asymptote at x = -2 and a jump or discontinuity at x = 2. Both of these points indicate that Function D is not continuous at x = -2 and x = 2.
5. Function E: The graph of Function E is continuous without any sudden jumps or gaps.
6. Function F: There is a vertical asymptote at x = 1 and a jump or discontinuity at x = 3. Therefore, Function F is not continuous at x = 1 and x = 3.
So, to answer your first question, the functions that are not continuous where they are defined are Function B, Function D, and Function F.
Moving on to the second question, which asks about differentiability. Remember that a function is differentiable at a point only if it is continuous at that point and does not have any sharp corners or cusps.
Let's analyze each function:
1. Function A: As we have determined earlier, Function A is continuous where it is defined. To check for differentiability, we need to examine if there are any sharp corners or cusps. From the given graph, there are no such points. Therefore, Function A is differentiable where it is defined.
2. Function B: From the graph, it is evident that Function B has a sharp corner at x = 0. Therefore, Function B is not differentiable at x = 0.
3. Function C: The graph suggests that Function C is continuous without any sharp corners or cusps. So, it is differentiable where it is defined.
4. Function D: The graph of Function D shows a sharp corner or cusp at x = 2. Therefore, Function D is not differentiable at x = 2.
5. Function E: Similar to the analysis for Function A and C, Function E is continuous and without any sharp corners or cusps, making it differentiable where it is defined.
6. Function F: From the graph, it is apparent that Function F has a sharp corner at x = 3. Hence, Function F is not differentiable at x = 3.
To summarize, the functions that are not differentiable where they are defined are Function B, Function D, and Function F.