  # Highlights of Calculus

## Video Lectures

Displaying all 18 video lectures.
I. Introduction
Lecture 1 Play Video
Faculty Introduction
In this video lecture, Prof. Gilbert Strang talks informally in his office at MIT about why he created this video series, and how MIT OpenCourseWare users can benefit from these materials.
II. Highlights of Calculus
Lecture 2 Play Video
Big Picture of Calculus
Calculus is about change. One function tells how quickly another function is changing. In this video lecture, Prof. Gilbert Strang shows how Calculus applies to ordinary life situations, such as:

- Driving a car
- Climbing a mountain
- Growing to full adult height
Lecture 3 Play Video
Big Picture: Derivatives
Calculus finds the relationship between the distance traveled and the speed — easy for constant speed, not so easy for changing speed. In this video lecture, Prof. Gilbert Strang finds the "rate of change" and the "slope of a curve" and the "derivative of a function."

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Lecture 4 Play Video
Max and Min and Second Derivative
In this video lecture, Prof. Gilbert Strang discusses Max and Min and Second Derivatives. At the top and bottom of a curve (Max and Min), the slope is zero. The "second derivative" shows whether the curve is bending down or up. Here is a real-world example of a minimum problem: What route from home to work takes the shortest time?
Lecture 5 Play Video
The Exponential Function
In this video lecture, Prof. Gilbert Strang explains how the "magic number e" connects to ordinary things like the interest on a bank account. The graph of y = e^x has the special property that its slope equals its height (it goes up "exponentially fast"). This is the great function of Calculus.
Lecture 6 Play Video
Big Picture: Integrals
The second half of calculus looks for the distance traveled even when the speed is changing. Finding this "integral" is the opposite of finding the derivative. In this video lecture, Prof. Gilbert Strang explains how the integral adds up little pieces to recover the total distance: "I know the speed at each moment of my trip, so how far did I go?"
III. Derivatives
Lecture 7 Play Video
Derivative of sin x and cos x
The two key functions of oscillation have specially neat derivatives:
- The slope of sin x is cos x
- The slope of cos x is - sin x

These come from one crucial fact: (sin x) / x approaches 1 at x = 0. This checks that the slope of sin x is cos 0 = 1 at the all-important point x = 0. In this video lecture, Prof. Gilbert Strang connects sine and cosine to moving around a circle, or up and down for a spring, or in and out for your lungs.
Lecture 8 Play Video
Product Rule and Quotient Rule
In this video lecture, Prof. Gilbert Strang discusses Product Rule and Quotient Rule.

How to find the slope of f(x) times g(x)?
Use the Product Rule.
The slope of f(x)g(x) has two terms:
f(x) times (slope of g(x)) PLUS g(x) times (slope of f(x))The Quotient Rule gives the slope of f(x) / g(x). That slope is[[ g(x) times (slope of f(x)) MINUS f(x) times (slope of g(x)) ]] / g squaredThese rules plus the Chain Rule will take you a long way.
Lecture 9 Play Video
Chains f(g(x)) and the Chain Rule
In this video lecture, Prof. Gilbert Strang discusses Chains f(g(x)) and the Chain Rule.
A chain of functions starts with y = g(x). Then it finds z = f(y). So z = f(g(x)). Very many functions are built this way, g inside f . So we need their slopes.The Chain Rule says: Multiply the Slopes of f and g.Find dy/dx for g(x). Then find dz/dy for f(y).Since dz/dy is found in terms of y, substitute g(x) in place of y.The way to remember the slope of the chain is dz/dx = dz/dy times dy/dx.Remove y to get a function of x. The slope of z = sin (3x) is 3 cos (3x).
Lecture 10 Play Video
Limits and Continuous Functions
In this video lecture, Prof. Gilbert Strang discusses Limits and Continuous Functions. What does it mean to say that a sequence of numbers a1, a2, ... approaches a LIMIT A? This means: For any little interval around A, the numbers eventually get in there and stay there.The numbers a1 = 1/2, a2 = 2/3, a3 = 3/4, ... approach the limit 1. The first a's don't matter. Change 2000 a's and the limit is still 1. What about powers of the a's like a1^b1 a2^b2...? If the b's approach B then those powers approach A^B except danger if B = 0 or infinity. For calculus the important case where you can't tell by just knowing A and B is A/B = 0/0. If f(x) and g(x) both get small (f/g looks like 0/0) then l'Hopital looks at slopes: f/g goes like f '/g'. When is f(x) continuous at x=a? This means: f(x) is close to f(a) when x is close to a.
Lecture 11 Play Video
Inverse Functions f ^-1 (y) and the Logarithm x = ln y
In this video lecture, Prof. Gilbert Strang discusses Inverse Functions f ^-1 (y) and the Logarithm x = ln y.

For the usual y = f(x), the input is x and the output is y. For the inverse function x = f^-1(y), the input is y and the output is x. If y equals x cubed, then x is the cube root of y: that is the inverse. If y is the great function e^x, then x is the natural logarithm ln y. Start at y, go to x = ln y, then back to y = e^(ln y). So the logarithm is the exponent that produces y. The logarithm of y = e^5 is ln y = 5. Logarithms grow very slowly.
Lecture 12 Play Video
Derivatives of ln y and sin ^-1 (y)
Make a chain of a function and its inverse: f^-1(f(x)) = x starts with x and ends with x.
Take the slope using the Chain Rule. On the right side the slope of x is 1.
Chain Rule: dx/dy dy/dx = 1
Here this says that df^-1/dy times df/dx equals 1.
So the derivative of f^-1(y) is 1/ (df/dx) but you have to write df/dx in terms of y.
The derivative of ln y is 1/ (derivative of f = e^x) = 1/e^x. This is 1/y, a neat slope. Changing letters is OK: The derivative of ln x is 1/x. Watch this video for graphs.
Lecture 13 Play Video
Growth Rate and Log Graphs
It is good to know how fast different functions grow. Professor Strang puts them in order from slow to fast: logarithm of x, powers of x, exponential of x, x factorial, x to the x power; what is even faster?

And it is good to know how graphs can show the key numbers in the growth rate of a function. A log-log graph plots log y against log x. If y = A x^n then log y = log A + n log x = line with slope nA semilog graph plots log y against x If y = A 10^cx then log y = log A + cx = line with slope c. You will never see y = 0 on these graphs because log 0 is minus infinity. But n and c jump out clearly.
Lecture 14 Play Video
Linear Approximation/Newton's Method
The slope of a function y(x) is the slope of its tangent line. Close to x=a, the line with slope y ' (a) gives a "linear" approximation.

y(x) is close to y(a) + (x - a) times y ' (a)
If you want to solve y(x) = 0, choose x so that y(a) + (x - a) y ' (a) = 0
This is a really fast way to get close to the exact solution to y(x) = 0:
"Newton's Method" -- x = a - y(a)/y '(a)
Lecture 15 Play Video
Power Series/Euler's Great Formula
In this course, Prof. Gilbert Strang discusses Power Series and Euler's Great Formula.

A special power series is e^x = 1 + x + x^2 / 2! + x^3 / 3! + ... + every x^n / n!
The series continues forever but for any x it adds up to the number e^x

If you multiply each x^n / n! by the nth derivative of f(x) at x = 0, the series adds to f(x). This is a Taylor Series. Of course, all those derivatives are 1 for e^x.

Two great series are cos x = 1 - x^2 / 2! + x^4 / 4!... and sin x = x - x^3 / 3!... cosine has even powers, sine has odd powers, both have alternating plus/minus signs. Fermat saw magic using i^2 = -1. Then e^ix exactly matches cos x + i sin x.
Lecture 16 Play Video
Differential Equations of Motion
These equations have 2nd derivatives because acceleration is in Newton's Law F = ma. The key model equation is (second derivative) y'' = minus y or y'' = minus a^2 y.
There are two solutions since the equation is second order. They are sine and cosine.
y = sin (at) and y = cos (at). Two derivatives bring back sine and cosine with minus a^2.
The next step allows damping (first derivative) as in my'' + dy' + ky = 0. How to solve? Just try y = e^at. You find that ma^2 + da + k = 0. Two a's give two solutions.
Lecture 17 Play Video
Differential Equations of Growth
In this video lecture, Prof. Gilbert Strang discusses Differential Equations of Growth.

The key model for growth (or decay when c < 0) is dy/dt = c y(t).
The next model allows a steady source (constant s in dy/dt = cy + s).
The solutions include an exponential e^ct (because its derivative brings down c).
So growth forever if c is positive and decay if c is negative.
A neat model for the population P(t) adds in minus sP^2 (so P won't grow forever).
This is nonlinear but luckily the equation for y = 1/P is linear and we solve it.

Population P follows an "S-curve" reaching a number like 10 or 11 billion. Great lecture but Professor Strang should have written e^-ct in the last formula
Lecture 18 Play Video
Six Functions, Six Rules, and Six Theorems
In this video lecture, Prof. Gilbert Strang discusses Six Functions, Six Rules, and Six Theorems.

This lecture compresses all the others into one fast video for review of derivatives.
Five of the 6 functions are old, the new one is a STEP function.

Slope = delta function.
The 6 rules cover f + g, f times g, f divided by g, chains f(g(x)), inverse of f, and then L'Hopital for 0/0.
The 6 theorems include the Fundamental Theorem of Calculus for Integral of Derivative of f(x).
Function 1 is f(x) Function 2 is its slope (rate of change). Add up those changes to recover f(x).
The Mean Value Theorem says that if your average speed is 70, then instant speed is 70 at least once. The Binomial Theorem tells you the series that adds up to the pth power f(x) = (1 + x)^p.