SECTION 2.1 - THE TANGENT AND VELOCITY PROBLEMS Tangent Problem Definition (Secant Line & Tangent Line)

Example 1 Find the equation of the tangent lines to the curve y =

2 at the points with x-coordinates 1 − 3x

x = 0 and x = −1 .

1

Example 2 Susie and Johnny are running a lemonade stand. The following table gives a running total of money they made as added each hour. Estimate the slope of the tangent line to the graph at t =3. # of hours total $

0 $0

1 $4.50

2 $10.25

3 $18.75

4 $26.50

5 $33.75

2

Velocity Problem Definition (Average Velocity & Instantaneous Velocity)

Example 3 A baseball is rolled down a ramp. The distance the ball has traveled, measured in feet, is t2 modeled by the function d(t ) = + 2t + 1 where time, t, is measured in seconds. Find the 4 average velocity of the ball over the following time periods. [1, 3]

[1, 2]

[1, 1.5]

[1, 1.1]

Find the instantaneous velocity of the ball when t = 1.

3

SECTION 2.2 – THE LIMIT OF A FUNCTION Definition (Limit)

Example 1 Find each limit or explain why it does not exist. lim f (x ) =

lim f (x ) =

x → −4

x → −2

lim f (x ) =

lim f (x ) =

x →0

x→4

Example 2 Estimate lim x 2 + 2x + 1 . x →6

(

)

4

Definition (One-Sided Limit)

Example 3  2x 2 if Let f (x ) =  x + 1 if

0≤x≤3 . Find lim+ f (x ) and lim− f (x ) . x →3 x →3 x>3

Definition (Infinite Limit)

5

Example 4 1 1 Find lim+ and lim− . x →0 x →0 x x

Example 5 Find the following limits. lim csc 2 x = x →0

lim x →2

1

(x − 2)2

=

Definition (Vertical Asymptote)

(See examples 4 & 5.)

6

Example 6 Find the vertical asymptotes of the function f (x ) =

x 2 + 3x + 7 . x −3

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SECTION 2.3 – CALCULATING LIMITS USING LIMIT LAWS Limit Laws Suppose c is a constant and the limits lim f (x ) and lim g(x ) exist. x→a

x→a

1.) lim (f (x ) + g(x )) = lim f (x ) + lim g(x ) x →a

x →a

x →a

2.) lim (f (x ) − g(x )) = lim f (x ) − lim g(x ) x →a

x →a

x →a

3.) lim cf (x ) = c lim f (x ) x →a

x →a

4.) lim (f (x )g(x )) = lim f (x ) * lim g(x ) x →a

5.) lim x →a

x →a

x →a

f (x ) f (x ) lim if lim g(x ) ≠ 0 = x →a g(x ) lim g(x ) x→a x →a

6.) lim (f (x )) = n

x →a

( lim x →a

)

f (x )

n

where n is a positive integer

7.) lim c = c x →a

8.) lim x = a x →a

9.) lim x n = a n where n is a positive integer x →a

10.) lim

n

x = n a where n is a positive integer. If n is even, we assume that a > 0 .

11.) lim

n

f (x ) = n lim f (x ) where n is a positive integer. If n is even, we assume that f (x ) > 0 .

x →a

x →a

x →a

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Example 1 Calculate the following limits. lim 3 x 2 + 5 x − 9 x → −2

lim

x →3

(

)

x2 − x + 3 2x + 1

x 2 + 4 x − 12 lim x →2 x 2 − 2x

lim

x → −8

x 2 + 6x − 4 x+8

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Direct Substitution Property

Example 2 Find the following limits. lim 2x 2 + 3 x + 4 = x →2

(

)

lim

x−9 = x + 2x − 3

lim

2(− 3 + h) − 18 = h

x→4

2

2

h →0

Theorem 1

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Example 3 Determine if the following limits exist. 4 x + 1 if x ≤ 4 lim f (x ) where f (x ) =  2 x→ 4  x + 1 if x > 4

lim x→4

4−x 4−x

Theorem 2

Squeeze Theorem

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Example 4 Use the squeeze theorem to find the following limits.  4 x 2 + 2x  lim  sin 2 x  x →0 3  

  1  lim  x 2 − 2x cos  x →2  x  

(

)

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SECTION 2.4 – THE PRECISE DEFINITION OF A LIMIT Definition (Limit – Precise Definition)

Definition (One Sided Limit – Precise Definition)

Definition (Infinite Limits – Precise Definition)

13

Example 1 Use the precise definition to prove the following statements. lim (7 x − 27 ) = 8 x →5

(

)

lim x 2 − 3 x = −2 x →2

(

)

lim x 3 + 2x 2 + 3 x − 4 = 2 x →1

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Example 2 Prove the following statement by using the precise definition of an infinite limit. 2 lim4 =∞ x→4 x−4

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SECTION 2.5 – CONTINUITY Definition (Continuous at a number)

Example 1  −x  Find all of the discontinuities of f (x ) =  3 − x (x − 3 )2 

if

x