Lecture 32: Implicit Differentiation
32.1 Implicit differentiation
Example
of x,
Let C be the circle with equation x2 + y 2 = 25. Then, treating y as a function
d
d 2
(x + y 2 ) =
(25)
dx
dx
dy
2x + 2y
=0
dx
dy
x
= , y 6= 0.
dx
y
x2 + y 2 = 25
For examp
Lecture 31: Functions defined implicitly
31.1 Functions defined implicitly
Example
Let C be the circle with equation
x2 + y 2 = 25.
Note that C is not the graph of a function;
indeed, for every
value of x with 5 < x < 5,
2
there correspond two values of
Lecture 35: Relating Variables
35.1 Relations between variables
Example A spherical balloon is inflated with air. If, at a given time t, V is the volume
of the balloon and r is the radius of the balloon, then
V =
4 3
r .
3
If S is the surface area of the
Lecture 22: Derivatives
22.1 Definition of the derivative
The derivative of a function f at a point a, denoted f 0 (a), is defined by
f (a + h) f (a)
f 0 (a) = lim
,
h0
h
provided the limit exists.
Definition
Example
If f (x) = x2 3x, then
f (1 + h) f (1)
Lecture 19: Rates of Change: Tangent Lines
19.1 Tangent lines revisited
Recall: If m is the slope of the line tangent to the graph of f at (a, f (a), then we should
have, for small values of h 6= 0,
m
f (a + h) f (a)
.
h
In fact, we should have
f (a + h)
Lecture 23: Derivatives as Functions
23.1 The derivative as a function
Note: Given a function f , the new function f 0 defined by
f (x + h) f (x)
h0
h
f 0 (x) = lim
for all points at which the limit exists is called the derivative of f . We say f is diffe
Lecture 20: Rates of Change: Velocity
20.1 Velocity revisited
Recall: If f (t) is the position of an object moving along a straight line at time t and v(t)
is the velocity of the object at time t, then, for small values of h 6= 0, we should have
v(a)
and
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Lecture 26: Area
26.1 Summation notation
It will be useful in our work to use the following notation: If a1 , a2 , . . . , an are numbers,
then
n
X
ai = a1 + a2 + + an .
i=1
Example
If a1 = 1, a2 = 3, a3 = 1, and a4 = 6, then
4
X
ai = a1 + a2 + a3 + a4 =
Lecture 28: The Chain Rule: Introduction
28.1 Examples
Example
Let h(x) = (3x2 + 1)2 . Then
h(x) = 9x4 + 6x2 + 1,
so
h0 (x) = 36x3 + 12x.
Note that if we let g(x) = 3x2 +1 and f (x) = x2 , then h(x) = f (g(x) = f g(x). Moreover,
f 0 (x) = 2x, g 0 (x) = 6x
Lecture 30: Rational Powers
30.1 Rational powers
Note: If y is a function of x, then, by the chain rule,
d
dy
f (y) = f 0 (y) .
dx
dx
In particular, for any real number n 6= 0,
dy
d n
(y ) = ny n1 .
dx
dx
With the chain rule, we can now verify that
d n
x
Lecture 26: Differentiation of Powers and Products
26.1 Differentiation of Powers
Proposition
If n < 0 is an integer, then
d n
(x ) = nxn1 .
dx
Proof
d
d n
(x ) =
dx
dx
1
xn
=
(xn )(0) (1)(nxn1 )
= nxn1+2n = nxn1 .
x2n
1
1
, then f 0 (x) = (1)x2 = 2 .
x
x
Lecture 33: Higher Order Derivatives
33.1 Higher order derivatives
If f is differentiable, then its derivative f 0 may also be differentiable. The derivative of f 0
is called the second derivative of f and is denoted f 00 .
Example
If f (x) = x3 6x5 , the
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Lecture 1: Two Fundamental Problems of Calculus
1.1 Problem of area
Problem: Find the area of a region in the plane.
Example Find the area of the unit circle. Idea: If An is the area of a regular nsided
polygon inscribed in the unit circle, and A is the
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Lecture 27: The Definite Integral
27.1 Definition of the definite integral
Definition
Suppose f is continuous on [a, b], n is a positive integer,
x =
ba
,
n
xi = a + ix for i = 1, 2, 3, . . ., and xi is a point in [xi1 , xi ]. Then the definite integral o
Lecture 25: Differentiation of Quotients
25.1 Differentiation of quotients
Suppose f and g are differentiable and k(x) =
0
k (x) = lim
f (x+h)
g(x+h)
f (x)
. Then
g(x)
f (x)
g(x)
h
f (x + h)g(x) f (x)g(x + h)
= lim
h0
hg(x + h)g(x)
f (x + h)g(x) f (x)g(x)
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Lecture 3: Types of Functions
3.1 Basic definitions
Definition
tion
If a0 , a1 , . . . , an are constants and n is a nonnegative integer, then the funcf (x) = an xn + an1 xn1 + + a1 x + a0
is called a polynomial.
Example
Definition
f (x) = 3x2 + 4x 6 is a
Lecture 34: Higher Order Derivatives: Acceleration
34.1 Position, velocity, and acceleration
Suppose s = f (t) gives the position, at time t, of an object moving along a straight line.
Then
ds
= f 0 (t) = velocity of the object at time t
v(t) =
dt
and
a(t
Lecture 24: Differentiation of Polynomials
24.1 Differentiation of polynomials
Suppose f (x) = ax + b. Then we have
(a(x + h) + b) (ax + b)
ah
= lim
= a.
h0
h0 h
h
f 0 (x) = lim
Proposition
If f (x) = ax + b, then f 0 (x) = a.
Note: If c is a constant and
Lecture 17: Properties of Continuous Functions
17.1 Properties of continuous functions
Proposition
at a:
If f and g are continuous at a, then each of the following is also continuous
(1) f (x) + g(x)
(2) f (x) g(x)
(3) cf (x) for any constant c
(4) f (x)g
Lecture 36: Related Rates
36.1 Related rates
Example A spherical balloon is being inflated at the rate of 100 cubic centimeters per
second. When the radius is 10 centimeters, how fast is the the radius of the balloon
increasing?
Let V be the volume and r
Lecture 16: Continuity on an Interval
16.1 Extending the definition of continuity
Definition
We say a function f is continuous from the right at a point a if lim+ f (x) =
xa
f (a) and continuous from the left at a if lim f (x) = f (a).
xa
Example
Let
g(t)
Lecture 21: Rates of Change
21.1 Rates of change in general
In general, if y = f (x), x represents a small change in x, and
y = f (x + x) f (x)
y
represents the corresponding change in y, then
is the average rate of change of y with
x
respect to x and
y
f
Lecture 37: Linear Approximations
37.1 Linear approximations
Definition
If f is differentiable at a, the function
L(x) = f (a) + f 0 (a)(x a)
is called the linearization of f at a.
Note: The graph of L is the tangent line to the graph of f at (a, f (a).
I
Lecture 27: More on Rates of Change
27.1 Velocity
Recall: If s = f (t) specifies the position of an object moving along a straight line at time
t, then
ds
= f 0 (t)
dt
is the velocity of the object at time t.
Example Suppose an object is thrown vertically
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Lecture 6: Properties of Limits
6.1 Basic properties of limits
Proposition
Suppose lim f (x) = L and lim g(x) = M . Then
xa
xa
1. lim (f (x) + g(x) = L + M ,
xa
2. lim (f (x) g(x) = L M ,
xa
3. lim cf (x) = cL for any constant c,
xa
4. lim (f (x)g(x) = LM
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Lecture 18: Maximum and Minimum Values
18.1 Maximum and minimum values
Definition Let D be the domain of the function f . If f (c) f (x) for all x in D, then
we say f has an absolute maximum at c and we call f (c) the maximum value of f on D. If
f (c) f (
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Lecture 17: Linear Approximations
17.1 Linear approximations
Definition
If f is differentiable at a, the function
L(x) = f (a) + f 0 (a)(x a)
is called the linearization of f at a.
Note: The graph of L is the tangent line to the graph of f at (a, f (a).
I
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Lecture 11: Derivatives as Functions
11.1 The derivative as a function
Note: Given a function f , the new function f 0 defined by
f (x + h) f (x)
h0
h
f 0 (x) = lim
for all points at which the limit exists is called the derivative of f . We say f is diffe
Lecture 29: The Chain Rule
29.1 The chain rule
Suppose f and g are differentiable and k(x) = f (g(x). Let u = g(x + h) g(x). We
assume g(x + h) 6= g(x) for all h sufficiently close to 0. Then
k(x + h) k(x)
h0
h
f (g(x + h) f (g(x)
= lim
h0
h
f (g(x + h) f