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Example Problems for Limits at Infinity
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- Intro
- Example I: Calculating Limits as x Goes to Infinity
- Example II: Calculating Limits as x Goes to Infinity
- Example III: Calculating Limits as x Goes to Infinity
- Example IV: Calculating Limits as x Goes to Infinity
- Example V: Calculating Limits as x Goes to Infinity
- Example VI: Calculating Limits as x Goes to Infinity
- Intro 0:00
- Example I: Calculating Limits as x Goes to Infinity 0:14
- Example II: Calculating Limits as x Goes to Infinity 3:27
- Example III: Calculating Limits as x Goes to Infinity 8:11
- Example IV: Calculating Limits as x Goes to Infinity 14:20
- Example V: Calculating Limits as x Goes to Infinity 20:07
- Example VI: Calculating Limits as x Goes to Infinity 23:36
AP Calculus AB Online Prep Course
Transcription: Example Problems for Limits at Infinity
Hello, welcome back to www.educator.com, welcome back to AP Calculus.0000
Today, I thought we would do some more example problems for limits at infinity0004
or as x goes to positive or negative infinity.0009
Let us jump right on in.0011
Evaluate the following limit x² + 6x - 12/ x³ + x² + x + 4, as the limit as x goes to negative infinity.0015
Here we have a rational function.0029
We know how to deal with rational functions.0031
We pretty much just divide the top and bottom, by the highest power of x in the denominator.0035
The first thing you do is basically plug in.0048
You evaluate the limit to see, before you actually have to manipulate.0050
In this case, when we put in negative infinity into here and here, what we are going to end up with is,0054
I will say plugging in, we get infinity/ negative infinity.0062
x², this term is going to dominate, this term is going to dominate.0083
Negative infinity², negative number² is positive.0089
Negative number³ is going to be negative.0092
We are going to get something like this which does not make sense.0094
This is going to be the limit as x goes to negative infinity of x² + 6x - 12/ the greatest power in the denominator which is x³.0100
That is our manipulation x³ + x² + x + 4/ x³.0112
This gives us the limit as x approaches negative infinity of, here we have 1/ x + 6/ x² – 12/ x³ / 1 + 1/ x + 1/ x² + 4/ x³.0121
Now when we take the limit as x goes to negative infinity, this is 0, this is 0, this is 0.0152
All of these go to 0 and you are left with 0/1 which is an actual finite number 0.0159
Our limit is 0.0166
Again, it is always nice to confirm this with a graph.0169
This is our function, down here a little table of values.0177
Basically, what this part of the table of values does is it confirms the fact that we start negative,0181
the function crosses the 0.0186
And then actually comes up and gets closer and closer to 0 which is the limit.0191
As x gets really big, the function gets close to 0 which is what we just calculated analytically.0199
Evaluate the following limit.0209
The limit as x goes to infinity of x + 4/ 8x³ + 1.0211
A couple of things to notice.0218
Again, instead of just launching right in this, you want to stop and ask yourself some questions.0219
Here the radical is the denominator.0223
It is a rational function.0226
We have to think about these things.0227
8x³ + 1, it is under the radical sign.0229
It, itself has to be greater than 0.0233
Let us see what the conditions are here.0236
Here the 8x³ + 1 is in the denominator.0241
We know that that value cannot be 0.0251
It cannot equal 0.0257
Since we cannot have the square root of a negative number, 8x³ + 1 has to be greater than 0.0266
It could be greater than or equal to 0.0275
But then, if it were equal to 0 then you have a 0 in the denominator.0277
That is why we have only the relation greater than.0279
I will write not greater than or equal to, which normally we could.0283
Let us go ahead and work this out first.0288
8x³ + 1 is greater than 0.0290
We have 8x³ greater than -1.0295
We have x³ greater than -1.0298
That is greater than -1/8, which implies that x itself has to be greater than -1/2.0307
That is our domain.0317
We said not that relation, greater than -1/2.0320
Here we do not have to worry about going to negative infinity.0324
All we have to worry about here is going to positive infinity.0328
Because x cannot be, x cannot go to negative infinity.0331
We do not have to worry about x going to negative infinity.0339
We saved ourselves a little bit of work.0352
Our x + 4/ 8x³ + 1, it is a rational function.0356
We want to divide it by the greatest power of x in the denominator.0368
This is going to be x + 4, x³/ √x³.0379
This is going to be x + 4/ x³/2.0396
The √x³ is x³/2 / 8x³ + 1/ x³.0405
That is going to equal, this is going to be 1/ x ^½.0418
This is going to be + 4/ x³/2.0424
This is going to be √8 + 1/ x³.0429
Now we go ahead and take the limit.0440
The limit as x goes to infinity of 1/ x ^½ + 4/ x³/2 / √8 + 1/ x³.0445
As x goes to infinity, this goes to 0, this goes to 0, this goes to 0.0463
We end up with 0/ √8 which is a finite number.0468
Our limit is 0.0472
The graph that we get is this.0477
We see as x gets really big, the function gets closer and closer and closer to 0.0480
There we go, confirmed it graphically.0487
Evaluate the following limit.0492
The limit of √9x ⁺10 – x³/ x + 5 + 100.0494
The best way to handle this is, let us try this.0503
Again, my way is not the right way.0511
It is just one way, you might come up with, 5 different people might come up with 5 different ways of doing this limit.0513
That is totally fine, that is the beauty of this.0518
Let us do the following.0531
Let us take this 9x ⁺10 – x³.0533
As x goes to infinity, here only 9x ⁺10 term is going to dominate.0541
I’m just going to deal with that term.0557
The same thing for the denominator.0559
For the denominator, the x⁵ is the one that I'm going to take.0560
Basically, what the limit that I'm going to take is the limit as x approaches infinity of √9x ⁺10/ x⁵.0569
Essentially, I just said that this does not matter and this does not matter.0582
The limit is going to be essentially the same.0585
I just deal with those.0587
This is going to equal to the limits as x approaches infinity, √9x ⁺10 = absolute value of 3x⁵.0591
Remember, the square root of something is the absolute value of something/ x⁵.0608
For x greater than 0, in other words x going to positive infinity.0618
The absolute value of 3x⁵/ x⁵, when x is greater than 0, this is just 3x⁵/ x⁵ = 3.0625
The limit as x approaches positive infinity of 3 = 3.0640
Now for x less than 0, this 3x⁵ absolute value/ x⁵, it is actually going to be -3x⁵/ x⁵ = -3.0648
The limit as x goes to negative infinity of -3 = -3.0668
Here you are going to end up with two different asymptotes, 3 and -3.0678
Now notice the difference between the following.0687
We finished the problem but we are going to talk about something.0693
Notice the difference between following.0700
When I take √x ⁺10, I get the absolute value of x⁵.0708
The absolute value of x⁵ is either going to be x⁵ because when x is greater than 0 or it is going to be,0717
I actually did it in reverse, that is why I got a little confused here.0740
It is going to be –x⁵ and that is when x is less than 0.0742
When x is greater than 0, it is going to be x⁵.0751
The reason is because this is an odd power.0755
x⁵ itself, depending on whether x is positive or negative, this inside is going to be a positive or negative number.0768
If what is in here is positive, then it is going to go one way.0780
If what is in here is negative, it is going to go the other way.0785
However, if I take something like x⁸, this is going to give me,0788
we said that the square root of a thing is going to be absolute value of x⁴.0798
Here it does not matter.0804
If x is positive or negative, it is an even power.0806
An even power is always going to be positive number.0812
Therefore, this is just going to be x⁴ because it is an even power.0816
Be careful of that, you have to watch the powers.0823
When you pass from the square root of something, we said the square root of something is the absolute of something.0825
But the power itself is going to make a difference on whether you separate or whether you do not.0831
Let us take a look at the graph of the function that we just did.0838
We said that as x goes to positive infinity, the function approaches 3.0842
As x goes to negative infinity, the function approaches -3.0849
I did not draw out the horizontal asymptotes here.0854
I just want you to see that, but it is essentially what we did.0855
Evaluate the following limit, the limit as x approaches positive infinity of 16x² + x.0863
You might be tempted to do something like this.0875
Let me write this down.0877
We are tempted to say as x goes to positive infinity, this term is going to dominate, which is true.0879
We are tempted to say that we can treat this as √16x² which is equal to 4x.0898
We can just say that 4x - 4x is equal to 0.0909
The limit is x approaches this is just 0.0913
That is not the case.0917
The problem is the x term may contribute to such a point that, what you end up with is infinity – infinity.0919
This will go to infinity, this will go to infinity.0944
But infinity – infinity, we are not exactly sure about the rates at which this goes to infinity and this goes to negative infinity.0946
Because we are not sure about how fast that happens,0955
we do not know if it goes to 0 or if it goes to infinity, or if it goes to some other number in between.0957
This is an indeterminate form, infinity – infinity.0964
We have to handle it differently.0967
Let us deal with the function itself, before we actually take the limit.0970
Here when we put the infinity in, we get infinity - infinity which does not make sense.0974
We have to manipulate it.0979
We have 16x² + x under the radical, -4x.0982
I’m going to go ahead and rationalize this out.0989
I’m going to multiply by its conjugate.0991
16x² + x, this is going to be + 4x/ 16x² + x + 4x.0994
When I multiply this out, I end up with 16x² + x.1007
This is going to be -16x²/ √16x² + x/ +4x.1013
Those go away, leaving me with just x/ 16x² + x + 4x.1027
We have a rational function, even though we have a square root in the denominator,1040
let us go ahead and divide by the largest power in the denominator which is what we always do in the denominator.1043
The largest power in the denominator is essentially going to be the √x².1060
It is going to be x, but it is going to be √x².1065
What we have is the following.1069
We are going to have x/x, that is the numerator.1072
We are going to have 16x² + x, all under the radical, + 4x all under x, which is going to equal 1/ √16x² + 4x/ x².1075
This one, I’m going to treat x as √x².1115
I’m going to get 16x² + 4x/ x², + 4x/ x, I’m going to leave this as x.1121
This x for these two, because under the radical I’m just going to treat it as √x².1132
That ends up equaling 1/ √16, this is x.1141
+ 1/ x and this is + 4.1155
There we go, now we can take the limit.1160
The limit as x approaches positive infinity of 1/ √16 + 1/ x + 4.1164
As x goes to infinity, the 1/x goes to 0.1175
We are left with 1/ 4 + 4.1181
√16 is 4, you will get 1/8.1184
Sure enough, that is what it looks like.1192
This is our asymptote, this is y = 1/8.1194
This is our origin, as x goes to positive infinity, the function itself gets closer and closer to 1/8.1198
That is the limit.1206
Evaluate the following limit as x goes to infinity, 1 – 5e ⁺x/ 1 + 3e ⁺x.1210
When we put x in, we are going to end up with -infinity/ infinity which is in indeterminate form.1220
We have to do something with it.1231
1 – 5e ⁺x/ 1 + 3e ⁺x, we can do the same thing that we did with rational functions.1235
This is going to be the same as, I’m going to divide everything by e ⁺x.1244
1 - 5e ⁺x, the top and the bottom, I mean, / 1 + 3e ⁺x/ e ⁺x.1249
What I end up with is 1/ e ⁺x - 5/ 1/ e ⁺x + 3.1258
I’m going to take the limit of that.1272
The limit as x goes to, I’m going to do positive infinity first.1274
1/ e ⁺x – 5, put the 1/ e ⁺x + 3.1283
As x goes to infinity, e ⁺x goes to infinity that means this thing goes to 0, this thing goes to 0.1291
I'm left with -5/3.1297
As x goes to positive infinity, my function actually goes to -5/3.1299
I have a horizontal asymptote at 5/3, -5/3.1306
Now for x going to negative infinity, I have the following.1311
The limit as x goes to negative infinity of 1 - 5e ⁺x/ 1 + 3e ⁺x.1318
x is a negative number, it is negative infinity.1332
e ⁺negative number is 1/ e ⁺positive number.1336
This is actually equivalent to the limit as x goes to positive infinity of 1 - 5/ e ⁺x.1339
It is e ⁻x is the same as e ⁻x is 1/ e ⁺x.1357
Because we are going to negative infinity, x is negative number.1366
Because it is a negative number, I can just drop it into the denominator and make it a positive number.1372
1 - 5 and then 1 + 3/ e ⁺x.1377
As x goes to infinity, this goes to 0, you are left with 1.1383
Sure enough, there you go.1392
As x goes to negative infinity, we approach y = 1.1394
As x goes to positive infinity, our function approaches y = -5/3.1401
That is it, just nice manipulation.1410
Let us see what we got.1417
Now the whole idea of a reachable finite numerical limit is as x gets closer and closer to a certain number or as x goes to infinity,1418
but f(x) gets closer and closer to a certain number like we just saw -5/3 or 1.1427
This latter number is the limit.1434
The question here is how big would x have to be, in order for the function f(x) = e ⁻x/25 + 21439
to be less than a distance of 0.001 away from its limit?1449
Closer and closer, closer and closer means we can take it as close as we want.1455
In this case, the tolerance that I'm looking for is 0.001 away from its limit.1459
The first thing we want to do, what is the limit?1467
What is the limit as x goes to infinity of e ⁻x/ 25 + 2.1474
Let us just deal with positive infinity here.1489
This is the same as e ⁻x/25.1507
This is the same as the limit as x goes to positive infinity of 1/ e ⁺x /25 + 2.1512
As x goes to infinity, e ⁺x/25 goes to infinity.1532
This goes to 0.1537
The limit is actually 2.1542
The limit of this function as x goes to infinity is equal to 2.1544
I probably going to need more room.1557
Let me go ahead and go and work in red.1559
Now e ⁻x/25 is always greater than 0.1566
e ^- x/ 25 + 2 is always going to be greater than 2.1576
f(x) which is equal to e ^- x/25 + 2, we said that the limit of this function as x goes to infinity is 2.1592
But we said that the function is always greater than 2 which means that1602
the function is actually approaching 2 from above.1606
It actually looks like this, this is our graph and this is our asymptote at 2.1610
The function is doing this.1619
The limit is 2, that is this dash line right here.1622
We know that the function itself, because this is always greater than 0, the function itself e ⁻x/25 + 2 is always going to be greater than 2.1627
It is always going to be above it.1635
It is above it, it is getting closer to it from above.1637
That is what is happening here physically, getting closer to the 2.1640
That is happening from above.1645
We want to make this distance, that distance right there.1648
We will call it d, we want that distance to be less than 0.001.1651
Our question is asking how far out do we have to go?1659
What x value passed which x value will this distance?1663
This distance between the function and limit be less than 0.001, that is what this is asking.1668
Again, we said we will call that d.1680
D is equal to the function itself - the limit.1685
Here was the limit, here was the function, this is the distance right there.1694
We want that distance, that distance is f(x) – l.1698
Here is our origin, this is 0,0.1704
This number - this number gives me the distance between them.1708
It is f(x) – l.1712
We know what f(x) is, that is just e ⁻x/25 + 2.1714
We know what l is, it is -2.1719
These go away, we want this distance which is e ⁻x/25, we want it to be less than 0.001.1724
Now we can solve this equation for x.1737
I’m going to go ahead and take the natlog of both sides.1744
I have -x/25 is less than the natlog of 0.001.1750
I’m going to make this a little more clear here, 0.001.1765
-x is less than 25 × the natlog of 0.001, that means x is greater than -25 × the natlog of 0.001.1771
Whenever I do, the natlog of 0.001 is going to be a negative number.1790
Negative × a negative, when I put this in the calculator, I get x has to be greater than 172.069.1795
The limit was this number.1808
The whole idea of the limit is we want to get closer and closer and closer.1810
In this particular case, we specified what we meant by closer and closer.1813
I want it closer than 0.001, the function to be less than not far away from the limit.1817
I knew that the function was approaching it from above.1823
The distance between the function and limit, that is what I want it to be, less than 0.01.1827
The distance between the function and limit is the function - the limit.1832
This distance, that distance right there.1836
I set it and I solve for x.1839
As long as x is bigger than 172.69, f(x) - l is going to be less than 0.001.1841
In other words, the function is going to be less than 0.001 units away from its limit.1853
What if f actually approached it from below?1865
What if we have something like this?1868
Let us say again, this was our limit.1874
This time let us say that the function came from below.1877
Now this is f(x) and this is the origin.1880
The limit is above the function.1884
The distance that we are interested in is this distance.1889
The distance between the function and limit.1891
Here the distance is going to equal the length - the function.1894
The length is a bigger number.1902
We want it to be positive.1905
It is going to be l – f(x).1907
Now we combine f(x) - l and l - f(x) as the absolute value of f(x) – l.1910
This distance, and if we are coming from above, this distance,1927
they will be the same if we use the absolute value because distance is a positive number.1930
You cannot have a negative distance.1936
We just combine those two, when we give the definition of a limit by using the absolute value sign.1938
It is that absolute value sign that has confounded and intimidated the students for about 150 years now.1944
Our formal mathematical definition of the limit.1955
We are concerned with the formal mathematical definition of the limit.1975
I told you not to believe about that, I do not believe that these kind of definitions,1977
these precise definitions do not belong in this level.1984
This level is about intuition.1986
By intuition, we mean the idea of how close can you get.1988
We speak of closer and closer and closer.1992
There is a way of describing that symbolically.1995
What do we mean by closer and closer, that is what I'm going to describe here.1997
Again, I just want you to see it because some of your classes will deal with it,2001
some of your classes would not deal with it.2004
But I wanted you to see the idea and where it actually came from.2006
Our formal mathematical definition of, when we say something like the limit as x approaches infinity of f(x) = l.2011
When we say that some function as x goes to infinity,2023
that the function actually approaches a finite limit, this symbol, here is what it means mathematically.2027
The formal definition is for any choice of a number that we will symbolize with ε,2035
which is going to be always greater than 0, there is an x value somewhere on the real line.2052
Such that the absolute value of f(x) - the limit is going to be less than this choice of ε,2065
whenever x is greater then x sub 0.2076
In the previous example, we found our x sub 0, that was our 172.2080
We found an x sub 0.2090
Our ε in that problem, we chose 0.001 as our ε.2093
We wanted to make the difference between the function and the limit less than 0.001.2100
We found an x sub 0 of 172.69.2104
Why do I keep writing 6, that is strange.2110
Any x value that is bigger than 172.69, we will make the difference between the function and the limit less than 0.001.2115
The precise general mathematical definition is, for any choice of the number ε greater than 0,2124
there exists an x sub 0 such that whenever x is bigger than x sub 0,2131
the difference between f(x) and its limit is going to be less than ε.2138
You can see why this stuff is confusing and why is it that it actually does not belong at this level.2142
Again, for those of you that go on in mathematics and taking analysis course, math majors mostly, this is what you will do.2147
You will go back and you actually work with epsilons, deltas, and x sub 0.2154
You will prove why certain things are the way they are.2159
At this level, we just want to be able to accept that those proofs had been done.2162
We want to be able to use it to solve problems.2166
We want to learn how to compute.2169
We want to use it as a tool.2171
We do not want to justify it.2173
Later, you can justify it, as a math major.2174
Now we just want to be able to use it.2177
This idea of closer and closer and closer to a limit is absolutely fine.2179
It is that intuition, if you want to do it.2185
Thank you so much for joining us here at www.educator.com.2188
We will see you next time, bye.2190

Raffi Hovasapian
Example Problems for Limits at Infinity
Slide Duration:Table of Contents
42m 8s
- Intro0:00
- Overview & Slopes of Curves0:21
- Differential and Integral0:22
- Fundamental Theorem of Calculus6:36
- Differentiation or Taking the Derivative14:24
- What Does the Derivative Mean and How do We Find it?15:18
- Example: f'(x)19:24
- Example: f(x) = sin (x)29:16
- General Procedure for Finding the Derivative of f(x)37:33
50m 53s
- Intro0:00
- Slope of the Secant Line along a Curve0:12
- Slope of the Tangent Line to f(x) at a Particlar Point0:13
- Slope of the Secant Line along a Curve2:59
- Instantaneous Slope6:51
- Instantaneous Slope6:52
- Example: Distance, Time, Velocity13:32
- Instantaneous Slope and Average Slope25:42
- Slope & Rate of Change29:55
- Slope & Rate of Change29:56
- Example: Slope = 233:16
- Example: Slope = 4/334:32
- Example: Slope = 4 (m/s)39:12
- Example: Density = Mass / Volume40:33
- Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change47:46
59m 12s
- Intro0:00
- Example I: Water Tank0:13
- Part A: Which is the Independent Variable and Which is the Dependent?2:00
- Part B: Average Slope3:18
- Part C: Express These Slopes as Rates-of-Change9:28
- Part D: Instantaneous Slope14:54
- Example II: y = √(x-3)28:26
- Part A: Calculate the Slope of the Secant Line30:39
- Part B: Instantaneous Slope41:26
- Part C: Equation for the Tangent Line43:59
- Example III: Object in the Air49:37
- Part A: Average Velocity50:37
- Part B: Instantaneous Velocity55:30
18m 43s
- Intro0:00
- Desmos Tutorial1:42
- Desmos Tutorial1:43
- Things You Must Learn To Do on Your Particular Calculator2:39
- Things You Must Learn To Do on Your Particular Calculator2:40
- Example I: y=sin x4:54
- Example II: y=x³ and y = d/(dx) (x³)9:22
- Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}13:15
51m 53s
- Intro0:00
- The Limit of a Function0:14
- The Limit of a Function0:15
- Graph: Limit of a Function12:24
- Table of Values16:02
- lim x→a f(x) Does not Say What Happens When x = a20:05
- Example I: f(x) = x²24:34
- Example II: f(x) = 727:05
- Example III: f(x) = 4.530:33
- Example IV: f(x) = 1/x34:03
- Example V: f(x) = 1/x²36:43
- The Limit of a Function, Cont.38:16
- Infinity and Negative Infinity38:17
- Does Not Exist42:45
- Summary46:48
24m 43s
- Intro0:00
- Example I: Explain in Words What the Following Symbols Mean0:10
- Example II: Find the Following Limit5:21
- Example III: Use the Graph to Find the Following Limits7:35
- Example IV: Use the Graph to Find the Following Limits11:48
- Example V: Sketch the Graph of a Function that Satisfies the Following Properties15:25
- Example VI: Find the Following Limit18:44
- Example VII: Find the Following Limit20:06
53m 48s
- Intro0:00
- Plug-in Procedure0:09
- Plug-in Procedure0:10
- Limit Laws9:14
- Limit Law 110:05
- Limit Law 210:54
- Limit Law 311:28
- Limit Law 411:54
- Limit Law 512:24
- Limit Law 613:14
- Limit Law 714:38
- Plug-in Procedure, Cont.16:35
- Plug-in Procedure, Cont.16:36
- Example I: Calculating Limits Mathematically20:50
- Example II: Calculating Limits Mathematically27:37
- Example III: Calculating Limits Mathematically31:42
- Example IV: Calculating Limits Mathematically35:36
- Example V: Calculating Limits Mathematically40:58
- Limits Theorem44:45
- Limits Theorem 144:46
- Limits Theorem 2: Squeeze Theorem46:34
- Example VI: Calculating Limits Mathematically49:26
21m 22s
- Intro0:00
- Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law0:16
- Example II: Evaluate the Following Limit1:51
- Example III: Evaluate the Following Limit3:36
- Example IV: Evaluate the Following Limit8:56
- Example V: Evaluate the Following Limit11:19
- Example VI: Calculating Limits Mathematically13:19
- Example VII: Calculating Limits Mathematically14:59
50m 1s
- Intro0:00
- Limit as x Goes to Infinity0:14
- Limit as x Goes to Infinity0:15
- Let's Look at f(x) = 1 / (x-3)1:04
- Summary9:34
- Example I: Calculating Limits as x Goes to Infinity12:16
- Example II: Calculating Limits as x Goes to Infinity21:22
- Example III: Calculating Limits as x Goes to Infinity24:10
- Example IV: Calculating Limits as x Goes to Infinity36:00
36m 31s
- Intro0:00
- Example I: Calculating Limits as x Goes to Infinity0:14
- Example II: Calculating Limits as x Goes to Infinity3:27
- Example III: Calculating Limits as x Goes to Infinity8:11
- Example IV: Calculating Limits as x Goes to Infinity14:20
- Example V: Calculating Limits as x Goes to Infinity20:07
- Example VI: Calculating Limits as x Goes to Infinity23:36
53m
- Intro0:00
- Definition of Continuity0:08
- Definition of Continuity0:09
- Example: Not Continuous3:52
- Example: Continuous4:58
- Example: Not Continuous5:52
- Procedure for Finding Continuity9:45
- Law of Continuity13:44
- Law of Continuity13:45
- Example I: Determining Continuity on a Graph15:55
- Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous17:57
- Example III: Is the Following Function Continuous at the Given Point?22:42
- Theorem for Composite Functions25:28
- Theorem for Composite Functions25:29
- Example IV: Is cos(x³ + ln x) Continuous at x=π/2?27:00
- Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?34:04
- Types of Discontinuity39:18
- Removable Discontinuity39:33
- Jump Discontinuity40:06
- Infinite Discontinuity40:32
- Intermediate Value Theorem40:58
- Intermediate Value Theorem: Hypothesis & Conclusion40:59
- Intermediate Value Theorem: Graphically43:40
- Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]47:46
40m 2s
- Intro0:00
- Derivative0:09
- Derivative0:10
- Example I: Find the Derivative of f(x)=x³2:20
- Notations for the Derivative7:32
- Notations for the Derivative7:33
- Derivative & Rate of Change11:14
- Recall the Rate of Change11:15
- Instantaneous Rate of Change17:04
- Graphing f(x) and f'(x)19:10
- Example II: Find the Derivative of x⁴ - x²24:00
- Example III: Find the Derivative of f(x)=√x30:51
53m 45s
- Intro0:00
- Example I: Find the Derivative of (2+x)/(3-x)0:18
- Derivatives II9:02
- f(x) is Differentiable if f'(x) Exists9:03
- Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other17:19
- Geometrically: Differentiability Means the Graph is Smooth18:44
- Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=020:53
- Example II: For x > 023:53
- Example II: For x < 025:36
- Example II: What is f(0) and What is the lim |x| as x→0?30:46
- Differentiability & Continuity34:22
- Differentiability & Continuity34:23
- How Can a Function Not be Differentiable at a Point?39:38
- How Can a Function Not be Differentiable at a Point?39:39
- Higher Derivatives41:58
- Higher Derivatives41:59
- Derivative Operator45:12
- Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³49:29
31m 38s
- Intro0:00
- Example I: Sketch f'(x)0:10
- Example II: Sketch f'(x)2:14
- Example III: Find the Derivative of the Following Function sing the Definition3:49
- Example IV: Determine f, f', and f'' on a Graph12:43
- Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value13:40
- Example VI: Distance vs. Time20:15
- Example VII: Displacement, Velocity, and Acceleration23:56
- Example VIII: Graph the Displacement Function28:20
47m 35s
- Intro0:00
- Differentiation of Polynomials & Exponential Functions0:15
- Derivative of a Function0:16
- Derivative of a Constant2:35
- Power Rule3:08
- If C is a Constant4:19
- Sum Rule5:22
- Exponential Functions6:26
- Example I: Differentiate7:45
- Example II: Differentiate12:38
- Example III: Differentiate15:13
- Example IV: Differentiate16:20
- Example V: Differentiate19:19
- Example VI: Find the Equation of the Tangent Line to a Function at a Given Point12:18
- Example VII: Find the First & Second Derivatives25:59
- Example VIII27:47
- Part A: Find the Velocity & Acceleration Functions as Functions of t27:48
- Part B: Find the Acceleration after 3 Seconds30:12
- Part C: Find the Acceleration when the Velocity is 030:53
- Part D: Graph the Position, Velocity, & Acceleration Graphs32:50
- Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents34:53
- Example X: Find a Point on a Graph42:31
47m 25s
- Intro0:00
- The Product, Power and Quotient Rules0:19
- Differentiate Functions0:20
- Product Rule5:30
- Quotient Rule9:15
- Power Rule10:00
- Example I: Product Rule13:48
- Example II: Quotient Rule16:13
- Example III: Power Rule18:28
- Example IV: Find dy/dx19:57
- Example V: Find dy/dx24:53
- Example VI: Find dy/dx28:38
- Example VII: Find an Equation for the Tangent to the Curve34:54
- Example VIII: Find d²y/dx²38:08
41m 8s
- Intro0:00
- Derivatives of the Trigonometric Functions0:09
- Let's Find the Derivative of f(x) = sin x0:10
- Important Limits to Know4:59
- d/dx (sin x)6:06
- d/dx (cos x)6:38
- d/dx (tan x)6:50
- d/dx (csc x)7:02
- d/dx (sec x)7:15
- d/dx (cot x)7:27
- Example I: Differentiate f(x) = x² - 4 cos x7:56
- Example II: Differentiate f(x) = x⁵ tan x9:04
- Example III: Differentiate f(x) = (cos x) / (3 + sin x)10:56
- Example IV: Differentiate f(x) = e^x / (tan x - sec x)14:06
- Example V: Differentiate f(x) = (csc x - 4) / (cot x)15:37
- Example VI: Find an Equation of the Tangent Line21:48
- Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?25:17
- Example VIII: Ladder Problem28:23
- Example IX: Evaluate33:22
- Example X: Evaluate36:38
24m 56s
- Intro0:00
- The Chain Rule0:13
- Recall the Composite Functions0:14
- Derivatives of Composite Functions1:34
- Example I: Identify f(x) and g(x) and Differentiate6:41
- Example II: Identify f(x) and g(x) and Differentiate9:47
- Example III: Differentiate11:03
- Example IV: Differentiate f(x) = -5 / (x² + 3)³12:15
- Example V: Differentiate f(x) = cos(x² + c²)14:35
- Example VI: Differentiate f(x) = cos⁴x +c²15:41
- Example VII: Differentiate17:03
- Example VIII: Differentiate f(x) = sin(tan x²)19:01
- Example IX: Differentiate f(x) = sin(tan² x)21:02
25m 32s
- Intro0:00
- Example I: Differentiate f(x) = sin(cos(tanx))0:38
- Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point2:25
- Example III: F(x) = f(g(x)), Find F' (6)4:22
- Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window5:35
- Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴10:18
- Example VI: Differentiate f(x) = sec²(12x)12:28
- Example VII: Differentiate14:41
- Example VIII: Differentiate19:25
- Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time21:13
52m 31s
- Intro0:00
- Implicit Differentiation0:09
- Implicit Differentiation0:10
- Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y12:15
- Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 1419:18
- Example III: Find (dy)/(dx) of x³y² + y³x² = 4x21:43
- Example IV: Find (dy)/(dx) of the Following Equation24:13
- Example V: Find (dy)/(dx) of 6sin x cos y = 129:00
- Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y31:02
- Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x37:36
- Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)41:03
- Example IX: Find (d²y)/(dx²) of x² + y² = 2544:05
- Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)47:48
47m 34s
- Intro0:00
- Linear Approximations & Differentials0:09
- Linear Approximations & Differentials0:10
- Example I: Linear Approximations & Differentials11:27
- Example II: Linear Approximations & Differentials20:19
- Differentials30:32
- Differentials30:33
- Example III: Linear Approximations & Differentials34:09
- Example IV: Linear Approximations & Differentials35:57
- Example V: Relative Error38:46
45m 33s
- Intro0:00
- Related Rates0:08
- Strategy for Solving Related Rates Problems #10:09
- Strategy for Solving Related Rates Problems #21:46
- Strategy for Solving Related Rates Problems #32:06
- Strategy for Solving Related Rates Problems #42:50
- Strategy for Solving Related Rates Problems #53:38
- Example I: Radius of a Balloon5:15
- Example II: Ladder12:52
- Example III: Water Tank19:08
- Example IV: Distance between Two Cars29:27
- Example V: Line-of-Sight36:20
37m 17s
- Intro0:00
- Example I: Shadow0:14
- Example II: Particle4:45
- Example III: Water Level10:28
- Example IV: Clock20:47
- Example V: Distance between a House and a Plane29:11
40m 44s
- Intro0:00
- Maximum & Minimum Values of a Function, Part 10:23
- Absolute Maximum2:20
- Absolute Minimum2:52
- Local Maximum3:38
- Local Minimum4:26
- Maximum & Minimum Values of a Function, Part 26:11
- Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min7:18
- Function with Local Max & Min but No Absolute Max & Min8:48
- Formal Definitions10:43
- Absolute Maximum11:18
- Absolute Minimum12:57
- Local Maximum14:37
- Local Minimum16:25
- Extreme Value Theorem18:08
- Theorem: f'(c) = 024:40
- Critical Number (Critical Value)26:14
- Procedure for Finding the Critical Values of f(x)28:32
- Example I: Find the Critical Values of f(x) x + sinx29:51
- Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]35:31
40m 44s
- Intro0:00
- Example I: Identify Absolute and Local Max & Min on the Following Graph0:11
- Example II: Sketch the Graph of a Continuous Function3:11
- Example III: Sketch the Following Graphs4:40
- Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²6:13
- Example V: Find the Critical Values of f(x) = |2x - 5|8:42
- Example VI: Find the Critical Values11:42
- Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]16:57
- Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]20:08
- Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]24:39
25m 54s
- Intro0:00
- Rolle's Theorem0:08
- Rolle's Theorem: If & Then0:09
- Rolle's Theorem: Geometrically2:06
- There May Be More than 1 c Such That f'( c ) = 03:30
- Example I: Rolle's Theorem4:58
- The Mean Value Theorem9:12
- The Mean Value Theorem: If & Then9:13
- The Mean Value Theorem: Geometrically11:07
- Example II: Mean Value Theorem13:43
- Example III: Mean Value Theorem21:19
25m 54s
- Intro0:00
- Using Derivatives to Graph Functions, Part I0:12
- Increasing/ Decreasing Test0:13
- Example I: Find the Intervals Over Which the Function is Increasing & Decreasing3:26
- Example II: Find the Local Maxima & Minima of the Function19:18
- Example III: Find the Local Maxima & Minima of the Function31:39
44m 58s
- Intro0:00
- Using Derivatives to Graph Functions, Part II0:13
- Concave Up & Concave Down0:14
- What Does This Mean in Terms of the Derivative?6:14
- Point of Inflection8:52
- Example I: Graph the Function13:18
- Example II: Function x⁴ - 5x²19:03
- Intervals of Increase & Decrease19:04
- Local Maxes and Mins25:01
- Intervals of Concavity & X-Values for the Points of Inflection29:18
- Intervals of Concavity & Y-Values for the Points of Inflection34:18
- Graphing the Function40:52
49m 19s
- Intro0:00
- Example I: Intervals, Local Maxes & Mins0:26
- Example II: Intervals, Local Maxes & Mins5:05
- Example III: Intervals, Local Maxes & Mins, and Inflection Points13:40
- Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity23:02
- Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity34:36
59m 1s
- Intro0:00
- Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes0:11
- Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes21:24
- Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D37:56
- Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes46:19
30m 9s
- Intro0:00
- L'Hospital's Rule0:19
- Indeterminate Forms0:20
- L'Hospital's Rule3:38
- Example I: Evaluate the Following Limit Using L'Hospital's Rule8:50
- Example II: Evaluate the Following Limit Using L'Hospital's Rule10:30
- Indeterminate Products11:54
- Indeterminate Products11:55
- Example III: L'Hospital's Rule & Indeterminate Products13:57
- Indeterminate Differences17:00
- Indeterminate Differences17:01
- Example IV: L'Hospital's Rule & Indeterminate Differences18:57
- Indeterminate Powers22:20
- Indeterminate Powers22:21
- Example V: L'Hospital's Rule & Indeterminate Powers25:13
38m 14s
- Intro0:00
- Example I: Evaluate the Following Limit0:17
- Example II: Evaluate the Following Limit2:45
- Example III: Evaluate the Following Limit6:54
- Example IV: Evaluate the Following Limit8:43
- Example V: Evaluate the Following Limit11:01
- Example VI: Evaluate the Following Limit14:48
- Example VII: Evaluate the Following Limit17:49
- Example VIII: Evaluate the Following Limit20:37
- Example IX: Evaluate the Following Limit25:16
- Example X: Evaluate the Following Limit32:44
49m 59s
- Intro0:00
- Example I: Find the Dimensions of the Box that Gives the Greatest Volume1:23
- Fundamentals of Optimization Problems18:08
- Fundamental #118:33
- Fundamental #219:09
- Fundamental #319:19
- Fundamental #420:59
- Fundamental #521:55
- Fundamental #623:44
- Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square24:36
- Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)35:13
- Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R43:10
55m 10s
- Intro0:00
- Example I: Optimization Problem0:13
- Example II: Optimization Problem17:34
- Example III: Optimization Problem35:06
- Example IV: Revenue, Cost, and Profit43:22
30m 22s
- Intro0:00
- Newton's Method0:45
- Newton's Method0:46
- Example I: Find x2 and x313:18
- Example II: Use Newton's Method to Approximate15:48
- Example III: Find the Root of the Following Equation to 6 Decimal Places19:57
- Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point23:11
55m 26s
- Intro0:00
- Antiderivatives0:23
- Definition of an Antiderivative0:24
- Antiderivative Theorem7:58
- Function & Antiderivative12:10
- x^n12:30
- 1/x13:00
- e^x13:08
- cos x13:18
- sin x14:01
- sec² x14:11
- secxtanx14:18
- 1/√(1-x²)14:26
- 1/(1+x²)14:36
- -1/√(1-x²)14:45
- Example I: Find the Most General Antiderivative for the Following Functions15:07
- Function 1: f(x) = x³ -6x² + 11x - 915:42
- Function 2: f(x) = 14√(x) - 27 4√x19:12
- Function 3: (fx) = cos x - 14 sinx20:53
- Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )22:10
- Function 5: f(x) = (3e^x) - 2/(1+x²)25:42
- Example II: Given the Following, Find the Original Function f(x)26:37
- Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 4027:55
- Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 530:34
- Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.232:54
- Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 737:54
- Example III: Falling Object41:58
- Problem 1: Find an Equation for the Height of the Ball after t Seconds42:48
- Problem 2: How Long Will It Take for the Ball to Strike the Ground?48:30
- Problem 3: What is the Velocity of the Ball as it Hits the Ground?49:52
- Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?50:46
51m 3s
- Intro0:00
- The Area Under a Curve0:13
- Approximate Using Rectangles0:14
- Let's Do This Again, Using 4 Different Rectangles9:40
- Approximate with Rectangles16:10
- Left Endpoint18:08
- Right Endpoint25:34
- Left Endpoint vs. Right Endpoint30:58
- Number of Rectangles34:08
- True Area37:36
- True Area37:37
- Sigma Notation & Limits43:32
- When You Have to Explicitly Solve Something47:56
33m 7s
- Intro0:00
- Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve0:10
- Example II: Using 5 Rectangles, Approximate the Area Under the Curve11:32
- Example III: Find the True Area by Evaluating the Limit Expression16:07
- Example IV: Find the True Area by Evaluating the Limit Expression24:52
43m 19s
- Intro0:00
- The Definite Integral0:08
- Definition to Find the Area of a Curve0:09
- Definition of the Definite Integral4:08
- Symbol for Definite Integral8:45
- Regions Below the x-axis15:18
- Associating Definite Integral to a Function19:38
- Integrable Function27:20
- Evaluating the Definite Integral29:26
- Evaluating the Definite Integral29:27
- Properties of the Definite Integral35:24
- Properties of the Definite Integral35:25
32m 14s
- Intro0:00
- Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals0:11
- Example II: Express the Following Limit as a Definite Integral5:28
- Example III: Evaluate the Following Definite Integral Using the Definition6:28
- Example IV: Evaluate the Following Integral Using the Definition17:06
- Example V: Evaluate the Following Definite Integral by Using Areas25:41
- Example VI: Definite Integral30:36
24m 17s
- Intro0:00
- The Fundamental Theorem of Calculus0:17
- Evaluating an Integral0:18
- Lim as x → ∞12:19
- Taking the Derivative14:06
- Differentiation & Integration are Inverse Processes15:04
- 1st Fundamental Theorem of Calculus20:08
- 1st Fundamental Theorem of Calculus20:09
- 2nd Fundamental Theorem of Calculus22:30
- 2nd Fundamental Theorem of Calculus22:31
25m 21s
- Intro0:00
- Example I: Find the Derivative of the Following Function0:17
- Example II: Find the Derivative of the Following Function1:40
- Example III: Find the Derivative of the Following Function2:32
- Example IV: Find the Derivative of the Following Function5:55
- Example V: Evaluate the Following Integral7:13
- Example VI: Evaluate the Following Integral9:46
- Example VII: Evaluate the Following Integral12:49
- Example VIII: Evaluate the Following Integral13:53
- Example IX: Evaluate the Following Graph15:24
- Local Maxs and Mins for g(x)15:25
- Where Does g(x) Achieve Its Absolute Max on [0,8]20:54
- On What Intervals is g(x) Concave Up/Down?22:20
- Sketch a Graph of g(x)24:34
34m 22s
- Intro0:00
- Example I: Evaluate the Following Indefinite Integral0:10
- Example II: Evaluate the Following Definite Integral0:59
- Example III: Evaluate the Following Integral2:59
- Example IV: Velocity Function7:46
- Part A: Net Displacement7:47
- Part B: Total Distance Travelled13:15
- Example V: Linear Density Function20:56
- Example VI: Acceleration Function25:10
- Part A: Velocity Function at Time t25:11
- Part B: Total Distance Travelled During the Time Interval28:38
27m 20s
- Intro0:00
- Table of Integrals0:35
- Example I: Evaluate the Following Indefinite Integral2:02
- Example II: Evaluate the Following Indefinite Integral7:27
- Example IIII: Evaluate the Following Indefinite Integral10:57
- Example IV: Evaluate the Following Indefinite Integral12:33
- Example V: Evaluate the Following14:28
- Example VI: Evaluate the Following16:00
- Example VII: Evaluate the Following19:01
- Example VIII: Evaluate the Following21:49
- Example IX: Evaluate the Following24:34
34m 56s
- Intro0:00
- Areas Between Two Curves: Function of x0:08
- Graph 1: Area Between f(x) & g(x)0:09
- Graph 2: Area Between f(x) & g(x)4:07
- Is It Possible to Write as a Single Integral?8:20
- Area Between the Curves on [a,b]9:24
- Absolute Value10:32
- Formula for Areas Between Two Curves: Top Function - Bottom Function17:03
- Areas Between Curves: Function of y17:49
- What if We are Given Functions of y?17:50
- Formula for Areas Between Two Curves: Right Function - Left Function21:48
- Finding a & b22:32
42m 55s
- Intro0:00
- Instructions for the Example Problems0:10
- Example I: y = 7x - x² and y=x0:37
- Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=26:25
- Example III: y=(1/x), y=(1/x³), and x=412:25
- Example IV: 15-2x² and y=x²-515:52
- Example V: x=(1/8)y³ and x=6-y²20:20
- Example VI: y=cos x, y=sin(2x), [0,π/2]24:34
- Example VII: y=2x², y=10x², 7x+2y=1029:51
- Example VIII: Velocity vs. Time33:23
- Part A: At 2.187 Minutes, Which care is Further Ahead?33:24
- Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?36:32
- Part C: At 4 Minutes Which Car is Ahead?37:11
- Part D: At What Time Will the Cars be Side by Side?37:50
34m 15s
- Intro0:00
- Volumes I: Slices0:18
- Rotate the Graph of y=√x about the x-axis0:19
- How can I use Integration to Find the Volume?3:16
- Slice the Solid Like a Loaf of Bread5:06
- Volumes Definition8:56
- Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation12:18
- Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation19:05
- Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation25:28
51m 43s
- Intro0:00
- Volumes II: Volumes by Washers0:11
- Rotating Region Bounded by y=x³ & y=x around the x-axis0:12
- Equation for Volumes by Washer11:14
- Process for Solving Volumes by Washer13:40
- Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis15:58
- Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis25:07
- Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis34:20
- Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis44:05
49m 36s
- Intro0:00
- Solids That Are Not Solids-of-Revolution0:11
- Cross-Section Area Review0:12
- Cross-Sections That Are Not Solids-of-Revolution7:36
- Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H10:54
- Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles20:39
- Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H29:27
- Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 6436:47
- Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis46:13
50m 2s
- Intro0:00
- Volumes by Cylindrical Shells0:11
- Find the Volume of the Following Region0:12
- Volumes by Cylindrical Shells: Integrating Along x14:12
- Volumes by Cylindrical Shells: Integrating Along y14:40
- Volumes by Cylindrical Shells Formulas16:22
- Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid18:33
- Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid25:57
- Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid31:38
- Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid38:44
- Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid44:03
32m 13s
- Intro0:00
- The Average Value of a Function0:07
- Average Value of f(x)0:08
- What if The Domain of f(x) is Not Finite?2:23
- Let's Calculate Average Value for f(x) = x² [2,5]4:46
- Mean Value Theorem for Integrate9:25
- Example I: Find the Average Value of the Given Function Over the Given Interval14:06
- Example II: Find the Average Value of the Given Function Over the Given Interval18:25
- Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]24:04
- Example IV: Find the Average Density of a Rod27:47
50m 32s
- Intro0:00
- Integration by Parts0:08
- The Product Rule for Differentiation0:09
- Integrating Both Sides Retains the Equality0:52
- Differential Notation2:24
- Example I: ∫ x cos x dx5:41
- Example II: ∫ x² sin(2x)dx12:01
- Example III: ∫ (e^x) cos x dx18:19
- Example IV: ∫ (sin^-1) (x) dx23:42
- Example V: ∫₁⁵ (lnx)² dx28:25
- Summary32:31
- Tabular Integration35:08
- Case 135:52
- Example: ∫x³sinx dx36:39
- Case 240:28
- Example: ∫e^(2x) sin 3x41:14
24m 50s
- Intro0:00
- Example I: ∫ sin³ (x) dx1:36
- Example II: ∫ cos⁵(x)sin²(x)dx4:36
- Example III: ∫ sin⁴(x)dx9:23
- Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx15:59
- #1: Power of sin is Odd16:00
- #2: Power of cos is Odd16:41
- #3: Powers of Both sin and cos are Odd16:55
- #4: Powers of Both sin and cos are Even17:10
- Example IV: ∫ tan⁴ (x) sec⁴ (x) dx17:34
- Example V: ∫ sec⁹(x) tan³(x) dx20:55
- Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx23:31
- #1: Power of sec is Odd23:32
- #2: Power of tan is Odd24:04
- #3: Powers of sec is Odd and/or Power of tan is Even24:18
22m 12s
- Intro0:00
- Trigonometric Integrals II0:09
- Recall: ∫tanx dx0:10
- Let's Find ∫secx dx3:23
- Example I: ∫ tan⁵ (x) dx6:23
- Example II: ∫ sec⁵ (x) dx11:41
- Summary: How to Deal with Integrals of Different Types19:04
- Identities to Deal with Integrals of Different Types19:05
- Example III: ∫cos(5x)sin(9x)dx19:57
17m 22s
- Intro0:00
- Example I: ∫sin²(x)cos⁷(x)dx0:14
- Example II: ∫x sin²(x) dx3:56
- Example III: ∫csc⁴ (x/5)dx8:39
- Example IV: ∫( (1-tan²x)/(sec²x) ) dx11:17
- Example V: ∫ 1 / (sinx-1) dx13:19
55m 12s
- Intro0:00
- Integration by Partial Fractions I0:11
- Recall the Idea of Finding a Common Denominator0:12
- Decomposing a Rational Function to Its Partial Fractions4:10
- 2 Types of Rational Function: Improper & Proper5:16
- Improper Rational Function7:26
- Improper Rational Function7:27
- Proper Rational Function11:16
- Proper Rational Function & Partial Fractions11:17
- Linear Factors14:04
- Irreducible Quadratic Factors15:02
- Case 1: G(x) is a Product of Distinct Linear Factors17:10
- Example I: Integration by Partial Fractions20:33
- Case 2: D(x) is a Product of Linear Factors40:58
- Example II: Integration by Partial Fractions44:41
42m 57s
- Intro0:00
- Case 3: D(x) Contains Irreducible Factors0:09
- Example I: Integration by Partial Fractions5:19
- Example II: Integration by Partial Fractions16:22
- Case 4: D(x) has Repeated Irreducible Quadratic Factors27:30
- Example III: Integration by Partial Fractions30:19
46m 37s
- Intro0:00
- Introduction to Differential Equations0:09
- Overview0:10
- Differential Equations Involving Derivatives of y(x)2:08
- Differential Equations Involving Derivatives of y(x) and Function of y(x)3:23
- Equations for an Unknown Number6:28
- What are These Differential Equations Saying?10:30
- Verifying that a Function is a Solution of the Differential Equation13:00
- Verifying that a Function is a Solution of the Differential Equation13:01
- Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation17:20
- General Solution22:00
- Particular Solution24:36
- Initial Value Problem27:42
- Example I: Verify that a Family of Functions is a Solution of the Differential Equation32:24
- Example II: For What Values of K Does the Function Satisfy the Differential Equation36:07
- Example III: Verify the Solution and Solve the Initial Value Problem39:47
28m 8s
- Intro0:00
- Separation of Variables0:28
- Separation of Variables0:29
- Example I: Solve the Following g Initial Value Problem8:29
- Example II: Solve the Following g Initial Value Problem13:46
- Example III: Find an Equation of the Curve18:48
51m 7s
- Intro0:00
- Standard Growth Model0:30
- Definition of the Standard/Natural Growth Model0:31
- Initial Conditions8:00
- The General Solution9:16
- Example I: Standard Growth Model10:45
- Logistic Growth Model18:33
- Logistic Growth Model18:34
- Solving the Initial Value Problem25:21
- What Happens When t → ∞36:42
- Example II: Solve the Following g Initial Value Problem41:50
- Relative Growth Rate46:56
- Relative Growth Rate46:57
- Relative Growth Rate Version for the Standard model49:04
24m 37s
- Intro0:00
- Slope Fields0:35
- Slope Fields0:36
- Graphing the Slope Fields, Part 111:12
- Graphing the Slope Fields, Part 215:37
- Graphing the Slope Fields, Part 317:25
- Steps to Solving Slope Field Problems20:24
- Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y22:38
45m 29s
- Intro0:00
- Exam Link0:10
- Problem #11:26
- Problem #22:52
- Problem #34:42
- Problem #47:03
- Problem #510:01
- Problem #613:49
- Problem #715:16
- Problem #819:06
- Problem #923:10
- Problem #1028:10
- Problem #1131:30
- Problem #1233:53
- Problem #1337:45
- Problem #1441:17
41m 55s
- Intro0:00
- Problem #150:22
- Problem #163:10
- Problem #175:30
- Problem #188:03
- Problem #199:53
- Problem #2014:51
- Problem #2117:30
- Problem #2222:12
- Problem #2325:48
- Problem #2429:57
- Problem #2533:35
- Problem #2635:57
- Problem #2737:57
- Problem #2840:04
58m 47s
- Intro0:00
- Problem #11:22
- Problem #24:55
- Problem #310:49
- Problem #413:05
- Problem #514:54
- Problem #617:25
- Problem #718:39
- Problem #820:27
- Problem #926:48
- Problem #1028:23
- Problem #1134:03
- Problem #1236:25
- Problem #1339:52
- Problem #1443:12
- Problem #1547:18
- Problem #1650:41
- Problem #1756:38
25m 40s
- Intro0:00
- Problem #1: Part A1:14
- Problem #1: Part B4:46
- Problem #1: Part C8:00
- Problem #2: Part A12:24
- Problem #2: Part B16:51
- Problem #2: Part C17:17
- Problem #3: Part A18:16
- Problem #3: Part B19:54
- Problem #3: Part C21:44
- Problem #3: Part D22:57
31m 20s
- Intro0:00
- Problem #4: Part A1:35
- Problem #4: Part B5:54
- Problem #4: Part C8:50
- Problem #4: Part D9:40
- Problem #5: Part A11:26
- Problem #5: Part B13:11
- Problem #5: Part C15:07
- Problem #5: Part D19:57
- Problem #6: Part A22:01
- Problem #6: Part B25:34
- Problem #6: Part C28:54
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