3.6
Chain Rule Formulas
The chain rule is an efficient method for differentiating composite
functions.
Example Version 1 of the Chain Rule For each of the following
composite functions, find the inner function u = g(x) and the
outer function y = f(u). The
3.1
DIFFERENTIAL CALCULUS
This is concerned with how one
quantity changes in relation to another
quantity.
The central concept of differential
calculus is the derivative.
DERIVATIVES AND RATES OF CHANGE
The problem of finding the tangent line to
a curve
3.8
The Derivative of y = ln x
The natural exponential function f ( x) e is a one-to-one
function on the interval (-, ), it has an inverse which is the
1
natural logarithmic function f 1 ( x) ln x. The domain of f is
the range of f, (0, ) The graphs of f
3.9
Derivatives of Inverse Trigonometric Functions
Recall the following:
The domain of the inverse
y sin 1 x x sin y
where y
2
y cos 1 x x cos y
where y
2
y tan 1 x x tan y
where y
2
y sec 1 x x sec y
where0 y and y
differentiate with respect to x
dy
3.7
Equations of the form y = f(x) is said to define y
explicitly as a function of x because the variable y
appears alone on one side of the equation. This is not
always the case, sometimes y is not alone. For
example,
yx + y + 1 = x
this is not in the fo
INFINITE LIMITS
2.4
Definition
Let f be a function, defined on both sides of a,
except possibly at a itself. Then
lim f x
x a
means that the values of f(x) can be made arbitrary
large by taking x sufficiently close to a, but not
equal to a.
Find
1
lim 2
LIMITS AT INFINITY
2.5
For Infinite Limits, the dependent variable becomes arbitrarily
large in magnitude as the independent variable becomes large,
whereas with Limits at infinity the independent variable becomes
large in magnitude as the dependent varia
Derivatives of
Trigonometric Functions
3.4
Proof of the first is on page 155 156, the
second is left as an exercise.
EXAMPLE Calculating trigonometric limits Evaluate the following limits.
lim
x 0
sin 5 x
3x
lim
0
cos 2 1
Derivatives of Sine and Cosine Fu
4.1
Absolute Maxima and Minima
Various cases for the function f ( x) x
2
Defining a function on a closed interval does not guarantee
the existence of absolute extreme values.
In order for there to be absolute minimum and maximum
values on an interval the
CONTINUITY
2.6
To go on without interruption is a simple ordinary definition of the
word continuous. In mathematics the meaning is basically the
same.
Continuity at a Point
a.
b.
y
y
4
4
3
3
Removable Discontinuities
2
2
1
1
x
-4
-3
-2
-1
1
2
c.
3
x
4
-4
4.3
Sketch the graph of
f
x intercept(s)
y intercept(s)
Vertical asymptotes
Horizontal asymptotes
First and second derivatives
Critical point(s)
Inflection point(s)
Test intervals
x
2 x2 9
x2 4
Sketch the graph of
f x x 4 x
Let f be a function that sati
Math 171
APPLICATION OF DERIVATIVES
OPTIMIZATION PROBLEMS
4.4
Min/Max Problems
The guidelines above provide a general
framework, however keep in mind that
details may vary depending upon the
problem.
A farmer has 2400 ft of fencing and wants to
fence off
Math 171
Calculus I
Limits
TECHNIQUES FOR COMPUTING LIMITS
2.3
Techniques for Computing Limits
The techniques used previously to estimate limits can sometimes lead to
incorrect results. Let us now look at analytical methods for evaluating
more precise lim
3.2
a.
d
( x12 )
dx
d
b.
( x)
dx
d
c.
( 25 )
dx
a.
d
3x 9
(
)
dx
5
d 2
b.
(
dx 7
t)
a.
d
(3x 4 5 x 3 x 2 9)
dx
d
(12e x - 6x )
dx
Example
Finding tangent lines
ex
x at x 0.
a. Write an equation of the line tangent to the graph of y
4
b. Find the point(s
LIMITS
2.7
Let f be a function defined on some open interval
that contains the number a, except possibly at a
itself and let L be a real number.
Then we say the limit of f(x) as x approaches
a is L and we write
lim f ( x) L
x a
i.e. for every number 0 the
MATH170 LECTURE NOTES Unit 2
P1
Limits An Intuitive Approach
The notion of a limit is fundamental to the study of calculus.
Frequently when studying a function y f x , we find ourselves interested in the functions
behaviour near a particular point c, but