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 f igure 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

Note that Right answer EX 3.3 Finding a Limit by Factoring

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

- discontinuous cases Continuity And Its Consequence (a) (d) (b) (c)

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 the line y = 2 is a horizontal asymptote. Since So

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) (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 Show Find δ such that

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 Exact Value 1 1 = 10,000,000 Calculate the following result by a

EX 7.5 Avoid A Loss-of-Significance Error Compute f (5 × 10 4 ) 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)