Since we are learning about limits and continuity, I want to see how much you have understood these concepts. Please post your comments to answer the following questions:
1) In the 1st problem of the homework assignment of Sections 2.1 and 2.2, why are the average velocities and the instantaneous velocity at time t = 2s NEGATIVE?
2) Geometrically, what is "the line tangent to a curve" at a particular point?
3) What is the difference between the statements "f(a) = L" and "lim_{x->a} f(x) = L"?
4) Assume that f(1) = -5 and f(3) = 5. Does there have to be a value of x, between 1 and 3, such that f(x) = 0? (Think about the Intermediate Value Theorem).
5) Using the point-slope form, derive the equation of the secant line going through two points A(1,0) and B(2,1) and having the slope m = 1. Show me two equations:
- one derived from choosing A as the point in the point-slope form,
- the other derived from choosing B as the point in the point-slope form.
Are they different?
The deadline to submit your comments is a week from now (i.e., Thursday, 09/13). Each question weights 20 points. Late submission will be graded zero.
PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING.
Subscribe to:
Post Comments (Atom)
29 comments:
1) Because the tangent line at that point is decreasing.
2) It is the slope at that particular point.
3) "f(a)=L" means that when you plug in the value "a" into the function "f(x)" you get "L".
"lim{x->a}f(x)=L" means that as you approach "a" from both sides, you will get to "L". It does not mean where "f(a)" is defined.
4) Yes, as long as the function "f(x)" is continuous.
5) A: y-0 = 1(x-1)
B: y-1 = 1(x-2)
They look different, but when you simplify them, by solving for "y", they both turn out as "y = x-1".
-Jordan Montgomery
1. The ball is thrown up in the air and so as it travels up, the velocity is positive however after two sec., the ball is falling back to the ground making the velocity negative because it is moving towards its origin.
2. A straight line that touches the graph at one specific point and has the same slope as the graph at that point.
3. The first is simply where the function is defined at the value a. The latter tells what point the function approaches from the left at x value=a. This point may or may not be defined in the function.
4. Assuming that the function is continuous on the interval [1,3], yes there must be some value, c, where f(c)=0.
5. m=(1-0)/(2-1)=1
A) y-0=1(x-1)
B) y-1=1(x-2)
They look different but when simplified both of these problems are the same. Both of them come out to be y=x-1.
Jim Prescott
1) The Average Velocity and the Instantaneous Velocity are negative because the slope of the secant and tangent lines are negative. And these are ngative because as the ball travels up it loses velocity in the upwards direction and gains it in the downwards.
2) The tangent line is a line that touches a specific point on the curve such that the tangent line and the curve have the exact same slope that that point.
3) f(a) = L means that the function "f" is defined at the given point "a" however it does not mean that "L" is the limit of f(x) as x->a. Lim_{x->a} f(x)=L means that the funtion "f" approaches point "L" as x approaches "a." The function however is not neccessarily defined there.
4)As long as the function f is continuous there is a value "x" between 1 and 3 such that f(x)=0.
5) a) y-0=1(x-1)
b) y-1=1(x-2)
Both of the equations, when simplified, become y=x-1.
Peter Harris
1. The average velocities are negative because after 2 seconds the ball is falling back to the ground.
2.The tangent line to a curve is a straight line that touches a curve at one and only one point and that is going in the same direction as the curve.
3. f(a)=L means f is defined at a. lim{x>a}f(x)=L means as x gets closer to a f(x) gets closer to L from either side but x does not equal a.
4.We can not assume that there will be a value of x between 1 and 3, so that f(a)=0 because we do not know if the function is continuous.
5. Slope= rise/run
m= 1-0/2-1 = 1
A. y-0=1(x-1)= x-1
B. y-1=1(x-2)= x-1
They are not different.
Karen Sanders
1. The reason the average velocities are negative is because the ball which is being measured is dropping after it was thrown into the air.
2. " The line tangent to a curve" is where the tangent touches the curve at only one point and has the same slope. The slope on the tangent at P is the limit of the slopes of the secant line as well.
3. The statement f(a)=L means that if "a" is placed into the equation it would equal L. The other statement lim as {x->a}f(x)=L means that as from either direction of "a" it gets closer and closer to L but it does not actually equal L.
4. Yes because somewhere between [1,3] f(x)=0 as long as there are no discontinuities and the graph is continuous.
5. No, from using the point-slope form, point A and B both give the final equation y=x-1 when the slope =1
A(1,0)
y-yp=m(x-xp)
y-0=1(x-1) -> y=1x-1 -> y=x-1
B(2,1)
y-1=1(x-2) -> y=x-2+1 -> y=x-1
y-0=1
Stephanie Wallace
1. The velocities are negative because the ball has already reached its peak and is coming back down. The velocity as the ball goes up is positive while the velocity as it comes back down is negative. The tangent line after the ball has reached its peak has a negative slope.
2. The tangent line is a line that touches a graph at only one point and represents the slope which the graph is approaching from each side of the point.
3. "f(a)=L" means that f(a) is defined, whereas "lim_{x->a} f(x)=L" means that the limit exists at f(x).
4. Yes, but only if the function is continuous on 1 < x < 3.
5. slope = (1-0)/(2-1) = (1/1) = 1
At point A: y-0 = 1(x-1)
y = x-1
At point B: y-1 = 1(x-2)
y-1 = x-2
y = x-1
No; the secant line is the same through both points.
- Levon Hoomes
1. It is negative because at the time t=2s the ball is past it's max height and is then traveling
downwards.
2. It is a line that touches a the line at one particular point
3. f(a)=L is the point at which the function is defined at a. The equation lim_{x->a} f(x) = L just states where it gets closer and closer to L, but doesn't have to actually be defined at L.
4. Yes, only if the function is continuous.
5. y-0=1(x-1) and y-1=1(x-2)
y=x-1 y-1=x-2
y=x-1 y=x-1
same
Trevor Hubbard
1.)After 2 seconds the ball returns to the origin or begins to fall. Because it is heading back to the origin downward it is negative.
2.)The tangent line of a curve is a straight line heading in the same direction of the curve and touches it at only one particular point.
3.)f(a)=L means that when you have "a" in your equation it equals or could be replace with the value of L. lim_{x->a}f(x)=L means that coming from the right or left of point "a" you get close to but never actually get to the value of L.
4.)If the graph is continuous then yes. If its not then no.
5.)
slope(m)= rise/run=1-0/2-1 = 1
a(1,0) y-0=1(x-1)= x-1
b(2,1) y-1=1(x-2)= x-1
They are the same
1. The average velocities and instantaneous velocity at time t=2s is negative because the slopes of both the tangent and secant lines are negative. After the 2 seconds the ball is falling back to the ground after reaching its peak in the air which is also why the average and instantaneous velocity at t=2s is negative.
2. "The line tangent to a curve" is a line that passes through only one specific point on a curve, having slope m:y-y1=m(x-x1), at that particular point.
3. The difference between "f(a)=L" and "lim_{x->a}f(x)=L" is that "f(a)=L" means that the function, f,is defined at the point "a" but does not imply that L is the limit of f(x) as x->a.
"lim_{x->a}f(x)=L" means that the values of f(x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x does not equal a.
4. According to the Intermediate Value Theorem, f has to be continuous on [1,3] in order to say about the existence of x between 1 and 3, such that f(x)=0.
So if we don't know about the continuity of f, we cannot say anything about this value x.
5. (A) y-0=1(x-1)
(B) y-1=1(x-2)
The equations, both A and B, when simplified by solving for y, become
y=x-1.
Rachel Walpole
1. The velocities are negative at time t=2 because at by that point, the ball has begun to fall back down.
2. The tangent line of a particular point is a line that touches only that point and has the same slope curve does at that point.
3. “f(a) = L” means that the function is defined at point a. “lim_{x->a} f(x) = L means that as x gets closer to the point a, the values of f(x) get closer to L.
4. Yes, because f(x) has to be equal to 0 somewhere between 1 and 3, but only if the function is continuous.
5. A: y - 0 = 1(x-1)
y = x - 1
B: y - 1 = 1(x-2)
y = x - 2 + 1
y = x - 1
They end up the same, both being y = x - 1
Rachel Discavage
1. The velocities are negative because the ball is falling.
2. The line that touches a point on the curve and has the same slope.
3. F(A) = L implies that the function is defined whereas {x->a} F(X) = L states that as x approaches it gets closer and closer but never actually touches.
4. Yes, if the function is continuous.
5. y-0 = 1(x-1)
y-1 = 1(x-2)
they look different; however, once you simplify both they turn out to be the same.
Bishop Toups
1) f(a+h) < f(a)
The function f(t) is decreasing at t=2s.
2) Geometrically, "the line tangent to a curve" at a particular point is a line with the same slope as the function it is tangent to at the point it touches on the graph.
3) lim_{x->a}f(x)=L. IF f(a) is defined and f(x) is continuous at a, then lim_{x->a}f(x)=f(a)=L.
4) 1< x <3
f(1) < f(x) < f(3)
-5< f(x) <5
Yes, there must be an x value between 1 and 3 such that f(x)=0.
5) m=(1-0)/(2-1)=1/1=1
a) Y-0=1(X-1)
Y=X-1
X-Y-1=0
b) Y-1=1(X-2)
Y-1=X-2
X-Y+1-2=0
X-Y-1=0
Conclusion: No, they are not different, the two equations are the same.
Katie Moore
1. The velocities are negative because the ball is falling.
2. The tangent line is a straight line that touches curve at a specific point and has the same slope ass the curve at that point.
3. The first one means that when you plug "a" into f(x) it will be equal to L. The second one means that as you approach "a" from both sides you get to L, but the function is not necessarily defined.
4. If the function is continuous then there must be a value of "0" between f(1)= -5 and f(3)=5. However, if the function is not continuous than we cannot assume that there is a value of "0".
5. A. Y=X-1
B. Y=X-1
They are the same.
-William Wigglesworth
1. The velocity is negative because the ball has a negative acceleration due to gravity.
2. A tangent line is a line thats has the same slope as a single point it touches.
3. F(a)=L is defining the function at a as L. lim (x->a) f(x)=L is stating that the limit as x approaches a is L, but does not prove it is defined.
4. Assuming that the function f(1)= -5 and f(3)=5 are continuous, than f(x) at zero does exist.
5.
(a): y - 0 = 1(x-1)
y = x - 1
(B): y - 1 = 1(x-2)
y = x - 2 + 1
y = x - 1
BY: Michelle Sokoll
1) Because after two seconds the ball is falling back towards the ground which gives the negative velocities. This falling gives a negative tangent slope and negative secant slope thus negative instantaneous and average velocities.
2) Geometrically, "the line tangent to a curve" is a line that has the same slope as the function in which it touches at a point on that curve.
3) f(a)=L, is stating that at a the function is L, whereas lim {x->a}f(x)=L means that as x is approaching a from either the left or right it could equal L but is not necessary existant.
4) If the function is continous than there has to be a value in which f(x)=0, otherwise it would be discontinous between 1 and 3.
5) y-y1=m(x-x1)
a) y-0= 1(x-1)
y= x-1
b) y-1= 1(x-2)
y=x-1
They are the same because it is the same line.
-Allison Moore
1)Because the ball is on its way down, making the velocity negative.
2)The "line tangent to a curve" is a line that touches the curve only once and represents the slope of the tangent line to the graph.
3) f(a) is defined at L. Lim f(x)=L(as x->a) means that as x approaches a from both sides you approach "L", but may not always be defined.
4)Because if f is continuous there is a value "x" between 1 and 3 suc that f(x)=0
5)slope=1
a)y-0=1(x-1)
b)y-1=1(x-2)
When simplified they both become y=x-1
1. When the ball is first thrown into the air it is a positive velocity, than when it begins to free fall (after 2 seconds) it becomes a negative velocity.
2. The line tangent to a curve at a particular point is a line that touches a point on the curve and has the same slope.
3. f(a) = L is stating that the point at a is L. lim{x->a}f(x)=L says that the limit as x approaches a is L, from the left and the right, but we don't know if they equal so it might not be defined.
4. If it is continuous between those points yes, other wise it would be discontinuous.
5. y - y1 = m (x - x1)
(a): y - 0 = 1 (x - 1)
y = x - 1
(B): y - 1 = 1 (x - 2)
y = x - 2 + 1
y = x - 1
they both equal y = x - 1
1. The average and instantaneous velocities are negative because the ball is falling, meaning the slopes need to be negative to reflect this action.
2. The line tangent to a curve refers to the slope of a line that is equivalent to the slope of the point it intersects on the curve.
3.f(a) = L, refers to in a function the point a exists at a point L on the curve. while lim_{x->a} f(x) = L means as x approaches a point a a function x equals a point L.
4.Based on the Intermediate Value Theorem the function is continuous meaning the interval between 1 and 3, where 1 and 3 are included, there needs to be and exact point in the function that equals zero between 1 and 3.
5.
A:y=x-1
B:y=x-1
No, both equations are the same meaning at a given moment in the curve it will equal another curve.
1) The average and instantaneous velocities are both negative at 2 seconds because at this point, the ball if falling back down to the ground, a negative direction. This means that the tangent line to the curve has a negative slope.
2) "The line tangent to the curve" is the line that touches the function at only one point, and has a slope equal to the slope at that point.
3) f(a)=L means that at the point x=a the function is defined as L, and the lim_{x->a} f(x)=L means that at x=a the function is approaching L from either one or both sides, but it is not necessarily defined.
4) If the function is continuous between 1 and 3, than yes there must be a value of x where f(x)=0. If the function is discontinuous there does not have to be an x value where f(x)=0, for example if there is a horizontal asymptote at y=0.
5)A - y-0=1x-1
y=x-1
B - y-1=1x-2
y=x-1
No, both equations are the same, y=x-1.
Barbra Giourgas
1. The reason it is negative is because it is being measured at after 2 seconds which is when the ball drops.
2. The line tangent to a curve is a line that touches a curve only once with equal slope.
3. f(a) = L is defined, however, the limit function is f(x)=L as x approaches a from both sides. it may be undefined.
4. Yes b/c according to the Intermediate value theorem there will be a number between -x1 and x2 that equals 0.
5. A) (y-0)=(x-1)
y=x-1
B) (y-1)=(x-2)
y=x-1
No they are the same
Jason Silvestre
A. The reason they are negative is because the slope of the tangent line is decreasing; and the ball is being dropped.
B. "The line tangent to a curve" means, that a line touches the function locally at one point
C. The statement "f(a) = L" means that a limit exists at that point, it is defined. The statement "lim_{x->a} f(x) = L, means as x approaches from the left or the right and it may not be defined in both cases (coming form the left, coming from the right)
D. Yes, the Intermediate Value Theorem states that since the function moves from the negative range to the positive range such a number f(x)= 0.
E. y - y1 = m (x - x1)
a. y-0= (x-1)---> y=x-1
b. y-1= (x-2) ---> y=x-1
no they are not different.
1) the velocity was negative because the ball was being dropped
2)this means is a line the touches the graph and only one point and has the same slope as the graph at that point
3)f(A)=L the limit exist at the point f(a) and is equal to f(a) and therefore continuous at that point and lim_{x->3}f(x)=L means the limit exist
4)yes b/c it goes from negative to positive
5)y=1(x-1)->y=x-1
y-1=1(x-2)->y=x-1
both the same
raquel cain
1. The velocities at t = 2, are negative values in this case because at this point the ball has begun its decent, and therefore creates a negative tangent line with respect to the x axis
2. The tangent line of the graph at any given point represents the slope of the line at that point, because if zoomed in far enough the tangent line and the point are indistinguishable at this point.
3. The statement "f(a) = L" is the same as saying that given a point 'a' in this function f, the point L exist, thus the point a is defined. where as the formula "lim_{x->a} f(x) = L" only represents that the function is approaching the point 'L' as x gets closer to a, not that it neccisarily equals to a
4. There will exist a point such that f(x) = 0 if the graph of the function is continuous, if it is not, the point may be within a jump, hole, ect.
5.
A:
y-0 = 1(x-1)
y = x-1
B:
y-1 = 1(x-2)
y-1 = x-2
y = x-1
No, they have the exact same graphs...
1. the values are negative for the problem because the ball is now losing speed and going down. Anything that is in the air must eventually come down, and the ball coming down makes these values negative.
2. the tangent line is simply the slope of that line at the given point. Due to the fact that the tangent line and the line it merges into are indistinguishable, they have the same slope.
3. f(a)=L means that the function (a) is defined at that point, where as the equation lim_{x->a} f(x) = L" only says that there is a limit approaching the point and not neccessarily defined at that given point.
4. yes, because the function of f(1) is negative and the function of f(3) is positive, thus according to the intermediate value theorem, there has to be a value between 1 thru 3 where f(x) is equal to 0
5. y - y1 = m (x - x1)
y - 0 = 1 (x - 1)
y = x - 1
y - 1 = 1 (x - 2)
y = x - 2 + 1
y = x - 1
As you can see they both come out to the same equation.
1. The velocities are negative because after the ball is thrown it reaches a point where it must start to decline in velocity as it declines in altitude. This is why the values are negative.
2.The tangent line at a particular point represents the slope of the line and the curve it attaches to at the certain point.
3. "F(a)=L" means that the function of "a" is defined at "L". On the other hand the "lim {x->a} F(x)=L" means that the function has a limit as it approaches "a" that limit is "L", but it does not tell whether or not the point is defined at "L" for that certain function.
4. Yes, according to the intermediate value theorem, since we have part of the function at a negative point (F(1)=-5) and the other part of the function at a positive point (f(3)=5), then there has to be a point that lies in between these values. Hence, F(x)=0.
5.Point A:
y-y1=m(x-x1)
y-0=1(x-1)
y=x-1
Point B:
y-y2=m(x-x2)
y-1=1(x-2)
y=x-1
Hence, the secant line has the same slope through both points.
David Schwartz
1. Velocity accounts for displacement and not distance and thus direction is of major importance, since the upwards y-axis was chosen as the positive direction, the opposite will logically become the negative direction and thus the values after t=2s become negative because the ball is falling towards the ground; opposite to the chosen y-axis upward positive values.
2. The line tangent to a curve touches a curve at only that point on that curve such that the slope of the line measures a particular value at that instance point.
3. f(a) = L states that the value of the function f at as is equal (and thus defined at L while the limit of said function states that the function approached L at the point a but does not necessarily have to be defined at a.
4.If the function f is continuous on the interval [1, 3] then there must be a value between f(1) and f(3) that equals zero.
5. Point A: y = x – 1
Point B: y -1 = x- 2 which when simplified equals y = x – 1
They are not different because you are finding the equation for the line and not for the tangent lines to those points.
1) Because the object is falling downward therefore giving it a negative velocity or slope.
2) The line tangent at a point is the change in x values over the change in y values.
3)F(a)= L is saying that the object is actually that point where as the limit one is saying that it comes very close to the point but never touches it.
4) Only if the function is defined and continuous between those points because according to the IVT. Therefore, the function must be continuous between the points of [1,3] for there to exist a value such that, f(x)=0.
5) For A y-0 = 1(x-1), which when simplified is y=x-1.
For B y-1 = 1(x-2), which also when simplified is y=x-1
-Jasper W. Yonker
1. The average velocity is negative at t=2s because the ball is falling back towards the ground.
2. A tangent line is a straight line that touches a graph at one point and has the same slope as the graph.
3. f(a)=L means that f(x)=L when x=a. lim{x->a} f(x)=L means that as x approaches a from both sides, f(x) moves closer to L.
4. If the functions are to be continuous there must be a value of x, between 1 and 3, such that f(x)=0. If there is not, the functions will be discontinuous.
5. m= 1-0/2-1 = 1
y-0=1(x-1)= x-1
y-1=1(x-2)= x-1
The equations are not different because they both become y=x-1
Michael Lyons
1. The average velocity in negative because the ball is moving back twoards the ground.
2. A tangent line is a straight line that touches the curve at on point, and has the same slope.
3. This means that f(a)=L means the function is defined at a,
The lim just means that as x approaces a, the function has a limit of a but may not be defined at a.
4. yes if the function is continous.
5. a. y-0=m(x-1)
b. y-1=m(x-2)
y = x - 1
They are the same when m = 1.
Post a Comment