Chapter 4
Applications of Derivatives
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4.1
Extreme Values of Functions
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How to find local extreme values
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Not every critical point or
endpoint signals the
presence of an extreme
value.
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4.2
The Mean Value Theorem
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Mean Value Theorem says
that some interior point the
instantaneous change must
equal the average change
over the entire interval.
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Proof: 𝑖𝑓 𝑥1 𝑎𝑛𝑑 𝑥2 𝑎𝑟𝑒 𝑎𝑛𝑦 𝑡𝑤𝑜 𝑝𝑜𝑖𝑛𝑡𝑠 𝑖𝑛 𝑎, 𝑏 , 𝑎𝑛𝑑 𝑥1 <
𝑥2 , 𝑡ℎ𝑒𝑛 𝑓 𝑥1 = 𝑓 𝑥2 .
𝑓 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒 𝑡ℎ𝑒𝑟𝑜𝑟𝑒𝑚 𝑜𝑛 𝑥1 , 𝑥2 ;
𝑓 𝑥2 − 𝑓(𝑥1 )
= 𝑓 ′ 𝑐 , = 0 𝑐 𝑖𝑠 𝑠𝑜𝑚𝑒 𝑝𝑜𝑖𝑛𝑡 𝑖𝑛 𝑥1 , 𝑥2
𝑥2 − 𝑥1
𝑓 𝑥1 = 𝑓(𝑥2 )
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4.3
Monotonic Functions and
The First Derivative Test
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4.4
Concavity and Curve Sketching
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4.6
Indeterminate Forms and
^
L’ Hopital’s
Rule
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4.8
Antiderivatives
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