Calculus One

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Calculus One

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About this course: Calculus is about the very large, the very small, and how things change. The surprise is that something seemingly so abstract ends up explaining the real world. Calculus plays a starring role in the biological, physical, and social sciences. By focusing outside of the classroom, we will see examples of calculus appearing in daily life. This course is a first and friendly introduction to calculus, suitable for someone who has never seen the subject before, or for someone who has seen some calculus but wants to review the concepts and practice applying those concepts to solve problems.

Created by:  The Ohio State University
  • Taught by:  Jim Fowler, PhD, P…

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When you enroll for courses through Coursera you get to choose for a paid plan or for a free plan

  • Free plan: No certicification and/or audit only. You will have access to all course materials except graded items.
  • Paid plan: Commit to earning a Certificate—it's a trusted, shareable way to showcase your new skills.

About this course: Calculus is about the very large, the very small, and how things change. The surprise is that something seemingly so abstract ends up explaining the real world. Calculus plays a starring role in the biological, physical, and social sciences. By focusing outside of the classroom, we will see examples of calculus appearing in daily life. This course is a first and friendly introduction to calculus, suitable for someone who has never seen the subject before, or for someone who has seen some calculus but wants to review the concepts and practice applying those concepts to solve problems.

Created by:  The Ohio State University
  • Taught by:  Jim Fowler, PhD, Professor

    Mathematics
Commitment 25 hours of videos and quizzes Language English How To Pass Pass all graded assignments to complete the course. User Ratings 4.8 stars Average User Rating 4.8See what learners said Trabajo del curso

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Syllabus


WEEK 1


Welcome to Calculus One
Welcome to Calculus! Join me on this journey through one of the great triumphs of human thought.


2 videos, 2 readings expand


  1. Video: Why is calculus going to be so much fun?
  2. Leyendo: How to Use Discussion Forums
  3. Video: How is this course structured?
  4. Leyendo: Get to Know Your Classmates


WEEK 2


Functions and Limits
Functions are the main star of our journey. Calculus isn't numbers: it's relationships between things, and how one thing changing affects something else.


13 videos, 5 practice quizzes expand


  1. Video: How do we get started with calculus?
  2. Video: What is a function?
  3. Video: When are two functions the same?
  4. Video: How can more functions be made?
  5. Cuestionario de práctica: Functions? What's a Function?
  6. Video: What are some real-world examples of functions?
  7. Video: What is the domain of square root?
  8. Cuestionario de práctica: Functions in the Real World
  9. Video: Morally, what is the limit of a sum?
  10. Video: What is the limit of sin (1/x)?
  11. Video: What is the limit of (sin x)/x?
  12. Video: What is the limit of (x^2 - 1)/(x-1)?
  13. Cuestionario de práctica: Limits? What's a Limit?
  14. Video: What is the limit of a product?
  15. Video: What is the limit of a quotient?
  16. Cuestionario de práctica: Working with Limits
  17. Video: How fast does a ball move?
  18. Cuestionario de práctica: Limits in Motion

Graded: Functions and Limits

WEEK 3


The End of Limits
People have thought about infinity for thousands of years; limits provide one way to make such ponderings precise. Continuity makes precise the idea that small changes in the input don't affect the output much.


15 videos, 4 practice quizzes expand


  1. Video: What else is there to study about functions and limits?
  2. Video: What is a one-sided limit?
  3. Video: What does "continuous" mean?
  4. Video: What is the intermediate value theorem?
  5. Video: How can I approximate root two?
  6. Cuestionario de práctica: Continuity
  7. Video: Why is there an x so that f(x) = x?
  8. Video: What does lim f(x) = infinity mean?
  9. Video: What is the limit f(x) as x approaches infinity?
  10. Video: Why is infinity not a number?
  11. Cuestionario de práctica: Infinity
  12. Video: What is the difference between potential and actual infinity?
  13. Video: What is the slope of a staircase?
  14. Video: How fast does water drip from a faucet?
  15. Cuestionario de práctica: Slope
  16. Video: What is the official definition of limit?
  17. Video: Why is the limit of x^2 as x approaches 2 equal to 4?
  18. Video: Why is the limit of 2x as x approaches 10 equal to 20?
  19. Cuestionario de práctica: Limit Definition

Graded: The End of Limits

WEEK 4


The Beginning of Derivatives
It is time to change topics, or rather, to study change itself! When we wiggle the input, the output value changes, and that ratio of output change to input change is the derivative.


13 videos, 4 practice quizzes expand


  1. Video: What comes next? Derivatives?
  2. Video: What is the definition of derivative?
  3. Video: What is a tangent line?
  4. Video: Why is the absolute value function not differentiable?
  5. Video: How does wiggling x affect f(x)?
  6. Cuestionario de práctica: What are derivatives?
  7. Video: Why is sqrt(9999) so close to 99.995?
  8. Video: What information is recorded in the sign of the derivative?
  9. Cuestionario de práctica: Why would I care to find the derivative?
  10. Video: Why is a differentiable function necessarily continuous?
  11. Video: What is the derivative of a constant multiple of f(x)?
  12. Cuestionario de práctica: How do differentiability and continuity relate?
  13. Video: Why is the derivative of x^2 equal to 2x?
  14. Video: What is the derivative of x^n?
  15. Video: What is the derivative of x^3 + x^2?
  16. Video: Why is the derivative of a sum the sum of derivatives?
  17. Cuestionario de práctica: How do I find the derivative?

Graded: The Beginning of Derivatives

WEEK 5


Techniques of Differentiation
With the product rule and the quotient rule, we can differentiate products and quotients. And since the derivative is a function, we can differentiate the derivative to get the second derivative.


14 videos, 5 practice quizzes expand


  1. Video: How do we compute derivatives?
  2. Video: What is the derivative of f(x) g(x)?
  3. Video: Morally, why is the product rule true?
  4. Video: How does one justify the product rule?
  5. Cuestionario de práctica: How do I differentiate a product?
  6. Video: What is the quotient rule?
  7. Video: How can I remember the quotient rule?
  8. Cuestionario de práctica: How do I differentiate a quotient?
  9. Video: What is the meaning of the derivative of the derivative?
  10. Video: What does the sign of the second derivative encode?
  11. Video: What does d/dx mean by itself?
  12. Cuestionario de práctica: Higher Derivatives
  13. Video: What are extreme values?
  14. Video: How can I find extreme values?
  15. Video: Do all local minimums look basically the same when you zoom in?
  16. Video: How can I sketch a graph by hand?
  17. Cuestionario de práctica: How do I sketch a graph without a computer?
  18. Video: What is a function which is its own derivative?
  19. Cuestionario de práctica: How do I differentiate e^x?

Graded: Techniques of Differentiation

WEEK 6


Chain Rule
The chain rule lets us differentiate the composition of two functions. The chain rule can be used to compute the derivative of inverse functions, too.


13 videos, 5 practice quizzes expand


  1. Video: Is there anything more to learn about derivatives?
  2. Video: What is the chain rule?
  3. Video: What is the derivative of (1+2x)^5 and sqrt(x^2 + 0.0001)?
  4. Cuestionario de práctica: What is the Chain Rule?
  5. Video: What is implicit differentiation?
  6. Video: What is the folium of Descartes?
  7. Cuestionario de práctica: How do I find the tangent line to a curve?
  8. Video: How does the derivative of the inverse function relate to the derivative of the original function?
  9. Video: What is the derivative of log?
  10. Video: What is logarithmic differentiation?
  11. Cuestionario de práctica: How do I find the derivative of an inverse function?
  12. Video: How can we multiply quickly?
  13. Cuestionario de práctica: How can I multiply quickly?
  14. Video: How do we justify the power rule?
  15. Video: How can logarithms help to prove the product rule?
  16. Video: How do we prove the quotient rule?
  17. Video: BONUS: How does one prove the chain rule?
  18. Cuestionario de práctica: How do I justify the derivative rules?

Graded: Chain Rule

WEEK 7


Derivatives of Transcendental (Trigonometric) Functions
So far, we can differentiate polynomials, exponential functions, and logarithms. Let's learn how to differentiate trigonometric functions.


14 videos, 5 practice quizzes expand


  1. Video: What are derivatives of transcendental functions?
  2. Video: What are transcendental functions?
  3. Video: Why does trigonometry work?
  4. Video: Why are there these other trigonometric functions?
  5. Cuestionario de práctica: What is trigonometry?
  6. Video: What is the derivative of sine and cosine?
  7. Video: What is the derivative of tan x?
  8. Video: What is the derivative of sin(x^2)?
  9. Video: What are the derivatives of the other trigonometric functions?
  10. Cuestionario de práctica: How can I differentiate trig functions?
  11. Video: What are inverse trigonometric functions?
  12. Video: What are the derivatives of inverse trig functions?
  13. Cuestionario de práctica: How can I differentiate inverse trig functions?
  14. Video: Why do sine and cosine oscillate?
  15. Video: How can we get a formula for sin(a+b)?
  16. Video: How can I approximate sin 1?
  17. Cuestionario de práctica: What can we learn from the derivatives?
  18. Video: How can we multiply numbers with trigonometry?
  19. Cuestionario de práctica: Multiplying Trigonometric Functions with Slide Rules

Graded: Derivatives of Transcendental (Trigonometric) Functions

WEEK 8


Derivatives in the Real World
Derivatives can be used to calculate limits via l'Hôpital's rule. Given a real-world equation involving two changing quantities, differentiating yields "related rates."


11 videos, 3 practice quizzes expand


  1. Video: Why would I ever want to take derivatives?
  2. Video: How can derivatives help us to compute limits?
  3. Video: How can l'Hôpital help with limits not of the form 0/0?
  4. Video: Why shouldn't I fall in love with l'Hôpital?
  5. Cuestionario de práctica: How can derivatives help with limits?
  6. Video: How long until the gray goo destroys Earth?
  7. Video: What does a car sound like as it drives past?
  8. Cuestionario de práctica: How can derivatives help me to understand rates of change in the real world?
  9. Video: How fast does the shadow move?
  10. Video: How fast does the ladder slide down the building?
  11. Video: How quickly does a bowl fill with green water?
  12. Video: How quickly does the water level rise in a cone?
  13. Video: How quickly does a balloon fill with air?
  14. Cuestionario de práctica: How do derivatives help me understand how two rates of change are related?

Graded: Derivatives in the Real World

WEEK 9


Optimization
In the real world, we must makes choices, and wouldn't it be great if we could make the best choice? Such optimization is made possible with calculus.


12 videos, 4 practice quizzes expand


  1. Video: Why is optimization part of this course?
  2. Video: What is the extreme value theorem?
  3. Video: What sorts of optimization problems will calculus help us solve?
  4. Cuestionario de práctica: Is optimization possible?
  5. Video: How do I find the maximum and minimum values of f on a given domain?
  6. Video: Why do we have to bother checking the endpoints?
  7. Video: Why bother considering points where the function is not differentiable?
  8. Video: How can you build the best fence for your sheep?
  9. Cuestionario de práctica: Okay, so if optimization is possible, how do I do it?
  10. Video: How large can xy be if x + y = 24?
  11. Video: How do you design the best soup can?
  12. Cuestionario de práctica: Why would I want to optimize a function?
  13. Video: Where do three bubbles meet?
  14. Video: How large of an object can you carry around a corner?
  15. Video: How short of a ladder will clear a fence?
  16. Cuestionario de práctica: Optimization in Action

Graded: Optimization

WEEK 10


Linear Approximation
Replacing the curved graph by a straight line approximation helps us to estimate values and roots.


13 videos, 5 practice quizzes expand


  1. Video: Why are we thinking about linear approximation?
  2. Video: What is up with all the numerical analysis this week?
  3. Video: Where does f(x+h) = f(x) + h f'(x) come from?
  4. Video: What is the volume of an orange rind?
  5. Cuestionario de práctica: What is linear approximation?
  6. Video: What happens if I repeat linear approximation?
  7. Video: Why is log 3 base 2 approximately 19/12?
  8. Cuestionario de práctica: What happens if I repeat linear approximation?
  9. Video: What does dx mean by itself?
  10. Cuestionario de práctica: What does dx mean by itself?
  11. Video: What is Newton's method?
  12. Video: What is a root of the polynomial x^5 + x^2 - 1?
  13. Video: How can Newton's method help me to divide quickly?
  14. Cuestionario de práctica: What is Newton's method?
  15. Video: What is the mean value theorem?
  16. Video: Why does f'(x) > 0 imply that f is increasing?
  17. Video: Should I bother to find the point c in the mean value theorem?
  18. Cuestionario de práctica: What is the mean value theorem?

Graded: Linear Approximation

WEEK 11


Antidifferentiation
Antidifferentiation is the process of untaking derivatives, of finding a function whose derivatives is a given function. Since it involves working backwards, antidifferentiation feels like "unbreaking a vase" and can be just as challenging.


14 videos, 4 practice quizzes expand


  1. Video: What does it mean to antidifferentiate?
  2. Video: How do we handle the fact that there are many antiderivatives?
  3. Cuestionario de práctica: But there are so many antiderivatives!
  4. Video: What is the antiderivative of a sum?
  5. Video: What is an antiderivative for x^n?
  6. Video: What is the most general antiderivative of 1/x?
  7. Video: What are antiderivatives of trigonometric functions?
  8. Video: What are antiderivatives of e^x and natural log?
  9. Cuestionario de práctica: How am I supposed to compute antiderivatives?
  10. Video: What is the antiderivative of f(mx+b)?
  11. Video: What is an antiderivative for e^(-x^2)?
  12. Video: How difficult is factoring compared to multiplying?
  13. Cuestionario de práctica: Why is this so hard?
  14. Video: Knowing my velocity, what is my position?
  15. Video: Knowing my acceleration, what is my position?
  16. Video: What is the antiderivative of sine squared?
  17. Video: What is a slope field?
  18. Cuestionario de práctica: Why would anybody want to do this?

Graded: Antidifferentiation

WEEK 12


Integration
By cutting up a curved region into thin rectangles and taking a limit of the sum of the areas of those rectangles, we compute (define!) the area of a curved region.


14 videos, 6 practice quizzes expand


  1. Video: If we are not differentiating, what are we going to do?
  2. Video: How can I write sums using a big Sigma?
  3. Video: What is the sum 1 + 2 + ... + k?
  4. Video: What is the sum of the first k odd numbers?
  5. Video: What is the sum of the first k perfect squares?
  6. Video: What is the sum of the first k perfect cubes?
  7. Cuestionario de práctica: What is summation notation?
  8. Video: What does area even mean?
  9. Cuestionario de práctica: What is area?
  10. Video: How can I approximate the area of a curved region?
  11. Video: What is the definition of the integral of f(x) from x = a to b?
  12. Cuestionario de práctica: So how do we calculate area precisely?
  13. Video: What is the integral of x^2 from x = 0 to 1?
  14. Video: What is the integral of x^3 from x = 1 to 2?
  15. Cuestionario de práctica: Can we compute any of these integrals?
  16. Video: What sorts of properties does the integral satisfy?
  17. Video: When is the accumulation function increasing? Decreasing?
  18. Cuestionario de práctica: Can we understand anything conceptually about integrals?
  19. Video: What is the integral of sin x dx from -1 to 1?
  20. Cuestionario de práctica: Can we compute any other integrals?

Graded: Integration

WEEK 13


Fundamental Theorem of Calculus
Armed with the Fundamental Theorem of Calculus, evaluating a definite integral amounts to finding an antiderivative.


13 videos, 4 practice quizzes expand


  1. Video: What is the big deal about the fundamental theorem of calculus?
  2. Video: What is the fundamental theorem of calculus?
  3. Cuestionario de práctica: What is the fundamental theorem of calculus?
  4. Video: How can I use the fundamental theorem of calculus to evaluate integrals?
  5. Video: What is the integral of sin x dx from x = 0 to x = pi?
  6. Video: What is the integral of x^4 dx from x = 0 to x = 1?
  7. Cuestionario de práctica: How am I supposed to use the fundamental theorem of calculus?
  8. Video: What is the area between the graphs of y = sqrt(x) and y = x^2?
  9. Video: What is the area between the graphs of y = x^2 and y = 1 - x^2?
  10. Video: What is the accumulation function for sqrt(1-x^2)?
  11. Cuestionario de práctica: What else can we compute this way?
  12. Video: Why does the Euler method resemble a Riemann sum?
  13. Video: In what way is summation like integration?
  14. Video: What is the sum of n^4 for n = 1 to n = k?
  15. Video: Physically, why is the fundamental theorem of calculus true?
  16. Video: What is d/da integral f(x) dx from x = a to x = b?
  17. Cuestionario de práctica: But why is the fundamental theorem true?

Graded: Fundamental Theorem of Calculus

WEEK 14


Substitution Rule
Substitution systematizes the process of using the chain rule in reverse. Considering how often we used the chain rule when differentiating, we will often want to use it in reverse to antidifferentiate.


12 videos, 3 practice quizzes expand


  1. Video: How can we compute more antiderivatives?
  2. Video: How does the chain rule help with antidifferentiation?
  3. Video: When I do u-substitution, what should u be?
  4. Video: How should I handle the endpoints when doing u-substitution?
  5. Video: Might I want to do u-substitution more than once?
  6. Cuestionario de práctica: What is the chain rule backwards?
  7. Video: What is the integral of dx / (x^2 + 4x + 7)?
  8. Video: What is the integral of (x+10)(x-1)^10 dx from x = 0 to x = 1?
  9. Video: What is the integral of x / (x+1)^(1/3) dx?
  10. Video: What is the integral of dx / (1 + cos x) ?
  11. Cuestionario de práctica: What are some tricks for doing substitutions?
  12. Video: What is d/dx integral sin t dt from t = 0 to t = x^2?
  13. Video: Formally, why is the fundamental theorem of calculus true?
  14. Video: Without resorting to the fundamental theorem, why does substitution work?
  15. Cuestionario de práctica: What if I differentiate an accumulation function?

Graded: Substitution Rule

WEEK 15


Techniques of Integration
Integration by parts is the product rule in reverse. Integrals of powers of trigonometric functions can be evaluated.


10 videos, 4 practice quizzes expand


  1. Video: How will we find antiderivatives for more complicated expressions?
  2. Video: What antidifferentiation rule corresponds to the product rule in reverse?
  3. Video: What is an antiderivative of x e^x?
  4. Video: How does parts help when antidifferentiating log x?
  5. Video: What is an antiderivative of e^x cos x?
  6. Cuestionario de práctica: How do I do integration by parts?
  7. Video: What is an antiderivative of e^(sqrt(x))?
  8. Cuestionario de práctica: How do I know when to use parts?
  9. Video: What is an antiderivative of sin^(2n+1) x cos^(2n) x dx?
  10. Video: What is the integral of sin^(2n) x dx from x = 0 to x = pi?
  11. Video: What is the integral of sin^n x dx in terms of sin^(n-2) x dx?
  12. Cuestionario de práctica: How do I integrate powers of sines and cosines?
  13. Video: Why is pi < 22/7?
  14. Cuestionario de práctica: How does long division help?

Graded: Techniques of Integration

WEEK 16


Applications of Integration
We have already used integrals to compute area; integration can also be used to compute volumes.


10 videos, 3 practice quizzes expand


  1. Video: What can we do with integrals besides calculating area?
  2. Video: What happens when I use thin horizontal rectangles to compute area?
  3. Video: When should I use horizontal as opposed to vertical pieces?
  4. Cuestionario de práctica: How else can I calculate area?
  5. Video: What does "volume" even mean?
  6. Video: What is the volume of a sphere?
  7. Video: How do washers help to compute the volume of a solid of revolution?
  8. Video: What is the volume of a thin shell?
  9. Video: What is the volume of a sphere with a hole drilled in it?
  10. Cuestionario de práctica: How can I calculate volume?
  11. Video: What does "length" even mean?
  12. Video: On the graph of y^2 = x^3, what is the length of a certain arc?
  13. Cuestionario de práctica: How can I calculate length?

Graded: Applications of Integration

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