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Get Full Access to UIC - MATH 165 - Study Guide - Final
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UIC / Mathematics / MATH 165 / Explain the use of the function value.

Explain the use of the function value. Description

Description: Detailed study guide for the final
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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

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