Mean Value Theorem PDF

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Mean Value Theorem...

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Using MVT to calculate error bounds •

15𝜋

1

Use the MVT to find an upper bound for the error involved if we approximate cos ( 21 ) by − . 2

𝑓(𝑥) = cos(𝑥)

1 15𝜋 ) − (− 2)| 21 2𝜋 15𝜋 |cos ( 21 ) − cos ( 3 )|

Evaluate 𝑞(𝑒𝑟𝑟𝑜𝑟𝑏𝑜𝑢𝑛𝑑) = |cos ( Using MVT:

2𝜋 15𝜋 𝑓: [ 3 , ] 21

=

→ ℝ with 𝑓(𝑥) = cos(𝑥).

2𝜋 15𝜋 , ] 3 21 15𝜋 2𝜋 𝑓( )−𝑓( ) 21 3

𝑓 is differentiable on [ By the MVT: ∴ 𝑓(

elementary (trigonometric)function is sin(x).

= 𝑓′(𝑐) for some

15𝜋 2𝜋 − 21 3

2𝜋 3...