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Week 15: Applications of integration
(collapsed, click to expand)- Completed 15.00 What application of integration will we consider? [1:45]
- Completed 15.01 What happens when I use thin horizontal rectangles to compute area? [6:37]
- Completed 15.02 When should I use horizontal as opposed to vertical pieces? [5:45]
- Completed 15.03 What does "volume" even mean? [4:47]
- Completed 15.04 What is the volume of a sphere? [6:03]
- Completed 15.05 How do washers help to compute the volume of a solid of revolution? [5:19]
- Completed 15.06 What is the volume of a thin shell? [7:48]
- Completed 15.07 What is the volume of a sphere with a hole drilled in it? [7:37]
- Completed 15.08 What does "length" even mean? [4:16]
- Completed 15.09 On the graph of y^2 = x^3, what is the length of a certain arc? [4:14]
- Completed 15.10 This title is missing a question mark. [1:15]
Week 14: Techniques of integration
(collapsed, click to expand)- Completed 14.00 What remains to be done? [1:29]
- Completed
14.01 What antidifferentiation rule corresponds to the product rule in reverse? [5:04]
Subtitles (text) for 14.01 What antidifferentiation rule corresponds to the product rule in reverse? [5:04]Subtitles (srt) for 14.01 What antidifferentiation rule corresponds to the product rule in reverse? [5:04]Video (MP4) for 14.01 What antidifferentiation rule corresponds to the product rule in reverse? [5:04]
- Completed 14.02 What is an antiderivative of x e^x? [4:13]
- Completed 14.03 How does parts help when antidifferentiating log x? [2:02]
- Completed 14.04 What is an antiderivative of e^x cos x? [6:12]
- Completed 14.05 What is an antiderivative of e^(sqrt(x))? [3:24]
- Completed 14.06 What is an antiderivative of sin^(2n+1) x cos^(2n) x dx? [5:50]
- Completed 14.07 What is the integral of sin^(2n) x dx from x = 0 to x = pi? [8:01]
- Completed 14.08 What is the integral of sin^n x dx in terms of sin^(n-2) x dx? [11:33]
- Completed 14.09 Why is pi < 22/7? [8:25]
Week 13: Substitution rule
(collapsed, click to expand)- Completed 13.00 How is this course structured?
- Completed 13.01 How does the chain rule help with antidifferentiation? [5:31]
- Completed 13.02 When I do u-substitution, what should u be? [7:09]
- Completed 13.03 How should I handle the endpoints when doing u-substitution? [5:13]
- Completed 13.04 Might I want to do u-substitution more than once? [4:22]
- Completed 13.05 What is the integral of dx / (x^2 + 4x + 7)? [9:04]
- Completed 13.06 What is the integral of (x+10)(x-1)^10 dx from x = 0 to x = 1? [5:36]
- Completed 13.07 What is the integral of x / (x+1)^(1/3) dx? [3:54]
- Completed 13.08 What is the integral of dx / (1 + cos x) ? [4:16]
- Completed 13.09 What is d/dx integral sin t dt from t = 0 to t = x^2? [3:51]
- Completed 13.10 Formally, why is the fundamental theorem of calculus true? [6:31]
- Completed
13.11 Without resorting to the fundamental theorem, why does substitution work? [3:47]
Subtitles (text) for 13.11 Without resorting to the fundamental theorem, why does substitution work? [3:47]Subtitles (srt) for 13.11 Without resorting to the fundamental theorem, why does substitution work? [3:47]Video (MP4) for 13.11 Without resorting to the fundamental theorem, why does substitution work? [3:47]
Week 12: Fundamental theorem of calculus
(collapsed, click to expand)- Completed 12.00 What is the big deal about the fundamental theorem of calculus? [2:13]
- Completed 12.01 What is the fundamental theorem of calculus? [5:32]
- Completed
12.02 How can I use the fundamental theorem of calculus to evaluate integrals? [6:06]
Subtitles (text) for 12.02 How can I use the fundamental theorem of calculus to evaluate integrals? [6:06]Subtitles (srt) for 12.02 How can I use the fundamental theorem of calculus to evaluate integrals? [6:06]Video (MP4) for 12.02 How can I use the fundamental theorem of calculus to evaluate integrals? [6:06]
- Completed 12.03 What is the integral of sin x dx from x = 0 to x = pi? [3:32]
- Completed 12.04 What is the integral of x^4 dx from x = 0 to x = 1? [4:15]
- Completed 12.05 What is the area between the graphs of y = sqrt(x) and y = x^2? [6:26]
- Completed 12.06 What is the area between the graphs of y = x^2 and y = 1 - x^2? [6:30]
- Completed 12.07 What is the accumulation function for sqrt(1-x^2)? [8:39]
- Completed 12.08 Why does the Euler method resemble a Riemann sum? [4:29]
- Completed 12.09 In what way is summation like integration? [2:31]
- Completed 12.10 What is the sum of n^4 for n = 1 to n = k? [9:24]
- Completed 12.11 Physically, why is the fundamental theorem of calculus true? [4:00]
- Completed 12.12 What is d/da integral f(x) dx from x = a to x = b? [5:06]
Week 11: Integrals
(collapsed, click to expand)- Completed 11.00 If we are not differentiating, what are we going to do? [2:57]
- Completed 11.01 How can I write sums using a big Sigma? [5:10]
- Completed 11.02 What is the sum 1 + 2 + ... + k? [6:11]
- Completed 11.03 What is the sum of the first k odd numbers? [4:15]
- Completed 11.04 What is the sum of the first k perfect squares? [6:47]
- Completed 11.05 What is the sum of the first k perfect cubes? [5:57]
- Completed 11.06 What does area even mean? [7:09]
- Completed 11.07 How can I approximate the area of a curved region? [9:57]
- Completed 11.08 What is the definition of the integral of f(x) from x = a to b? [5:48]
- Completed 11.09 What is the integral of x^2 from x = 0 to 1? [8:08]
- Completed 11.10 What is the integral of x^3 from x = 1 to 2? [8:35]
- Completed 11.11 When is the accumulation function increasing? Decreasing? [4:44]
- Completed 11.12 What sorts of properties does the integral satisfy? [4:42]
- Completed 11.13 What is the integral of sin x dx from -1 to 1? [3:15]
Week 10: Antiderivatives
(collapsed, click to expand)- Completed 10.00 What does it mean to antidifferentiate? [2:20]
- Completed 10.01 How do we handle the fact that there are many antiderivatives? [5:26]
- Completed 10.02 What is the antiderivative of a sum? [3:42]
- Completed 10.03 What is an antiderivative for x^n? [7:36]
- Completed 10.04 What is the most general antiderivative of 1/x? [4:14]
- Completed 10.05 What are antiderivatives of trigonometric functions? [5:44]
- Completed 10.06 What are antiderivatives of e^x and natural log? [2:44]
- Completed 10.07 How difficult is factoring compared to multiplying? [5:30]
- Completed 10.08 What is an antiderivative for e^(-x^2)? [4:49]
- Completed 10.09 What is the antiderivative of f(mx+b)? [5:18]
- Completed 10.10 Knowing my velocity, what is my position? [3:16]
- Completed 10.11 Knowing my acceleration, what is my position? [4:24]
- Completed 10.12 What is the antiderivative of sine squared? [3:18]
- Completed 10.13 What is a slope field? [4:56]
Week 9: Linear approximation
(collapsed, click to expand)- Completed 9.00 What is up with all the numerical analysis this week? [1:34]
- Completed 9.01 Where does f(x+h) = f(x) + h f'(x) come from? [5:59]
- Completed 9.02 What is the volume of an orange rind? [6:40]
- Completed 9.03 What happens if I repeat linear approximation? [10:33]
- Completed 9.04 Why is log 3 base 2 approximately 19/12? [10:21]
- Completed 9.05 What does dx mean by itself? [5:38]
- Completed 9.06 What is Newton's method? [9:55]
- Completed 9.07 What is a root of the polynomial x^5 + x^2 - 1? [6:55]
- Completed 9.08 How can Newton's method help me to divide quickly? [7:24]
- Completed 9.09 What is the mean value theorem? [6:51]
- Completed 9.10 Why does f'(x) > 0 imply that f is increasing? [5:10]
- Completed 9.11 Should I bother to find the point c in the mean value theorem? [4:27]
Week 8: Optimization
(collapsed, click to expand)- Completed 8.00 What sorts of optimization problems will calculus help us solve? [1:38]
- Completed 8.01 What is the extreme value theorem? [8:56]
- Completed 8.02 How do I find the maximum and minimum values of f on a given domain? [9:06]
- Completed 8.03 Why do we have to bother checking the endpoints? [4:15]
- Completed
8.04 Why bother considering points where the function is not differentiable? [7:17]
Subtitles (text) for 8.04 Why bother considering points where the function is not differentiable? [7:17]Subtitles (srt) for 8.04 Why bother considering points where the function is not differentiable? [7:17]Video (MP4) for 8.04 Why bother considering points where the function is not differentiable? [7:17]
- Completed 8.05 How can you build the best fence for your sheep? [8:49]
- Completed 8.06 How large can xy be if x + y = 24? [5:42]
- Completed 8.07 How do you design the best soup can? [10:32]
- Completed 8.08 Where do three bubbles meet? [12:45]
- Completed
8.09 How large of an object can you carry around a corner? [10:32]
Model Hallway for 8.09 How large of an object can you carry around a corner? [10:32]Subtitles (text) for 8.09 How large of an object can you carry around a corner? [10:32]Subtitles (srt) for 8.09 How large of an object can you carry around a corner? [10:32]Video (MP4) for 8.09 How large of an object can you carry around a corner? [10:32]
- Completed 8.10 How short of a ladder will clear a fence? [4:03]
Week 7: Applications of differentiation
(collapsed, click to expand)- Completed 7.00 What applications of the derivative will we do this week? [1:22]
- Completed 7.01 How can derivatives help us to compute limits? [9:26]
- Completed 7.02 How can l'Hôpital help with limits not of the form 0/0? [14:43]
- Completed 7.03 Why shouldn't I fall in love with l'Hôpital? [8:14]
- Completed 7.04 How long until the gray goo destroys Earth? [3:46]
- Completed 7.05 What does a car sound like as it drives past? [3:57]
- Completed 7.06 How fast does the shadow move? [5:11]
- Completed 7.07 How fast does the ladder slide down the building? [3:50]
- Completed
7.08 How quickly does a bowl fill with green water? [4:07]
Volume of Water for 7.08 How quickly does a bowl fill with green water? [4:07]Water Height for 7.08 How quickly does a bowl fill with green water? [4:07]Radius of Bowl for 7.08 How quickly does a bowl fill with green water? [4:07]Subtitles (text) for 7.08 How quickly does a bowl fill with green water? [4:07]Subtitles (srt) for 7.08 How quickly does a bowl fill with green water? [4:07]Video (MP4) for 7.08 How quickly does a bowl fill with green water? [4:07]
- Completed 7.09 How quickly does the water level rise in a cone? [7:00]
- Completed 7.10 How quickly does a balloon fill with air? [3:45]
Week 6: Derivatives of transcendental functions
(collapsed, click to expand)- Completed 6.00 What are transcendental functions? [2:03]
- Completed 6.01 Why does trigonometry work? [3:12]
- Completed 6.02 Why are there these other trigonometric functions? [4:48]
- Completed 6.03 What is the derivative of sine and cosine? [10:04]
- Completed 6.04 What is the derivative of tan x? [9:25]
- Completed 6.05 What are the derivatives of the other trigonometric functions? [5:35]
- Completed 6.06 What is the derivative of sin(x^2)? [4:36]
- Completed 6.07 What are inverse trigonometric functions? [4:32]
- Completed
6.08 What are the derivatives of inverse trig functions? [10:26]
Table of Arccosines for 6.08 What are the derivatives of inverse trig functions? [10:26]Table of Cosines for 6.08 What are the derivatives of inverse trig functions? [10:26]Subtitles (text) for 6.08 What are the derivatives of inverse trig functions? [10:26]Subtitles (srt) for 6.08 What are the derivatives of inverse trig functions? [10:26]Video (MP4) for 6.08 What are the derivatives of inverse trig functions? [10:26]
- Completed 6.09 Why do sine and cosine oscillate? [4:39]
- Completed 6.10 How can we get a formula for sin(a+b)? [4:15]
- Completed 6.11 How can I approximate sin 1? [3:25]
- Completed
6.12 How can we multiply numbers with trigonometry? [4:11]
Table of Arccosines for 6.12 How can we multiply numbers with trigonometry? [4:11]Table of Cosines for 6.12 How can we multiply numbers with trigonometry? [4:11]Subtitles (text) for 6.12 How can we multiply numbers with trigonometry? [4:11]Subtitles (srt) for 6.12 How can we multiply numbers with trigonometry? [4:11]Video (MP4) for 6.12 How can we multiply numbers with trigonometry? [4:11]
Week 5: Chain Rule
(collapsed, click to expand)- Completed 5.00 Is there anything more to learn about derivatives? [1:00]
- Completed 5.01 What is the chain rule? [10:32]
- Completed 5.02 What is the derivative of (1+2x)^5 and sqrt(x^2 + 0.0001)? [7:04]
- Completed 5.03 What is implicit differentiation? [5:34]
- Completed 5.04 What is the folium of Descartes? [4:40]
- Completed
5.05 How does the derivative of the inverse function relate to the derivative of the original function? [10:20]
Subtitles (text) for 5.05 How does the derivative of the inverse function relate to the derivative of the original function? [10:20]Subtitles (srt) for 5.05 How does the derivative of the inverse function relate to the derivative of the original function? [10:20]Video (MP4) for 5.05 How does the derivative of the inverse function relate to the derivative of the original function? [10:20]
- Completed 5.06 What is the derivative of log? [6:55]
- Completed 5.07 What is logarithmic differentiation? [4:24]
- Completed
5.08 How can we multiply quickly? [8:48]
Slide Rule for 5.08 How can we multiply quickly? [8:48]Quartersquares for 5.08 How can we multiply quickly? [8:48]Log Table for 5.08 How can we multiply quickly? [8:48]Subtitles (text) for 5.08 How can we multiply quickly? [8:48]Subtitles (srt) for 5.08 How can we multiply quickly? [8:48]Video (MP4) for 5.08 How can we multiply quickly? [8:48]
- Completed 5.09 How do we justify the power rule? [11:17]
- Completed 5.10 How can logarithms help to prove the product rule? [3:28]
- Completed 5.11 How do we prove the quotient rule? [5:01]
- Completed 5.12 BONUS How does one prove the chain rule? [6:48]
Week 4: Techniques of differentiation
(collapsed, click to expand)- Completed 4.00 What will Week 4 bring us? [1:21]
- Completed 4.01 What is the derivative of f(x) g(x)? [6:46]
- Completed 4.02 Morally, why is the product rule true? [6:15]
- Completed 4.03 How does one justify the product rule? [6:10]
- Completed 4.04 What is the quotient rule? [4:09]
- Completed 4.05 How can I remember the quotient rule? [5:57]
- Completed 4.06 What is the meaning of the derivative of the derivative? [11:03]
- Completed 4.07 What does the sign of the second derivative encode? [4:26]
- Completed 4.08 What does d/dx mean by itself? [4:05]
- Completed 4.09 What are extreme values? [7:22]
- Completed 4.10 How can I find extreme values? [9:54]
- Completed 4.11 Do all local minimums look basically the same when you zoom in? [3:55]
- Completed 4.12 How can I sketch a graph by hand? [7:28]
- Completed 4.13 What is a function which is its own derivative? [9:01]
Week 3: Derivatives
(collapsed, click to expand)- Completed 3.00 What comes next? Derivatives? [1:37]
- Completed 3.01 What is the definition of derivative? [6:34]
- Completed 3.02 What is a tangent line? [3:28]
- Completed 3.03 Why is the absolute value function not differentiable? [2:38]
- Completed 3.04 How does wiggling x affect f(x)? [3:29]
- Completed 3.05 Why is sqrt(9999) so close to 99.995? [5:43]
- Completed 3.06 What information is recorded in the sign of the derivative? [4:13]
- Completed 3.07 Why is a differentiable function necessarily continuous? [6:01]
- Completed 3.08 What is the derivative of a constant multiple of f(x)? [4:53]
- Completed 3.09 Why is the derivative of x^2 equal to 2x? [12:21]
- Completed 3.10 What is the derivative of x^n? [7:31]
- Completed 3.11 What is the derivative of x^3 + x^2? [5:07]
- Completed 3.12 Why is the derivative of a sum the sum of derivatives? [4:48]
Week 2: Infinity and continuity
(collapsed, click to expand)- Completed 2.00 Where are we in the course? [1:22]
- Completed 2.01 What is a one-sided limit? [3:45]
- Completed 2.02 What does "continuous" mean? [5:01]
- Completed 2.03 What is the intermediate value theorem? [2:23]
- Completed 2.04 How can I approximate root two? [10:20]
- Completed 2.05 Why is there an x so that f(x) = x? [5:12]
- Completed 2.06 What does lim f(x) = infinity mean? [5:24]
- Completed 2.07 What is the limit f(x) as x approaches infinity? [4:43]
- Completed 2.08 Why is infinity not a number? [6:21]
- Completed 2.09 What is the difference between potential and actual infinity? [2:49]
- Completed 2.10 What is the slope of a staircase? [6:50]
- Completed 2.11 How fast does water drip from a faucet? [5:21]
- Completed 2.12 BONUS What is the official definition of limit? [3:34]
- Completed 2.13 BONUS Why is the limit of x^2 as x approaches 2 equal to 4? [4:59]
- Completed 2.14 BONUS Why is the limit of 2x as x approaches 10 equal to 20? [2:17]
Week 1: Functions and limits
(collapsed, click to expand)- Completed 1.00 Who will help me? [1:46]
- Completed 1.01 What is a function? [11:19]
- Completed 1.02 When are two functions the same? [5:57]
- Completed 1.03 How can more functions be made? [3:25]
- Completed 1.04 What are some real-world examples of functions? [6:56]
- Completed 1.05 What is the domain of square root? [15:56]
- Completed 1.06 What is the limit of (x^2 - 1)/(x-1)? [8:48]
- Completed 1.07 What is the limit of (sin x)/x? [6:10]
- Completed 1.08 What is the limit of sin (1/x)? [8:17]
- Completed 1.09 Morally, what is the limit of a sum? [6:14]
- Completed 1.10 What is the limit of a product? [2:13]
- Completed 1.11 What is the limit of a quotient? [9:17]
- Completed 1.12 How fast does a ball move? [16:42]