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so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | so by side-side-side congruency , we now know -- and we want to be careful to get our corresponding sides right -- we now know that triangle cde is congruent to triangle dbf . i want to get the corresponding sides . i 'm looking at the colors . | why would you want to create medial triangles anyway ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | now let 's think about this triangle up here . we could call it bdf . so first of all , if we compare triangle bdf to the larger triangle , they both share this angle right over here , angle abc . | could n't you just keep drawing out triangles over and over again like the koch snowflake ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | cd over cb is 1/2 , ce over ca is 1/2 , and the angle in between is congruent . so by sas similarity , we know that triangle cde is similar to triangle cba . and just from that , you can get some interesting results . | what is sas similarity and what does it stand for ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | so let 's go about proving it . so first , let 's focus on this triangle down here , triangle cde . and it looks similar to the larger triangle , to triangle cba . but let 's prove it to ourselves . | it looks like the triangle is an equilateral triangle , so it makes 4 smaller equilateral triangles , but can you do the same to isoclines triangles ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | so now let 's go to this third triangle . i think you see the pattern . i 'm sure you might be able to just pause this video and prove it for yourself . | do medial triangles count as fractals because you can always continue the pattern ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | cd over cb is 1/2 , ce over ca is 1/2 , and the angle in between is congruent . so by sas similarity , we know that triangle cde is similar to triangle cba . and just from that , you can get some interesting results . | sal says sas similarity , but is n't it supposed to be sas `` congruency '' ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | so we 'd have that yellow angle right over here . and this triangle right over here was also similar to the larger triangle . so it will have that same angle measure up here . | also , if you were only given the medial triangle , how would one figure out the points of the whole triangle ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | and so that 's pretty cool . we just showed that all three , that this triangle , this triangle , this triangle , and that triangle are congruent . and also , we can look at the corresponding -- and that they all have ratios relative to -- they 're all similar to the larger triangle , to triangle abc . | is the centroid of a triangle always the the circumcenter ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | and so that 's pretty cool . we just showed that all three , that this triangle , this triangle , this triangle , and that triangle are congruent . and also , we can look at the corresponding -- and that they all have ratios relative to -- they 're all similar to the larger triangle , to triangle abc . | if so , and the centroid of the main triangle is congruent to the centroid of the medial triangle , would that make the incircle of the main triangle and the circumcircle of the medial triangle identical ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | so we know that this length right over here is going to be the same as fa or fb . and we get that straight from similar triangles . because these are similar , we know that de over ba has got to be equal to these ratios , the other corresponding sides , which is equal to 1/2 . | can their be multiple medial triangles in one problem ? |
so i 've got an arbitrary triangle here . we 'll call it triangle abc . and what i want to do is look at the midpoints of each of the sides of abc . so this is the midpoint of one of the sides , of side bc . let 's call that point d. let 's call this midpoint e. and let 's call this midpoint right over here f. and sinc... | so if you connect three non-linear points like this , you will get another triangle . and this triangle that 's formed from the midpoints of the sides of this larger triangle -- we call this a medial triangle . and that 's all nice and cute by itself . | wait , so the medial triangle is the one in the middle ; the one formed by connecting the midpoints of the sides of the triangle ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | instead , we 're multiplying or dividing by the same amount each time . in this case , we 're multiplying by 1.4 , by 1.4 each time . so we are clearly geometric . | how did 40 % calculate in to 1.4 ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | so , when n is equal to one , when n is equal to one , g of n is going to be , or g of one is going to be the number of party favors seo-yun had before the first guest . well , before the first guest , she had 50 party favors . she had 50 party favors . | in the 2nd question why would the first guest have 50 favors ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | so because the difference between successive terms is the same , we know this is an arithmetic sequence . this is an arithmetic sequence and then they say write an explicit formula for the sequence . so let 's think about this . | what advantages does recursive notation have over explicit notation ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | so , she would have 44 , and i think you see the pattern . for every time n , when n equals one , g of n is 50 , and every time we increase n by one , every time we increment n , we are increasing g of n by plus three , by minus three , i should say because she 's giving away party favors , minus three . minus three . | how come sal writes 50-3 ( n-1 ) instead of 50- ( 3 ) ^n-1 like we have been learning for geometric sequences ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | then in year three , we 're going to grow by 40 % of 700 , which is 280 , so it 's going to grow to 980 . notice it 's definitely not an arithmetic sequence . an arithmetic sequence , we would be adding or subtracting the same amount every time , but we 're not . | one small thing : would the definition of the following sequence be arithmetic or geometric ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | in this case , we 're multiplying by 1.4 , by 1.4 each time . so we are clearly geometric . depending on your answer to the question above , the recursive definition of the sequence can have one of the following two forms . | the question : geometric or explicit ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | so , table . so , this is n and this is f of n. so when n is equal to one , the first year , n equals one , there were 500 leaves . f of n is 500 . | when should you put n-1 instead of n in your explicit formula ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | so because the difference between successive terms is the same , we know this is an arithmetic sequence . this is an arithmetic sequence and then they say write an explicit formula for the sequence . so let 's think about this . | can we apply geometry sequence into finance to calculate compound interest/percent ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | so , table . so , this is n and this is f of n. so when n is equal to one , the first year , n equals one , there were 500 leaves . f of n is 500 . | how do i know if ( n-1 ) should be an exponent or not ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | then in year three , we 're going to grow by 40 % of 700 , which is 280 , so it 's going to grow to 980 . notice it 's definitely not an arithmetic sequence . an arithmetic sequence , we would be adding or subtracting the same amount every time , but we 're not . | could someone please clarify the difference between an arithmetic and geometric sequence ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . | if the first term in an arithmetic series is 3 , the last term is 136 , and the sum is 1390 , how do i get the first 3 terms ? |
mohamed decides to track the number of leaves on the tree in his backyard each year . the first year , there were 500 leaves . each year thereafter , the number of leaves was 40 % more than the year before . let n be a positive integer , and let f of n denote the number of leaves on the tree in mohamed 's back yard in... | minus three . so because the difference between successive terms is the same , we know this is an arithmetic sequence . this is an arithmetic sequence and then they say write an explicit formula for the sequence . so let 's think about this . | so , in simple terms , do you add/subtract in an arithmetic sequence ; and multiply in a geometric sequence ? |
so systolic heart failure , your heart ca n't pump as hard as it used to and so it does n't squeeze as much blood out and it does n't meet the body 's demands . it 's called systolic heart failure because we 're talking about systole , which is the phase of the cardiac cycle where the heart contracts and ejects blood ... | so if you just look at this heart and compare it to a healthy heart , you 'll see that these walls are way thinner and these ventricles are way bigger . these are classic signs of a heart with systolic heart failure . so what are some of these underlying diseases ? | is the ejection fraction reduced only for systolic heart failure , and not diastolic heart failure ? |
so systolic heart failure , your heart ca n't pump as hard as it used to and so it does n't squeeze as much blood out and it does n't meet the body 's demands . it 's called systolic heart failure because we 're talking about systole , which is the phase of the cardiac cycle where the heart contracts and ejects blood ... | the second category is n't as straight forward . it 's going to be reduced blood supply , but i 'm not talking about the blood supply to the body , i 'm actually talking about to the heart itself . and so coronary artery disease is a huge cause of systolic heart failure because the coronaries are what supply your heart... | would reduced blood supply to the body from plaques in arteries or veins also cause heart failure even if the coronary arteries are n't affected by lowering the amount of blood that goes to the heart which then lowers the amount of blood that goes to the lungs ? |
- [ sal ] which of the differential equations are separable ? and i encourage you to pause this video and see which of these are actually separable . now , the way that i approach this is i try to solve for the derivative , and if when i solve for the derivative , if i get dy , dx is equal to some function of y times s... | if i multiply both sides by dx and divide both sides by this right over here , i would get one over y squared plus y dy is equal to x squared plus x dx . so clearly separable . alright now this last choice , this is interesting , they 've essentially distributed the derivative right over here . | if the criteria for an equation to be separable is to isolate x and y as stated in previous videos , why does it have to be a product ? |
- [ sal ] which of the differential equations are separable ? and i encourage you to pause this video and see which of these are actually separable . now , the way that i approach this is i try to solve for the derivative , and if when i solve for the derivative , if i get dy , dx is equal to some function of y times s... | so i 'm gon na factor it out . i 'm gon na get dy dx times x plus y , x plus y , is equal to x . now if i were to divide both sides by x plus y , i 'm gon na get dy dx is equal to x over x plus y . | in other words , why couldnt we manipulate # 2 to be dy + y = dx - x + ( 1/2 ) and that be separable ? |
- [ sal ] which of the differential equations are separable ? and i encourage you to pause this video and see which of these are actually separable . now , the way that i approach this is i try to solve for the derivative , and if when i solve for the derivative , if i get dy , dx is equal to some function of y times s... | so i 'm gon na factor it out . i 'm gon na get dy dx times x plus y , x plus y , is equal to x . now if i were to divide both sides by x plus y , i 'm gon na get dy dx is equal to x over x plus y . | can anyone help me figure out how to solve the differential equation ( 2+2y^2 ) y'=e^x ( y ) in terms of x ? |
- [ sal ] which of the differential equations are separable ? and i encourage you to pause this video and see which of these are actually separable . now , the way that i approach this is i try to solve for the derivative , and if when i solve for the derivative , if i get dy , dx is equal to some function of y times s... | if i multiply both sides by dx and divide both sides by this right over here , i would get one over y squared plus y dy is equal to x squared plus x dx . so clearly separable . alright now this last choice , this is interesting , they 've essentially distributed the derivative right over here . | how can be solved an equation that is not separable ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | in this case , the limit is n't . i think now you 're starting to see why the limit is a slightly different concept than just evaluating the function at that point because you have functions where , for whatever reason at a certain point , either the function might not be defined or the function kind of jumps up or dow... | does the hole represent discontinuity in the function ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | what is the purpose of limits ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | i should -- it does n't look completely right below it , but i think you got to get the picture . see , this graph is x squared . it 's exactly x squared until we get to x equals 2 . | can a function have a circular graph ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | i should -- it does n't look completely right below it , but i think you got to get the picture . see , this graph is x squared . it 's exactly x squared until we get to x equals 2 . | what happens when you graph x^3 ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | let me draw . so now it 's almost the same as this curve , except something interesting happens at x equals 2 . so it 's just like this . | what happens when 0 is multiplied by a undefined number ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | how is it useful in future studies ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | what are some applications of limits ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | is there a formulaic definition for limits ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | i think as you do more and more problems , you 'll get more and more of an intuition as to what a limit is . and then as we go into drill derivatives and integrals , you 'll actually understand why people probably even invented limits to begin with . we 'll see you in the next presentation . | who invented calculus and when ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | but the way you write it is you say the limit -- oh , my color is on the wrong -- ok , let me use the pen and yellow . ok , the limit as x approaches 2 of x squared . now , all this is saying is what value does the expression x squared approach as x approaches 2 ? | if you have lim x- > 0 of 1/x , does it become infinity or negative infinity ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | see , this graph is x squared . it 's exactly x squared until we get to x equals 2 . at x equals 2 , we have a grap -- no , not a grap . | so ... what exactly is calculus and its applications in real life ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | what exactly is the definition of a limit ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | let me pick a different color . so as x approaches 2 from this side , from the left-hand side or from numbers less than 2 , f of x is approaching values approaching 4 , right ? f of x is approaching 4 as x approaches 2 , right ? | do we need to consider the left and right hand side behavior of a function to determine its limit ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | but the way you write it is you say the limit -- oh , my color is on the wrong -- ok , let me use the pen and yellow . ok , the limit as x approaches 2 of x squared . now , all this is saying is what value does the expression x squared approach as x approaches 2 ? | does it make sense to take the limit of a function when approaching a y value instead of an x value ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | the expression is equal to 4 . the way i think about it is as you move on the curve closer and closer to the expression 's value , what does the expression equal ? in this case , it equals 4 . | `` as you move closer and closer to the expressions value , what does the expression equal ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | and when x is equal to 2 -- let 's say this is 3 . when x is equal to 2 , f of x is equal to 3 . this is actually right below this . | if what is an interval and a point for x-2 over x-5 , where the average rate of change and the instantaneous rate of change are equal ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so let me draw . x squared looks something like -- let me use a different color . x square looks something like this , right ? and when x is equal to 2 , y , or the expression -- because we do n't say what this is equal to . | what a graph of your statement looks like ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | what is the difference between limits and piece-wise functions ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | so does that mean that technically all graphs have limits at all possible points , in other words all being evidence of zeno 's paradox ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | would a finite limit be considered a regular limit ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | what is -- my pen still works -- what is the limit -- i used cursive this time -- what is the limit as x -- that 's an x -- as x approaches 2 of f of x ? that 's an x . it says x approaches 2 . it 's just like that . | why is the graph of x^2 not have a value at x=2 ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | what is the correlation between limits and derivatives ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | what is the easiest way to calculate the limit ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | what is the benifit in solving the limit ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | and when x is equal to 2 -- let 's say this is 3 . when x is equal to 2 , f of x is equal to 3 . this is actually right below this . | why do you write a function or an equation with equal sign ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so as x approaches 2 from this side , from the left-hand side or from numbers less than 2 , f of x is approaching values approaching 4 , right ? f of x is approaching 4 as x approaches 2 , right ? i think you see that . | lim x^2=4 x- > 2 a x^2 will never be 4 , because x is approaching 2 , but it will never be 2 , so why it is not written as : lim x^2 - > 4 x- > 2 any intuition on writing it with equals sign , and are there cases when limit of something will also be approaching ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | but the way you write it is you say the limit -- oh , my color is on the wrong -- ok , let me use the pen and yellow . ok , the limit as x approaches 2 of x squared . now , all this is saying is what value does the expression x squared approach as x approaches 2 ? | how come when you are asked to find a limit for a linear function you just plug in the value as x approaches to find the limit , but for a quadratic or rational function its much harder ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | but the way you write it is you say the limit -- oh , my color is on the wrong -- ok , let me use the pen and yellow . ok , the limit as x approaches 2 of x squared . now , all this is saying is what value does the expression x squared approach as x approaches 2 ? well , this is pretty easy . | let say you are asked to find the limit of a piece-wise function where x approaches a y value that is undefined but defined at a different y value how come the limit is n't that other y value but at the y value where it is undefined ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so it 's just like this . it 's like an x squared curve like that . but at x equals 2 and f of x equals 4 , we draw a little hole . we draw a hole because it 's not defined at x equals 2 . | could a function like f ( x ) have more than one gap ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | if you just follow along the curve , as you approach f of 2 , you get closer and closer to 4 . similarly , as you go from the right-hand side -- make sure my thing 's still working . as you go from the right-hand side , you go along the curve , and f of x is also slowly approaching 4 . | however i 'm still stuck on one thing , what happens when 0 is multiplied by an undefined number ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | in this case , the limit as you approach the expression is equal to evaluating the expression of that value . in this case , the limit is n't . i think now you 're starting to see why the limit is a slightly different concept than just evaluating the function at that point because you have functions where , for whateve... | how is limit different from a function ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so that 's an equally neat-looking graph as the one i just drew . let me draw . so now it 's almost the same as this curve , except something interesting happens at x equals 2 . | how to draw graph , how you know it is curve ... i dont understand this thing ... .. can you please explain ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | but the way you write it is you say the limit -- oh , my color is on the wrong -- ok , let me use the pen and yellow . ok , the limit as x approaches 2 of x squared . now , all this is saying is what value does the expression x squared approach as x approaches 2 ? | could someone help explain what is x squared ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | let me pick a different color . so as x approaches 2 from this side , from the left-hand side or from numbers less than 2 , f of x is approaching values approaching 4 , right ? f of x is approaching 4 as x approaches 2 , right ? | why is the second bracket given only in one side ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | how do you find limits numerically ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | where does the limits have its application in our surroundings ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | hi , why the limit still exists when there is an undefined area on the graph ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | do one-sided limits count as a real limit or is it just a concept that is really never applied ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | what is a practical application of limits ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | so if a limit is reached what happens ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so let me ask you a question . what is -- my pen still works -- what is the limit -- i used cursive this time -- what is the limit as x -- that 's an x -- as x approaches 2 of f of x ? that 's an x . | f ( x ) is undefined at valve limit exists , is it always true ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | in this case , the limit is n't . i think now you 're starting to see why the limit is a slightly different concept than just evaluating the function at that point because you have functions where , for whatever reason at a certain point , either the function might not be defined or the function kind of jumps up or dow... | so when the graph is approaching 3 , the value of the function has two different values ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | that means is 1/0 is undefined or simply infinity ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | but the way you write it is you say the limit -- oh , my color is on the wrong -- ok , let me use the pen and yellow . ok , the limit as x approaches 2 of x squared . now , all this is saying is what value does the expression x squared approach as x approaches 2 ? | why would n't the f ( x ) limit be 2 because as the dot on the line `` the open circle dot '' was thought to be non-existent thusly meaning that the filled in dot at 3 would make the lim x- > 2 = 3 ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | in this case , the limit as you approach the expression is equal to evaluating the expression of that value . in this case , the limit is n't . i think now you 're starting to see why the limit is a slightly different concept than just evaluating the function at that point because you have functions where , for whateve... | what is the difference between value of a function at a point and the limit at a point ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . | the first principle of limits video is n't ter ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | at x equals 2 , we have a grap -- no , not a grap . we have a gap in the graph , which maybe could be called a grap . we have a gap in the graph , and then we keep -- and then after x equals 2 , we keep moving on . | could someone show me what proper notation is ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | how do i type the limit notation ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | how would you resolve an absolute value while computing limits ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | in what sort of situation would limits be used in the real world ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | can a function have more than one limit ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | well , it essentially equals 4 , right ? the expression is equal to 4 . the way i think about it is as you move on the curve closer and closer to the expression 's value , what does the expression equal ? | how do you know what type of graph and expression forms , given the function ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | in this case , the limit as you approach the expression is equal to evaluating the expression of that value . in this case , the limit is n't . i think now you 're starting to see why the limit is a slightly different concept than just evaluating the function at that point because you have functions where , for whateve... | or a limit of a function does not inherit the continuous or discontinuous property of a function ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | what is the use of limit in our daily life ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so this graph kind of goes -- it 's just like x squared , but instead of f of 2 being 4 , f of 2 drops down to 3 , but then we keep on going . so going back to the limit problem , what is the limit as x approaches 2 ? now , well , let 's think about the same thing . | what exactly is the definition of a limit ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | as you go from the right-hand side , you go along the curve , and f of x is also slowly approaching 4 . so , as you can see , as we go closer and closer and closer to x equals 2 , f of whatever number that is approaches 4 , right ? so , in this case , the limit as x approaches 2 is also equal to 4 . | is infinity a number with digits ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | i think this will give you intuition for what a limit is . in another presentation , i 'll give you the more formal mathematical , you know , the delta-epsilon definition of a limit . and actually , in the very next module , i 'm now going to do a bunch of problems involving the limit . | what the is delta symbol ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so let me draw . x squared looks something like -- let me use a different color . x square looks something like this , right ? | what can you use the delta symbol for ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . | how do you tinker with a code ? |
welcome to the presentation on limits . let 's get started with some -- well , first an explanation before i do any problems . so let 's say i had -- let me make sure i have the right color and my pen works . ok , let 's say i had the limit , and i 'll explain what a limit is in a second . but the way you write it is y... | so as x approaches 2 from this side , from the left-hand side or from numbers less than 2 , f of x is approaching values approaching 4 , right ? f of x is approaching 4 as x approaches 2 , right ? i think you see that . | f ( x ) = { x^2 , x < 2 : x^3 , x > 2 } , the limit approaching from less than 2 would be 4 , but approaching from greater than 2 would be 8 , but how do you express that in notation ? |
we have three plus blank is equal to 10 . so pause the video and try to figure out what the blank is . what do i have to add to three to get to 10 ? so you 've had a go at it . now let 's see if we can do this together . so over here , i have three of these , i guess you could say purple , these purple pinkish circles... | we have three plus blank is equal to 10 . so pause the video and try to figure out what the blank is . | is multiplying negative numbers possible ? |
we have three plus blank is equal to 10 . so pause the video and try to figure out what the blank is . what do i have to add to three to get to 10 ? so you 've had a go at it . now let 's see if we can do this together . so over here , i have three of these , i guess you could say purple , these purple pinkish circles... | we have three plus blank is equal to 10 . so pause the video and try to figure out what the blank is . | is there such thing as negative zero ? |
we have three plus blank is equal to 10 . so pause the video and try to figure out what the blank is . what do i have to add to three to get to 10 ? so you 've had a go at it . now let 's see if we can do this together . so over here , i have three of these , i guess you could say purple , these purple pinkish circles... | well , three plus one , two , three , four , five , six , seven . is equal to 10 . let 's do one more of these . | how many ways can i make 10 without using fractions ? |
the world islam can best be translated into english as meaning surrender , and the context of the islamic faith is referring to a surrender to the will of god . now , a muslim is someone who practices islam , one who submits to the will of god , and the central text in islam is the quran , which muslims believe is the... | they believe him to be a very significant prophet , the prophet before muhammad came to reveal to the quran . now , for a practicing muslim , there are often considered to be five pillars , and this is especially the case for the majority of muslims , for sunni muslims . shia muslims have a slightly different combinati... | are we to accept that sunni has a different set of pillars than all other muslims ? |
the world islam can best be translated into english as meaning surrender , and the context of the islamic faith is referring to a surrender to the will of god . now , a muslim is someone who practices islam , one who submits to the will of god , and the central text in islam is the quran , which muslims believe is the... | so once again , a very clear message , in at least the islamic tradition , that this is the same faith or tradition as that of moses . and now here 's reference to jesus . `` then we caused our messengers to follow in their footsteps `` and we caused jesus , son of mary , `` to follow and gave him the gospel `` and pla... | wait , wasnt jesus born in the year 0. why is it saying he was born in 4bc ? |
the world islam can best be translated into english as meaning surrender , and the context of the islamic faith is referring to a surrender to the will of god . now , a muslim is someone who practices islam , one who submits to the will of god , and the central text in islam is the quran , which muslims believe is the... | `` and who is better in religion than `` one who submits himself to god while being a doer of good `` and follows the religion of abraham , `` inclining toward truth ? `` and god took abraham as a friend . '' and abraham in particular plays a very central role . | how would abraham have a `` connection '' with god ? |
the world islam can best be translated into english as meaning surrender , and the context of the islamic faith is referring to a surrender to the will of god . now , a muslim is someone who practices islam , one who submits to the will of god , and the central text in islam is the quran , which muslims believe is the... | `` and this is a confirming book in an arabic tongue `` to warn those who have wronged `` and as good tidings to the doers of good . '' so once again , a very clear message , in at least the islamic tradition , that this is the same faith or tradition as that of moses . and now here 's reference to jesus . | we 're the byzantine , western european , and islamic empires relatively equal in power a 1000 years ago ? |
the world islam can best be translated into english as meaning surrender , and the context of the islamic faith is referring to a surrender to the will of god . now , a muslim is someone who practices islam , one who submits to the will of god , and the central text in islam is the quran , which muslims believe is the... | now , muslims are very sensitive to this , because they do n't view muhammad as a divine figure the way that christians view christ . they view muhammad as a human , a human whose practices and whose life they view , they revere , but they do n't view him as a divine figure . they view him as the messenger who revealed... | how do they ( beliefs , religions ) hold in the light of reason and understanding of current human knowledge ? |
the world islam can best be translated into english as meaning surrender , and the context of the islamic faith is referring to a surrender to the will of god . now , a muslim is someone who practices islam , one who submits to the will of god , and the central text in islam is the quran , which muslims believe is the... | `` then we caused our messengers to follow in their footsteps `` and we caused jesus , son of mary , `` to follow and gave him the gospel `` and placed compassion and mercy `` in the hearts of those who followed him . '' the key difference between muslims and christians in terms of the life of jesus , is that muslims d... | why muslims believe in allah ? |
( gentle piano music ) we 're looking at the bas-reliefs of the arch of titus , the most famous of which show the spoils of jerusalem being brought into rome in the great triumphal parade honoring the general , soon to be , emperor , titus , at his great victory at destroying jerusalem . a triumphal arch is something... | but there were things in the temple . there were holy objects , and that 's what we see here , being carried into rome as spoils , in the arch of titus . so we have the menorah , a very important symbol in jewish history , especially in the roman period , but we see other holy objects that were in the temple , like th... | did the arch of titus inspire the arc de triumph ? |
( gentle piano music ) we 're looking at the bas-reliefs of the arch of titus , the most famous of which show the spoils of jerusalem being brought into rome in the great triumphal parade honoring the general , soon to be , emperor , titus , at his great victory at destroying jerusalem . a triumphal arch is something... | ( gentle piano music ) we 're looking at the bas-reliefs of the arch of titus , the most famous of which show the spoils of jerusalem being brought into rome in the great triumphal parade honoring the general , soon to be , emperor , titus , at his great victory at destroying jerusalem . a triumphal arch is something... | what does the pink and green coloring signify in the map 8 ? |
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