.lst-kix_vhbdxs1aubef-0 > li:before{content:"● "}.lst-kix_vhbdxs1aubef-1 > li:before{content:"○ "}ul.lst-kix_vhbdxs1aubef-1{list-style-type:none}ul.lst-kix_vhbdxs1aubef-0{list-style-type:none}.lst-kix_vhbdxs1aubef-3 > li:before{content:"● "}ul.lst-kix_vhbdxs1aubef-3{list-style-type:none}ul.lst-kix_vhbdxs1aubef-2{list-style-type:none}.lst-kix_vhbdxs1aubef-4 > li:before{content:"○ "}.lst-kix_vhbdxs1aubef-5 > li:before{content:"■ "}.lst-kix_vhbdxs1aubef-6 > li:before{content:"● "}.lst-kix_5c5zs4ie9t1h-0 > li:before{content:"● "}.lst-kix_vhbdxs1aubef-8 > li:before{content:"■ "}.lst-kix_4lphzo1lwim2-2 > li:before{content:"■ "}.lst-kix_4lphzo1lwim2-3 > li:before{content:"● "}.lst-kix_vhbdxs1aubef-7 > li:before{content:"○ "}.lst-kix_4lphzo1lwim2-0 > li:before{content:"● "}.lst-kix_4lphzo1lwim2-4 > li:before{content:"○ "}ul.lst-kix_vhbdxs1aubef-5{list-style-type:none}ul.lst-kix_vhbdxs1aubef-4{list-style-type:none}ul.lst-kix_vhbdxs1aubef-7{list-style-type:none}ul.lst-kix_vhbdxs1aubef-6{list-style-type:none}ul.lst-kix_vhbdxs1aubef-8{list-style-type:none}.lst-kix_4lphzo1lwim2-1 > li:before{content:"○ "}.lst-kix_5c5zs4ie9t1h-8 > li:before{content:"■ "}.lst-kix_s5omiecv4is4-8 > li:before{content:"■ "}ul.lst-kix_5c5zs4ie9t1h-7{list-style-type:none}ul.lst-kix_5c5zs4ie9t1h-8{list-style-type:none}ul.lst-kix_5c5zs4ie9t1h-5{list-style-type:none}ul.lst-kix_5c5zs4ie9t1h-6{list-style-type:none}.lst-kix_s5omiecv4is4-6 > li:before{content:"● "}.lst-kix_s5omiecv4is4-7 > li:before{content:"○ "}ul.lst-kix_5c5zs4ie9t1h-3{list-style-type:none}.lst-kix_5c5zs4ie9t1h-6 > li:before{content:"● "}ul.lst-kix_5c5zs4ie9t1h-4{list-style-type:none}ul.lst-kix_5c5zs4ie9t1h-1{list-style-type:none}.lst-kix_5c5zs4ie9t1h-7 > li:before{content:"○ "}ul.lst-kix_5c5zs4ie9t1h-2{list-style-type:none}ul.lst-kix_5c5zs4ie9t1h-0{list-style-type:none}ul.lst-kix_4lphzo1lwim2-0{list-style-type:none}.lst-kix_5c5zs4ie9t1h-1 > li:before{content:"○ "}ul.lst-kix_4lphzo1lwim2-1{list-style-type:none}.lst-kix_s5omiecv4is4-2 > li:before{content:"■ "}.lst-kix_s5omiecv4is4-3 > li:before{content:"● "}.lst-kix_5c5zs4ie9t1h-2 > li:before{content:"■ "}ul.lst-kix_5qnhlgam42kb-7{list-style-type:none}ul.lst-kix_5qnhlgam42kb-8{list-style-type:none}.lst-kix_s5omiecv4is4-0 > li:before{content:"● "}.lst-kix_s5omiecv4is4-1 > li:before{content:"○ "}.lst-kix_s5omiecv4is4-4 > li:before{content:"○ "}.lst-kix_s5omiecv4is4-5 > li:before{content:"■ "}.lst-kix_5c5zs4ie9t1h-5 > li:before{content:"■ "}ul.lst-kix_4lphzo1lwim2-8{list-style-type:none}.lst-kix_5qnhlgam42kb-1 > li:before{content:"○ "}ul.lst-kix_5qnhlgam42kb-1{list-style-type:none}ul.lst-kix_5qnhlgam42kb-2{list-style-type:none}.lst-kix_5c5zs4ie9t1h-4 > li:before{content:"○ "}ul.lst-kix_4lphzo1lwim2-6{list-style-type:none}.lst-kix_5qnhlgam42kb-0 > li:before{content:"● "}.lst-kix_5qnhlgam42kb-2 > li:before{content:"■ "}ul.lst-kix_4lphzo1lwim2-7{list-style-type:none}ul.lst-kix_5qnhlgam42kb-0{list-style-type:none}.lst-kix_5c5zs4ie9t1h-3 > li:before{content:"● "}ul.lst-kix_4lphzo1lwim2-4{list-style-type:none}ul.lst-kix_5qnhlgam42kb-5{list-style-type:none}.lst-kix_vhbdxs1aubef-2 > li:before{content:"■ "}ul.lst-kix_4lphzo1lwim2-5{list-style-type:none}ul.lst-kix_5qnhlgam42kb-6{list-style-type:none}ul.lst-kix_4lphzo1lwim2-2{list-style-type:none}ul.lst-kix_5qnhlgam42kb-3{list-style-type:none}ul.lst-kix_4lphzo1lwim2-3{list-style-type:none}ul.lst-kix_5qnhlgam42kb-4{list-style-type:none}.lst-kix_5qnhlgam42kb-5 > li:before{content:"■ "}.lst-kix_5qnhlgam42kb-4 > li:before{content:"○ "}.lst-kix_5qnhlgam42kb-6 > li:before{content:"● "}.lst-kix_5qnhlgam42kb-3 > li:before{content:"● "}.lst-kix_5qnhlgam42kb-7 > li:before{content:"○ "}.lst-kix_5qnhlgam42kb-8 > li:before{content:"■ "}.lst-kix_4lphzo1lwim2-8 > li:before{content:"■ "}.lst-kix_4lphzo1lwim2-6 > li:before{content:"● "}.lst-kix_4lphzo1lwim2-7 > li:before{content:"○ "}.lst-kix_4lphzo1lwim2-5 > li:before{content:"■ "}ul.lst-kix_s5omiecv4is4-0{list-style-type:none}ul.lst-kix_s5omiecv4is4-1{list-style-type:none}ul.lst-kix_s5omiecv4is4-2{list-style-type:none}ul.lst-kix_s5omiecv4is4-3{list-style-type:none}ul.lst-kix_s5omiecv4is4-8{list-style-type:none}ul.lst-kix_s5omiecv4is4-4{list-style-type:none}ul.lst-kix_s5omiecv4is4-5{list-style-type:none}ul.lst-kix_s5omiecv4is4-6{list-style-type:none}ul.lst-kix_s5omiecv4is4-7{list-style-type:none}

MATH FINALWe also discuss several other topics like What is the definition of human nature?
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1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7,  2.8, 3.1, 3.2, 3.3, 3.5, 4.1, 4.2, 4.3, 4.4, 4.5, 5.1, 5.2 6.1, 6.2, 6.3, 6.5

1.1. Limits: A Numerical and Graphical Approach

• Definition

• As x approaches a, the limit of f(x) is L, written

• 
• “x → a”: This indicates that x is approaching a from both sides
  

• Theorem

• As x approaches a, the limit of f(x) is L if the limit from the left exists and the limit from the right exists and both limits are L that is

• If,  =  then,

• Function value and limit are not the same

• The limit of f(x) at a number a does not depend on the function value at a or whether a exists
  

1.2. Algebraic Limits and Continuity

• Theorem Limits of Rational Functions

• For any rational function f, with a in the domain of f:

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• Definition

• A function f is continuous at x = a if:

• 1. f(a) exists
• 2.  exists                (continuous)
• 3.                 (continuous)
  

• Marginal cost is different from average cost

• DIfference quotient

•  

• 
• 
• ;
• 
• 

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• Ex. 2. Find Δ y in each case

• a. y= x 2, x= 4, and Δ x= 0.1

• Δ y = (4+0.1) 2 - 4 2
• (4.1)2 - 4 2 = 16.81 - 16 = 0.81

• b. y= x 3, x=2, and Δ x= -0.1

• (2-0.1)3 - 23 = -1.141

• f’(x) = ⇒
•  ; ;

• Ex. 3

• y = f(x) =
• a. Find dy when x= 4 and dx= 1
• b. Compare dy and 0 Δ y
• a. use the results of a and b to estimate value of
  
• a. ⇒ ⇒
• b. The value of Δ y is the actual change in y between x= 4 and x= 10.

• Δ y =

• c.

1.6. Differentiation Techniques: the Product and Quotient Rule

• f(x) = u*v
• f’(x) = u * v’ +u’ * v
• Quotient rule:

• Q(x) = ;  
  

1.7. The Chain Rule

•   

2.1. Using first derivative to classify maximum and minimum values and sketch graphs

References: