RES.18-005 Highlights of Calculus | Derivatives (12 videos)

Derivative of sin x and cos x

Strang, Gilbert — 2010-04-30T12:11:42+05:00

The two key functions of oscillation have specially neat derivatives: The slope of sin x is cos x ! And 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.

Professor Strang connects sine and cosine to moving around a circle,
or up and down for a spring, or in and out for your lungs.

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
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Video - download: Internet Archive (MP4)
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Product Rule and Quotient Rule

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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 squared

These rules plus the CHAIN RULE will take you a long way.

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

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Chains f(g(x)) and the Chain Rule

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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).

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Limits and Continuous Functions

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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. See end of video

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Inverse Functions f ^-1 (y) and the Logarithm x = ln y

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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.......

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Derivatives of ln y and sin ^-1 (y)

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Growth Rate and Log Graphs

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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 n

A 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.

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Linear Approximation/Newton's Method

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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) SEE THE EXAMPLES

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Power Series/Euler's Great Formula

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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.

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Differential Equations of Motion

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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: good

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Differential Equations of Growth

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)

Six Functions, Six Rules, and Six Theorems

Strang, Gilbert — 2010-04-30T12:11:42+05:00

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

Professor Strang's Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Transcript: PDF (English - US)
Subtitles: SRT (English - US)
Thumbnail - JPG (OCW)
Video - download: Internet Archive (MP4)
Video - download: iTunes U (MP4)
Video - stream: YouTube

(CC BY-NC-SA)