This blog created by Fariel Mohan is to assist MATH110D students in understanding foundational mathematical concepts.
Wednesday, January 16, 2008
Rate of change
What is gradient? What is the problem with finding gradient in a straight line? What is the problem with finding gradient in a curve? Is gradient in any way connected to rate of change?
imagine your are walking up a hill and you feel as if you are going to roll back down, this effect is due to its gradient or the rate of change in height of the hill
GRADIENT OF A SLOPE REFERS TO THE STEEPNESS OF THE SLOPE. AND ACCORDING TO HOW STEEP, I.E "GRADIENT",THE SLOPE IS, THE GREATER THE "PULL" OF GRAVITY WILL ACT ON WHAT EVER OBJECT ON THAT SLOPE. NOW COMING TO RATES OF CHANGE.... IMAGINE IF A BALL WAS PLACED ON A SO CALL "GENTLE" SLOPE AND WAS LEFT TO ROLL, DUE TO ACCELERATION CAUSED BY GRAVITY, THE BALL'S VELOCITY WILL INCREASE AFTER A CURTAIN TIME.BUT IF THIS SAME BALL WAS SET TO ROLL ON A STEEPER SLOPE, SINCE THE "GRADIENT" WAS INCREASED, THE PULL OF GRAVITY ON THE BALL WILL ALSO INCREASE AND THERFORE THE VELOCITY OF THE BALL, ROLLING DOWN THIS STEEPER SLOPE WOULD BE GREATER THAN THE VELOCITY THE BALL HAD ON THE GENTLE SLOPE.AND THIS THEREBY END MY EXPLAINATION ON RATE OF CHANGE.COMMENTS WILL BE ACCEPTED.
In maths gradient of a straight line can be found using 2 coordinates. It is the ratio of the change on the y-axis(y2-y1)to the change on the x-axis(x2-x1). Gradient can also relate to other things for example colours, different gradients of colours can mean different shades.
Gradient is another word for slope.The higher the gradient of a graph at a point,the steeper the line is at that point, negative gradient means that the line slopes downwards.The gradient = (change in y-coordinate)/(change in x-coordinate).
i agree with u dogman but u forgot about the curves an to find the greadent of a curve you can draw a tangant alongside a curve and then take two points on which the tangent and the curve makes contact.
i must say that i agree with everyone so far. everyone has the gist of it. the gradient of a line is the measure of the slope of that line and can be calculated given two coordinates on the line. however, the gradient of a curve is not so easy to determine seeing that the gradient is always changing. also, at any point on the curve there is only one x and y coordinate which makes the gradient impossible to find
i know a curve don't ever have a constant gradient, the gradient of the curve can't be found because the rate of change on a curve would be different at every point
a problem with finding the gradient of a straight line is that is that not all the points plotted on the graph may lie on this line since a best fit line may be drawn and so the line plotted from the graph will be inaccurate, hence the gradient would also be inaccurate
yes... rate of change, when compared to the gradient of a graph are connected in that; if the increase or decrease in the value of a given quantity is plotted a against the timetaken in a graph (e.g velocity/time gragh) the rate of change value would also be equal to the gradient of that graph
i must agree with tweety and suybzero as the gradient is basically the slope in a given line,however the gradient of a straight line graph is always the same at any point but on a curve the gradient is always changing at different points.
Gradient is the degree of a slope, which could be founded by the change in height divided by the distance travelled. Gradient in a straight line is easily found by using any two points on that line to find the same value but in a curve gradient has different values at different points. Gradient is connected to the rate of change which gives differing gradient values by the speed of the change.
I understand that indeed the gradient of a line is the rate of change as the relationship between the two axes moves from one point to another.To some extent I can also agree with people who refer to the gradient as the slope of sometheing because in visualizing that concept, one can deduce that a slope of any kind does not involve the position of just one point.I see slope as elevation,ie.the distance "covered" from one point to another.That too is a rate of change.Problems in finding the gradients can be: 1.In a straight line, the line may not always pass through all the plotted points to connect them,so then obtaining the values to find the gradient is not so straightfoward.You have to examine the area in the graph well to get the accurate values on the 'x'and 'y' axes. 2.In a curve,there are no two points that are the same on that graph.You can't just take up two points from the curve and find the gradient. All points will have to be examined.If it is found that two points along the curve have close values,that may then lead to a start in attempting to find the gradient.
wel first of all wat dis gradient thing is about?? i using dis word all throughout my skool life an still doh noe wat it mean..i de bounce up dis word innah topic called cordinate geometry..an i use some kinda formula wat de techer give me..if i remember well i substituted x and y values into de formular and i obtained a result called the gradient...or hor and in physics it had a graph with velocity and time and i used values from de graph to obtain de gradient .this gradient was actually de accleration.....so to my understanding the gradient is used to show de relation ship of two things.plz correct me if im wrong!!!
the gradient of a curve is not fixed and changes which each new point on the curve, sometimes to find the gradient of a specific point on the curve, one must calculated the gradient of other points on that are very close that point and from that deduce its gradient
the gradient of a curve can be difficult to calculate since at every new point a change occurs, but the gradient of a line really implies to a rate of change of the height with respect to each new point following.that is it is a measure of its steepness.
problem with finding the gradiant of curve...a curve does not have a constant gradiant but many gradiants.. it may prove to be very tedious and time consuming in finding the gradiant at several points.
gradient is vertical rise over horizontal shift meaning it is just a value for steepness of a line or curve and i agree it could be the slope as well. in finding gradient of a straight line i think it may not always be accurate since sometimes the line might not pass through all the points like say a line of best fit has to be drawn and points are left out. please tell me if i'm wrong cause i'm not quite sure. for a curve it will be difficult to find the gradient because each point on the curve will have a different gradient however points very very close to each other will have a gradient value that is very close.
ratmam know what you mean i think you are right in a way since the relationship between two things could result in a value for rate of change such as accelleration as lilo explained about the velocity time graph. i also believe that gradient is connected to rate of change infact it is rate of change. like say you are driving up a steep hill when you are now about to go up the hill you will press on the gas a little but when you continue to go up and the hill gets steeper you will press the accelerator even more and harder for more power and the car to go, this is due to the change in steepness which is the rate of change or gradient of the hill.
yes j joker j!!! wat do u really mean by "choosing bad points" im confused. i tink the problem with finding the gradiant of a straight is that the line does not always pess through all the points. hence different lines of different gradiants would be obtained. True???
I am very disappointed in the comments. I have summarized how many comments by user ids I am to classify whether they are valid or invalid
Out of 70 students I have 26 users in this response. For this response the 44 others have until Friday. The day I check a comment, I will not check that comment again.
moose 1 boo boo boy 1 subyzero 1 tweety 1 don cristobal 1 master_cookie 1 dogman 1 apocalypse 1 who 1 ichigo 1 j joker j 2 sleepy 1 dark angel 1 wong fei hong 1 xm2 2 lilo 4 supermuffinz 1 cruiser 2 poison 1 sparkle 1 cokebaby 3 sweediekinks 1 ratman 2 tinkerbell 1 tigger 1 kelz 2
In general a rate of change may be the change in anything divided by the corresponding change in a related variable. The "slope" of a graph of y vs. x for example is the change in y divided by the corresponding change in x. It is called the rate of change in y with respect to x.this is called the gradient.For each of the chosen y values there will be a corresponding value of x. Linear Rate of Change.It is harder to obtain two possible values of x; therefore x + delta x represents the x2 value or the neighbor of x.Not only does the rate of change depend on where along the x axis we look, it also depends on how we select the interval over which to measure it.
ker vin i know that the gradient is the rate of change but it is not completely clear to me. i dont really follow wat u said in your last comment can any1 eni=lighten me.
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41 comments:
gradient out of any mathematical context would be the slope something like a hill for example but in maths it equals y2-y1 divided by x2-x1
the gradiant of a line is the extent to which a line is sloped or more generally the rate of change of the line as the line progresses
the gradient of a line to my own understanding is the steepness of a slope. which shows the rate of change of the line.
i agree with suybzero gradient is the rate of change of a slope/steepness of something
imagine your are walking up a hill and you feel as if you are going to roll back down, this effect is due to its gradient or the rate of change in height of the hill
GRADIENT OF A SLOPE REFERS TO THE STEEPNESS OF THE SLOPE. AND ACCORDING TO HOW STEEP, I.E "GRADIENT",THE SLOPE IS, THE GREATER THE "PULL" OF GRAVITY WILL ACT ON WHAT EVER OBJECT ON THAT SLOPE. NOW COMING TO RATES OF CHANGE.... IMAGINE IF A BALL WAS PLACED ON A SO CALL "GENTLE" SLOPE AND WAS LEFT TO ROLL, DUE TO ACCELERATION CAUSED BY GRAVITY, THE BALL'S VELOCITY WILL INCREASE AFTER A CURTAIN TIME.BUT IF THIS SAME BALL WAS SET TO ROLL ON A STEEPER SLOPE, SINCE THE "GRADIENT" WAS INCREASED, THE PULL OF GRAVITY ON THE BALL WILL ALSO INCREASE AND THERFORE THE VELOCITY OF THE BALL, ROLLING DOWN THIS STEEPER SLOPE WOULD BE GREATER THAN THE VELOCITY THE BALL HAD ON THE GENTLE SLOPE.AND THIS THEREBY END MY EXPLAINATION ON RATE OF CHANGE.COMMENTS WILL BE ACCEPTED.
In maths gradient of a straight line can be found using 2 coordinates. It is the ratio of the change on the y-axis(y2-y1)to the change on the x-axis(x2-x1). Gradient can also relate to other things for example colours, different gradients of colours can mean different shades.
missippii think a gradient is a variation of two pointson a graph.
gradient measures the steepness of a line
Gradient is another word for slope.The higher the gradient of a graph at a point,the steeper the line is at that point, negative gradient means that the line slopes downwards.The gradient = (change in y-coordinate)/(change in x-coordinate).
i agree with u dogman but u forgot about the curves an to find the greadent of a curve you can draw a tangant alongside a curve and then take two points on which the tangent and the curve makes contact.
you could find the gradient like that, dogman, but you could also differentiate the equation to get the gradient , if given an equation.
yes dogman i do agree with you but the gradiant could be easier stated as the verticalrise divide by the horizontal shift
i must say that i agree with everyone so far. everyone has the gist of it. the gradient of a line is the measure of the slope of that line and can be calculated given two coordinates on the line. however, the gradient of a curve is not so easy to determine seeing that the gradient is always changing. also, at any point on the curve there is only one x and y coordinate which makes the gradient impossible to find
the gradient is a value obtained from a line or slope which the line or slope increases or decreases at constant rate.
by the way how did the formula or the concept of a gradient come into play???????
how does the gradient affect everyday lives?????
i know a curve don't ever have a constant gradient, the gradient of the curve can't be found because the rate of change on a curve would be different at every point
a problem with finding the gradient of a straight line is that is that not all the points plotted on the graph may lie on this line since a best fit line may be drawn and so the line plotted from the graph will be inaccurate, hence the gradient would also be inaccurate
yes... rate of change, when compared to the gradient of a graph are connected in that; if the increase or decrease in the value of a given quantity is plotted a against the timetaken in a graph (e.g velocity/time gragh) the rate of change value would also be equal to the gradient of that graph
i must agree with tweety and suybzero as the gradient is basically the slope in a given line,however the gradient of a straight line graph is always the same at any point but on a curve the gradient is always changing at different points.
A gradient is the steepness or slope of a line at an incline
gradient is simply the slope of a line, which is calculated by the change in y divided by the change in x. it is also called the rate of change
Gradient is the degree of a slope, which could be founded by the change in height divided by the distance travelled. Gradient in a straight line is easily found by using any two points on that line to find the same value but in a curve gradient has different values at different points. Gradient is connected to the rate of change which gives differing gradient values by the speed of the change.
gradiant of a line is the rate of change in hte variable on the y axis divided by the rate of change in the variable on the x axis.
I understand that indeed the gradient of a line is the rate of change as the relationship between the two axes moves from one point to another.To some extent I can also agree with people who refer to the gradient as the slope of sometheing because in visualizing that concept, one can deduce that a slope of any kind does not involve the position of just one point.I see slope as elevation,ie.the distance "covered" from one point to another.That too is a rate of change.Problems in finding the gradients can be:
1.In a straight line, the line may not always pass through all the plotted points to connect them,so then obtaining the values to find the gradient is not so straightfoward.You have to examine the area in the graph well to get the accurate values on the 'x'and 'y' axes.
2.In a curve,there are no two points that are the same on that graph.You can't just take up two points from the curve and find the gradient. All points will have to be examined.If it is found that two points along the curve have close values,that may then lead to a start in attempting to find the gradient.
wel first of all wat dis gradient thing is about?? i using dis word all throughout my skool life an still doh noe wat it mean..i de bounce up dis word innah topic called cordinate geometry..an i use some kinda formula wat de techer give me..if i remember well i substituted x and y values into de formular and i obtained a result called the gradient...or hor and in physics it had a graph with velocity and time and i used values from de graph to obtain de gradient .this gradient was actually de accleration.....so to my understanding the gradient is used to show de relation ship of two things.plz correct me if im wrong!!!
the gradient of a curve is not fixed and changes which each new point on the curve, sometimes to find the gradient of a specific point on the curve, one must calculated the gradient of other points on that are very close that point and from that deduce its gradient
the gradient of a curve can be difficult to calculate since at every new point a change occurs, but the gradient of a line really implies to a rate of change of the height with respect to each new point following.that is it is a measure of its steepness.
the problem of finding the gradient of a straight line may be reading the graph wrong or chosing bad points!
problem with finding the gradiant of curve...a curve does not have a constant gradiant but many gradiants.. it may prove to be very tedious and time consuming in finding the gradiant at several points.
gradient is the rate of change of one variable with respect to another.
the gradient of a straight line can be used to find the an unknown value of x or y provided that value on the other axis is known
gradient is vertical rise over horizontal shift meaning it is just a value for steepness of a line or curve and i agree it could be the slope as well. in finding gradient of a straight line i think it may not always be accurate since sometimes the line might not pass through all the points like say a line of best fit has to be drawn and points are left out. please tell me if i'm wrong cause i'm not quite sure. for a curve it will be difficult to find the gradient because each point on the curve will have a different gradient however points very very close to each other will have a gradient value that is very close.
ratmam know what you mean i think you are right in a way since the relationship between two things could result in a value for rate of change such as accelleration as lilo explained about the velocity time graph. i also believe that gradient is connected to rate of change infact it is rate of change. like say you are driving up a steep hill when you are now about to go up the hill you will press on the gas a little but when you continue to go up and the hill gets steeper you will press the accelerator even more and harder for more power and the car to go, this is due to the change in steepness which is the rate of change or gradient of the hill.
j joker wat u mean by choosing bad points? u mean to say coordinates along the straight line?
yes j joker j!!! wat do u really mean by "choosing bad points" im confused. i tink the problem with finding the gradiant of a straight is that the line does not always pess through all the points. hence different lines of different gradiants would be obtained. True???
gradient of a line can be defined as the steepness . I learnt some thing new today. when x has a limit of delta H .
Class
I am very disappointed in the comments. I have summarized how many comments by user ids
I am to classify whether they are valid or invalid
Out of 70 students I have 26 users in this response. For this response the 44 others have until Friday.
The day I check a comment, I will not check that comment again.
moose 1
boo boo boy 1
subyzero 1
tweety 1
don cristobal 1
master_cookie 1
dogman 1
apocalypse 1
who 1
ichigo 1
j joker j 2
sleepy 1
dark angel 1
wong fei hong 1
xm2 2
lilo 4
supermuffinz 1
cruiser 2
poison 1
sparkle 1
cokebaby 3
sweediekinks 1
ratman 2
tinkerbell 1
tigger 1
kelz 2
In general a rate of change may be the change in anything divided by the corresponding change in a related variable. The "slope" of a graph of y vs. x for example is the change in y divided by the corresponding change in x. It is called the rate of change in y with respect to x.this is called the gradient.For each of the chosen y values there will be a corresponding value of x. Linear Rate of Change.It is harder to obtain two possible values of x; therefore x + delta x represents the x2 value or the neighbor of x.Not only does the rate of change depend on where along the x axis we look, it also depends on how we select the interval over which to measure it.
the problem with finding gradient in a curve is that d curve has no 2 points in a straight line there4 the gradient cant be found
ker vin i know that the gradient is the rate of change but it is not completely clear to me. i dont really follow wat u said in your last comment can any1 eni=lighten me.
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