If a graph is continuous is it differentiable? Why or Why not? Give examples to support your conclusion. If a graph is differentiable is it continuous? Why or Why not? Give examples to support your conclusion.
If a graph is continuous it is sometimes differentiable, because a continuous graph will be differentiable as long as the graph does not conflict with the properties of differentiability.
Why or why not?
A graph can be continuous when there is a corner, a vertical tangent line, or a cusp, however, the properties of differentiability state that if there is a corner, a vertical tangent line, or a cusp in the graph the function is not differentiable. Because a graph can be continuous with a corner, a vertical tangent line, or a cusp but the function cannot be differentiable at a corner, vertical tangent line, or a cusp the graph is not always continuous if it is differentiable.
Give examples to support your conclusion.
If given a graph of X^odd/odd there would be a vertical tangent line this graph would be continuous but it would not be differentiable because the properties of differentiability state that a vertical tangent line is not differentiable. If given a graph of X^even/even the graph would be continuous and also differentiable because X^even/even does not conflict with the properties of differentiability.
If a graph is differentiable is it continuous? Yes, because for a graph to be differentiable it must be continuous.
Why or Why not?
The properties of differentiability state that for a graph a function is not differentiable when the graph is not continuous. Therefore, in order for the graph to be differentiable the graph must be continuous.
Give examples to support your conclusion.
One of the properties of differentiability state that if there is a jump, a removable, or a vertical asymptote the graph will not be differentiable and the only occasions where a graph is not continuous is when there is a jump, a removable, or a vertical asymptote. Therefore, the only way the graph can be differentiable is if the graph is continuous. For example, if the graph has a jump there is no possible way for the graph to be differentiable. Likewise, if the graph is considered differentiable there is no way there is a jump, a removable, or a vertical asymptote and the graph is thereby not able to be discontinuous.
If a graph is continuous is it differentiable? Why or why not?
A graph is sometimes continuous if it is differentiable. If a graph is not differentiable at a point you can take a derivative, but you cannot plug in x. Graphs that are differentiable will be continuous until there is a conflict at any point. Some of these conflicts include absolute value or a piecewise. When graphing an absolute value, there will be a corner which makes the graph not differentiable. The graph also must not have a vertical tangent line. Lastly, if a graph isn’t continuous there cannot be a jump, removable, or vertical asymptote. A graph has to be fully continuous everywhere for it to be differentiable. A good example would be sin x. Sin x is differentiable because when you graph it there are no breaks or bends.
If a graph is differentiable is it continuous? Why or why not?
If the graph is differentiable at a point, then it must also be continuous at that point. Pretty much any differentiable function must be continuous at every point in its domain. On a graph a differentiable function will not have a corner, or absolute value, or a piece wise. They cannot have a vertical tangent line, which are exponents that are odd/odd. And obviously if there is a jump, removable, or vertical asymptote the graph cannot be continuous. If a graph is differentiable the graph has a non-vertical tangent line at a point, therefore the graph cannot have a break, bend, or cusp at that point.
**If a graph is continuous, is it differentiable? NO, not necessarily. In simple cases, a function that is continuous will also be differentiable. In other cases, a continuous function may not be differentiable. In "simple" cases (as I would say), a continous function that is also differentiable would be a polynomial (linear/quadratic/etc.) On the other hand, a continuous function will NOT be differentiable if the function includes an absolute value, or a variable with a fraction as an exponent. Here are a few examples:
1.) abs(x+6) Is this function differentiable? *NO, it is not. You should recognize that it's not differentiable once you see the absolute value symbols..Even though the graph looks continuous in your calculator, in absolute values there are "corners." With corners, your function CANNOT be differentiable.
2.) (x-5)^3/5 Is this function differentiable? *The first thing you should notice is the exponent..Since it's a fraction, it can be one of two things (both of which are NOT differentiable). It can either be a vertical tangent line (odd#/odd#) or a cusp (even#/odd#) *In this case, it looks like there will be a vertical tangent line, so the function is not differentiable.
**If a graph is differentiable, is it continuous? Now automatically, one would think that these two questions mean the same thing even though the words are flipped around...maybe not, but I sure thought that at first. However, the answer to this question is YES, because the question means--if you can take the derivative of a function, then that function must be continuous. And that is trueeeeeee! Because you cannot take the derivative of something that isn't continous..such as a function that has a jump, removable, or vertical asymptote. Here are some examples:
1.) 4x^3 + 13x^2 - 2x *This function is a polynomial, so therefore it is continuous. *So when you take the derivative you would get: 12x^2 + 26x - 2
2.) abs(x+4)/(x+4) *Hmmm...looka that. An absolute value AND it's in a fraction. I probably stumped you there, huh?..I know *Well it's pretty simple actually. We had a few problems like this in our limits packets. Although the absolute value is deceiving, I didn't say anything about them being in a fraction now did I? :P ..So all you really have to do is type the equation into your calculator to see if it's continuous. Andddddddd, it's not. There is a jump at x=-4, so therefore the function is NOT differentiable because it is not continuous. ;D
“If a graph is continuous is it differentiable?” No, because a graph can be continuous and still not differentiable. It’s not differentiable if there is a corner on the graph. This happens with absolute values, but absolute values are also always continuous. Ex. 4) absolute value of (x-2) There is an absolute value, therefore this equation is not differentiable at x = 2, but the equation is continuous because there are no jumps or vertical asymptotes. You also can’t take the derivative of a vertical tangent line, but vertical tangent lines are also continuous. Ex. 5) (x+5)^3/5 Because of the exponent 3/5, this equation is not differentiable at x = -5, but the equation is continuous because there are no jumps or vertical asymptotes. And lastly, you can’t take the derivative of a cusp, but once again, cusps are continuous Ex. 6) x^(6/7) Because of the exponent 6/7, this equation is not differentiable at x = 0, but the equation is continuous because there are no jumps or vertical asymptotes.
“If a graph is differentiable is it continuous?” Yes, because in order for a graph to be continuous, it can not have a jump or a vertical asymptote with un-matching sides and a non-continuous graph is not differentiable. Therefore, if you can take the derivative of a graph, it will always be continuous because you can not take the derivative of a non-continuous graph. Ex. (The following equation is a piecewise): f(x) = x, x < 2 & x^2, x > (or equal to) 2 When you plug in x(2), you get different answers, therefore the sides do not match. This means there is a jump and the graph is not continuous. Because this graph is not continuous, it is also not differentiable.
If a graph is continuous, is it dfferentiable? No, a graph can be continuous and not differentiable, examples such as corners (abs. value/piecewise), cusps, and vertical tangent lines. the function is considered continuous if there are no jumps, removables, or vertical asypmtotes, however, it is not differentiable. |5x| is continuous, but is not differetniable at x=5
If a graph is differentiable, is it continous? Yes because the graph is continuous if there are no discontinuities (jumps, asymptotes, etc.) it is continuous because there is no derivative of uncontinuous graphs. x2 is continuous (parablola), therefore it is differentiable
A continuous graph can sometimes be differentiable, but not always. A continuous graph may include a corner (created by absolute value), a vertical asymptote, or a cusp. According to the properties of differentiability, if the continuous graph contains a corner, vertical asymptote, or cusp, it is not differentiable.
"Why or why not?"
A graph can be continuous when there is a corner, a vertical tangent line, or a cusp. However, the properties of differentiability state that if there is a corner, a vertical tangent line, or a cusp in the graph the function is not differentiable. Because a graph can be continuous with a corner, a vertical tangent line, or a cusp but the function cannot be differentiable at a corner, vertical tangent line, or a cusp the graph is not always differentiable if it is continuous.
"Give examples to support your answer."
1.) is abs(x+4) differentiable? -No, because there is an absolute value in the equation, there is a corner. Although the graph is continuous, the properties of differentiablility do not allow this equation to be differentiable.
2.) is (x+5)^3/5 differentiable? -No, because the exponent is a negative over a negative, i.e. 3/5, it is not differentiable at x=-5. Although, the equation is continuous because there are no jumps or removables. Also, the derivative of this equation cannot be taken.
"If a graph is differentiable, is it continuous?"
Yes, a non-continuous graph is not differentiable. Therefore, if you can take the derivative of the graph, it will always be continuous. -you cannot take the derivative of a non-continuous graph. One of the properties of differentiability state that if there is a jump, a removable, or a vertical asymptote the graph will not be differentiable and the only occasions where a graph is not continuous is when there is a jump, a removable, or a vertical asymptote. Therefore, the only way the graph can be differentiable is if the graph is continuous.
1.) abs(x+2)/(x+2) -although there is an absolute value, it is in a fraction. You just type the equation into your calculator and see whether it is or is not continuous. (its not). There is a jump at x=2, and since the graph is not continuous, it is not differentiable.
If a graph is continuous, is it dfferentiable? No, a graph can be continuous and not differentiable: corners, cusps, and vertical tangent lines; the function is considered continuous if there are no jumps, removables, or vertical asypmtotes, but it is not differentiable. |7x| is continuous, but is not differetniable at x=7
If a graph is differentiable, is it continous? Yes, the graph is continuous if there arent any discontinuities, such as jumps or asymptotes; it is continuous because there is no derivative of uncontinuous graphs. x2 is continuous as a parabola, so it is differentiable
The answer to that question is sometimes. A continuous graph is sometimes differentiable, because even if the graph is continuous, there can be a corner, vertical tangent line, or a cusp to make it not differentiable. |2x+1| is continuous, but is not differentiable at x=(-1/2)
If a graph is differentiable, is it continuous?
Yes. If a graph is not continuous then it is not differentiable. So the graph has to be continuous if it is differentiable. An example for this would be: x^2+3 It is continuous because it is a parabola. It is differentiable because there are no forms of discontinuity, no cusps, no corners, or no vertical tangent lines.
Sometimes, because a graph is continnous when they have a cusp, or a corner, or a vertical tangent line. But if there is a cusp, a corner, or a vertical tangent line then the graph is not differentiable.
Ex. if the function has a cusp then it is not differentiable but it can be continuous. Ex. if the function has a removable then it is not differentiable and it is not continuous
Always, if a function is differentiable then it does not have a cusp, corner, vertical tangent line, jump, removable, or vertical asymptote. And if a function does not have any of those, then it is continuous.
Ex.If a function does not have a cusp, a vertical tangent, a corner, a removable, a jump, or a vertical asymptote then the function is both differentiable and continuous.
If a graph is continuous, it is sometimes differentiable. Sometimes the graph of a continuous function is differentiable, depending on the rules of differentiability (or non-differentiability for that matter). If there is a vertical tangent line, a corner, jump, removable, or vertical asymptotes on a continuous graph, it is not differentiable because those are all parts of rules of non differentiability. Example: tan x. It is continuous, but however there is a vertical tangent line, so it not differentiable.
If a graph is differentiable, is it continuous?
If a graph is differentiable, it is continuous. One of the rules of non-differentiability is that if it is not continuous, it is not differentiable. Therefore, if it is differentiable, it should be continuous. An example could be x^3. It is continuous, and it is differentiable.
There are many rules of differentiability and non-differentiability that you just have to pay attention to and understand.
Fawkin internet.....finally got it fixed, and it's like midnight.....ain't that just fan-fawkin-tastic?!.....
If a graph is continuous, is it differentiable?
Sometimes.....that's literally all of an answer that I can think of right now.....umm.....yea......sooo......o wait.....BRAINBLAST!!....if there's a vertical tangent line, corner, jump, removable, or vertical asymptote on a continuous graph, it's not differentiable because of a rule we learned.....like tan x, it's continuous, but there's a vertical tangent line in it, so it's not differentiable
If a graph is differentiable, is it continuous?
hmm....that's a hard one...(TWSS).....lol, had to.....well, I do know that if a graph is differentiable, it's continuous.....yet again, there were some rules we learned last week about stuff like this...sadly, I can't remember them at all.....well, if its differentiable, it should be continuous.....something like x^3 is continuous and differentiable AT THE SAME TIME......I KNOW....IT'S MINDBLOWING.....
That's all I've got...it's coming in a little late because of my internet, so don't shoot me......PEACE
If a graph is continuous is it differentiable?
ReplyDeleteIf a graph is continuous it is sometimes differentiable, because a continuous graph will be differentiable as long as the graph does not conflict with the properties of differentiability.
Why or why not?
A graph can be continuous when there is a corner, a vertical tangent line, or a cusp, however, the properties of differentiability state that if there is a corner, a vertical tangent line, or a cusp in the graph the function is not differentiable. Because a graph can be continuous with a corner, a vertical tangent line, or a cusp but the function cannot be differentiable at a corner, vertical tangent line, or a cusp the graph is not always continuous if it is differentiable.
Give examples to support your conclusion.
If given a graph of X^odd/odd there would be a vertical tangent line this graph would be continuous but it would not be differentiable because the properties of differentiability state that a vertical tangent line is not differentiable.
If given a graph of X^even/even the graph would be continuous and also differentiable because X^even/even does not conflict with the properties of differentiability.
If a graph is differentiable is it continuous?
Yes, because for a graph to be differentiable it must be continuous.
Why or Why not?
The properties of differentiability state that for a graph a function is not differentiable when the graph is not continuous. Therefore, in order for the graph to be differentiable the graph must be continuous.
Give examples to support your conclusion.
One of the properties of differentiability state that if there is a jump, a removable, or a vertical asymptote the graph will not be differentiable and the only occasions where a graph is not continuous is when there is a jump, a removable, or a vertical asymptote. Therefore, the only way the graph can be differentiable is if the graph is continuous.
For example, if the graph has a jump there is no possible way for the graph to be differentiable. Likewise, if the graph is considered differentiable there is no way there is a jump, a removable, or a vertical asymptote and the graph is thereby not able to be discontinuous.
Prompt #3
ReplyDeleteIf a graph is continuous is it differentiable? Why or why not?
A graph is sometimes continuous if it is differentiable. If a graph is not differentiable at a point you can take a derivative, but you cannot plug in x. Graphs that are differentiable will be continuous until there is a conflict at any point. Some of these conflicts include absolute value or a piecewise. When graphing an absolute value, there will be a corner which makes the graph not differentiable. The graph also must not have a vertical tangent line. Lastly, if a graph isn’t continuous there cannot be a jump, removable, or vertical asymptote. A graph has to be fully continuous everywhere for it to be differentiable. A good example would be sin x. Sin x is differentiable because when you graph it there are no breaks or bends.
If a graph is differentiable is it continuous? Why or why not?
If the graph is differentiable at a point, then it must also be continuous at that point. Pretty much any differentiable function must be continuous at every point in its domain. On a graph a differentiable function will not have a corner, or absolute value, or a piece wise. They cannot have a vertical tangent line, which are exponents that are odd/odd. And obviously if there is a jump, removable, or vertical asymptote the graph cannot be continuous. If a graph is differentiable the graph has a non-vertical tangent line at a point, therefore the graph cannot have a break, bend, or cusp at that point.
**If a graph is continuous, is it differentiable?
ReplyDeleteNO, not necessarily. In simple cases, a function that is continuous will also be differentiable. In other cases, a continuous function may not be differentiable. In "simple" cases (as I would say), a continous function that is also differentiable would be a polynomial (linear/quadratic/etc.) On the other hand, a continuous function will NOT be differentiable if the function includes an absolute value, or a variable with a fraction as an exponent. Here are a few examples:
1.) abs(x+6) Is this function differentiable?
*NO, it is not. You should recognize that it's not differentiable once you see the absolute value symbols..Even though the graph looks continuous in your calculator, in absolute values there are "corners." With corners, your function CANNOT be differentiable.
2.) (x-5)^3/5 Is this function differentiable?
*The first thing you should notice is the exponent..Since it's a fraction, it can be one of two things (both of which are NOT differentiable). It can either be a vertical tangent line (odd#/odd#) or a cusp (even#/odd#)
*In this case, it looks like there will be a vertical tangent line, so the function is not differentiable.
**If a graph is differentiable, is it continuous?
Now automatically, one would think that these two questions mean the same thing even though the words are flipped around...maybe not, but I sure thought that at first. However, the answer to this question is YES, because the question means--if you can take the derivative of a function, then that function must be continuous. And that is trueeeeeee! Because you cannot take the derivative of something that isn't continous..such as a function that has a jump, removable, or vertical asymptote. Here are some examples:
1.) 4x^3 + 13x^2 - 2x
*This function is a polynomial, so therefore it is continuous.
*So when you take the derivative you would get:
12x^2 + 26x - 2
2.) abs(x+4)/(x+4)
*Hmmm...looka that. An absolute value AND it's in a fraction. I probably stumped you there, huh?..I know
*Well it's pretty simple actually. We had a few problems like this in our limits packets. Although the absolute value is deceiving, I didn't say anything about them being in a fraction now did I? :P ..So all you really have to do is type the equation into your calculator to see if it's continuous. Andddddddd, it's not. There is a jump at x=-4, so therefore the function is NOT differentiable because it is not continuous. ;D
“If a graph is continuous is it differentiable?”
ReplyDeleteNo, because a graph can be continuous and still not differentiable.
It’s not differentiable if there is a corner on the graph. This happens with absolute values, but absolute values are also always continuous.
Ex. 4) absolute value of (x-2)
There is an absolute value, therefore this equation is not differentiable at x = 2, but the equation is continuous because there are no jumps or vertical asymptotes.
You also can’t take the derivative of a vertical tangent line, but vertical tangent lines are also continuous.
Ex. 5) (x+5)^3/5
Because of the exponent 3/5, this equation is not differentiable at x = -5, but the equation is continuous because there are no jumps or vertical asymptotes.
And lastly, you can’t take the derivative of a cusp, but once again, cusps are continuous
Ex. 6) x^(6/7)
Because of the exponent 6/7, this equation is not differentiable at x = 0, but the equation is continuous because there are no jumps or vertical asymptotes.
“If a graph is differentiable is it continuous?”
Yes, because in order for a graph to be continuous, it can not have a jump or a vertical asymptote with un-matching sides and a non-continuous graph is not differentiable. Therefore, if you can take the derivative of a graph, it will always be continuous because you can not take the derivative of a non-continuous graph.
Ex. (The following equation is a piecewise): f(x) = x, x < 2 & x^2, x > (or equal to) 2
When you plug in x(2), you get different answers, therefore the sides do not match. This means there is a jump and the graph is not continuous. Because this graph is not continuous, it is also not differentiable.
If a graph is continuous, is it dfferentiable?
ReplyDeleteNo, a graph can be continuous and not differentiable, examples such as corners (abs. value/piecewise), cusps, and vertical tangent lines. the function is considered continuous if there are no jumps, removables, or vertical asypmtotes, however, it is not differentiable.
|5x| is continuous, but is not differetniable at x=5
If a graph is differentiable, is it continous?
Yes because the graph is continuous if there are no discontinuities (jumps, asymptotes, etc.) it is continuous because there is no derivative of uncontinuous graphs.
x2 is continuous (parablola), therefore it is differentiable
"If a graph is continuous, is it differentiable?"
ReplyDeleteA continuous graph can sometimes be differentiable, but not always. A continuous graph may include a corner (created by absolute value), a vertical asymptote, or a cusp. According to the properties of differentiability, if the continuous graph contains a corner, vertical asymptote, or cusp, it is not differentiable.
"Why or why not?"
A graph can be continuous when there is a corner, a vertical tangent line, or a cusp. However, the properties of differentiability state that if there is a corner, a vertical tangent line, or a cusp in the graph the function is not differentiable. Because a graph can be continuous with a corner, a vertical tangent line, or a cusp but the function cannot be differentiable at a corner, vertical tangent line, or a cusp the graph is not always differentiable if it is continuous.
"Give examples to support your answer."
1.) is abs(x+4) differentiable?
-No, because there is an absolute value in the equation, there is a corner. Although the graph is continuous, the properties of differentiablility do not allow this equation to be differentiable.
2.) is (x+5)^3/5 differentiable?
-No, because the exponent is a negative over a negative, i.e. 3/5, it is not differentiable at x=-5. Although, the equation is continuous because there are no jumps or removables. Also, the derivative of this equation cannot be taken.
"If a graph is differentiable, is it continuous?"
Yes, a non-continuous graph is not differentiable. Therefore, if you can take the derivative of the graph, it will always be continuous. -you cannot take the derivative of a non-continuous graph. One of the properties of differentiability state that if there is a jump, a removable, or a vertical asymptote the graph will not be differentiable and the only occasions where a graph is not continuous is when there is a jump, a removable, or a vertical asymptote. Therefore, the only way the graph can be differentiable is if the graph is continuous.
1.) abs(x+2)/(x+2)
-although there is an absolute value, it is in a fraction. You just type the equation into your calculator and see whether it is or is not continuous. (its not). There is a jump at x=2, and since the graph is not continuous, it is not differentiable.
If a graph is continuous, is it dfferentiable?
ReplyDeleteNo, a graph can be continuous and not differentiable: corners, cusps, and vertical tangent lines; the function is considered continuous if there are no jumps, removables, or vertical asypmtotes, but it is not differentiable.
|7x| is continuous, but is not differetniable at x=7
If a graph is differentiable, is it continous?
Yes, the graph is continuous if there arent any discontinuities, such as jumps or asymptotes; it is continuous because there is no derivative of uncontinuous graphs.
x2 is continuous as a parabola, so it is differentiable
If a graph is continuous, is it differentiable?
ReplyDeleteThe answer to that question is sometimes. A continuous graph is sometimes differentiable, because even if the graph is continuous, there can be a corner, vertical tangent line, or a cusp to make it not differentiable.
|2x+1| is continuous, but is not differentiable at x=(-1/2)
If a graph is differentiable, is it continuous?
Yes. If a graph is not continuous then it is not differentiable. So the graph has to be continuous if it is differentiable. An example for this would be:
x^2+3
It is continuous because it is a parabola. It is differentiable because there are no forms of discontinuity, no cusps, no corners, or no vertical tangent lines.
Sometimes, because a graph is continnous when they have a cusp, or a corner, or a vertical tangent line. But if there is a cusp, a corner, or a vertical tangent line then the graph is not differentiable.
ReplyDeleteEx. if the function has a cusp then it is not differentiable but it can be continuous.
Ex. if the function has a removable then it is not differentiable and it is not continuous
Always, if a function is differentiable then it does not have a cusp, corner, vertical tangent line, jump, removable, or vertical asymptote. And if a function does not have any of those, then it is continuous.
Ex.If a function does not have a cusp, a vertical tangent, a corner, a removable, a jump, or a vertical asymptote then the function is both differentiable and continuous.
If a graph is continuous, is it differentiable?
ReplyDeleteIf a graph is continuous, it is sometimes differentiable. Sometimes the graph of a continuous function is differentiable, depending on the rules of differentiability (or non-differentiability for that matter). If there is a vertical tangent line, a corner, jump, removable, or vertical asymptotes on a continuous graph, it is not differentiable because those are all parts of rules of non differentiability. Example: tan x. It is continuous, but however there is a vertical tangent line, so it not differentiable.
If a graph is differentiable, is it continuous?
If a graph is differentiable, it is continuous. One of the rules of non-differentiability is that if it is not continuous, it is not differentiable. Therefore, if it is differentiable, it should be continuous. An example could be x^3. It is continuous, and it is differentiable.
There are many rules of differentiability and non-differentiability that you just have to pay attention to and understand.
Fawkin internet.....finally got it fixed, and it's like midnight.....ain't that just fan-fawkin-tastic?!.....
ReplyDeleteIf a graph is continuous, is it differentiable?
Sometimes.....that's literally all of an answer that I can think of right now.....umm.....yea......sooo......o wait.....BRAINBLAST!!....if there's a vertical tangent line, corner, jump, removable, or vertical asymptote on a continuous graph, it's not differentiable because of a rule we learned.....like tan x, it's continuous, but there's a vertical tangent line in it, so it's not differentiable
If a graph is differentiable, is it continuous?
hmm....that's a hard one...(TWSS).....lol, had to.....well, I do know that if a graph is differentiable, it's continuous.....yet again, there were some rules we learned last week about stuff like this...sadly, I can't remember them at all.....well, if its differentiable, it should be continuous.....something like x^3 is continuous and differentiable AT THE SAME TIME......I KNOW....IT'S MINDBLOWING.....
That's all I've got...it's coming in a little late because of my internet, so don't shoot me......PEACE