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Tangent Lines
EX 1.2 Estimating the Slope of a Curve
Estimate the slope of y = sin x at x = 0.
The Length Of a Curve
EX 1.3 Estimating the Length of a Curve
Estimate the length of the curve y = sin x for 0 ≤ x ≤ π
The Concept Of Limit
Consider the functions
Both functions are undefined at x = 2.
These two functions look quite different in the vicinity of x = 2.
The Concept Of Limit (Cont’d)
Consider the function
The limit of f (x) as x approaches 2 from
the left is 4.
The limit of f (x) as x approaches 2 from
the right is 4.
The limit of f (x) as x approaches 2 is 4.
The Concept Of Limit (Cont’d)
Consider the function
The Concept Of Limit (Cont’d)
A limit exists if and only if both one-sided limits exist and
are equal.
EX 2.1 Determining Limits Graphically
Use the graph in the following figure to determine
EX 2.2 A Limit Where Two Factors Cancel
EX 2.3 A Limit That Does Not Exist
EX 2.4 Approximating the Value of a Limit
EX 2.5 A Case Where One-Sided Limits Are Needed
Note that
Computation Of Limits
Computation Of Limits (Cont’d)
Application Example
EX 3.1 Finding the Limit of a Polynomial
EX 3.2 Finding the Limit of a Rational Function
EX 3.3 Finding a Limit by Factoring
Note that
Right answer
Computation Of Limits (Cont’d)
Computation Of Limits (Cont’d)
EX 3.5 Finding a Limit By Rationalizing
EX 3.6 Evaluating a Limit of an Inverse Trigonometric Function
EX 3.7 A Limit of a Product That Is Not the product of the Limits
EX 3.7 (Cont’d)
Computation Of Limits (Cont’d)
EX 3.8 Using the Squeeze Theorem to Verify the Value of a Limit
EX 3.8 (Cont’d)
By graphing
By squeeze theorem
EX 3.9 A Limit for a Piecewise-Defined Function
Concept of velocity
For an object moving in a straight line, whose position
at time t is given by the function f (t),
The instantaneous velocity of that object at time t = a
Concept of velocity (Cont’d)
EX 3.10 Evaluating a Limit Describing Velocity
we can’t simply
substitute h = 0
Continuity And Its Consequence
- discontinuous cases
(a)
(c)
(b)
(d)
Continuity And Its Consequence (Cont’d)
EX 4.1 Finding Where a Rational Function Is Continuous
EX 4.2 Removing a Discontinuity
Make the function from example 4.1 continuous everywhere by
redefining it at a single point.
EX 4.3 Nonremovable Discontinuities
There is no way to redefine either function at x = 0 to
make it continuous there.
Continuity And Its Consequence (Cont’d)
EX 4.4 Continuity for a Rational Function
 f will be continuous at all x where
the denominator is not zero.
Think about why you didn’t see anything
peculiar about the graph at x =-1.
Continuity And Its Consequence (Cont’d)
EX 4.5 Continuity for a Composite Function
Determine where
is continuous.
Continuity And Its Consequence (Cont’d)
EX 4.6 Intervals Where a Function Is Continuous
EX 4.7 Interval of Continuity for a Logarithm
EX 4.8 Federal Tax Table
EX 4.8 Federal Tax Table (Cont’d)
Continuity And Its Consequence (Cont’d)
EX 4.9 Finding Zeros by the Method of Bisections
Limits Involving Infinity
EX 5.1 A Simple Limit Revisited
Remark
EX 5.2 A Function Whose One-Sided Limits Are Both Infinite
EX 5.5 A Limit Involving a Trigonometric Function
Limits at Infinity
EX 5.6 Finding Horizontal Asymptotes
Since
So
the line y = 2 is a horizontal asymptote.
EX 5.7 A Limit of a Quotient That Is Not the Quotient of the Limits
EX 5.8 Finding Slant Asymptotes
y→
as
EX 5.9 Two Limits of an Exponential Function
Evaluate
(a)
and
(b)
and
EX 5.10 Two Limits of an Inverse TrigonometricF unction
Note the graph of
In similar way,
EX 5.11 Finding the Size of an Animal’s Pupils
f (x) :the diameter of pupils
x : the intensity of light
find the diameter of the pupils with (a) minimum and (b) maximum
light.
EX 5.12 Finding the Limiting Velocity of a Falling Object
Note
So
Formal Definition Of Limits
EX 6.2 Verifying a Limit
(3x  4)  10
Show lim
x2
Find δ such that
x2 





   3
Formal Definition Of Limits (Cont’d)
Formal Definition Of Limits (Cont’d)
Formal Definition Of Limits (Cont’d)
Formal Definition Of Limits (Cont’d)
Limits And Loss-Of-Significance Errors
EX 7.1 A Limit with Unusual Graphical and Numerical Behavior
Incorrect calculated value ?!
Computer Representation of Real Numbers
All computing devices have finite memory and
consequently have limitations on the size mantissa and
exponent that they can store.
This is called finite precision.
Most calculator carry a 14-digit mantissa and a 3-digit
exponent.
EX 7.2 Computer Representation of a Rational Number
Determine how
is stored internally on a 10-digit computer and
how
is stored internally on a 14-digit computer.
EX 7.3 A Computer Subtraction of Two “Close” Numbers
A computer with 14-digit mantissa
= 0.
Exact value
EX 7.4 Another Subtraction of Two “Close” Numbers
Calculate the following result by a
1
1
Exact Value
= 10,000,000
EX 7.5 Avoid A Loss-of-Significance Error
Compute f (5×104) with 14-digit mantissa for
EX 7.5 (Cont’d)
Rewrite the function as follows:
EX 7.6 Loss-of-Significance Involving a Trigonometric Function
EX 7.7 A Loss-of-Significance Error Involving a Sum
Directly Calculate:
EX 7.7(Cont’d)