Fundamental Theorem of Calculus Part 2 Fri Jan 22

1
Fundamental Theorem of Calculus Part 2Fri Jan 22

• Do Now
• Evaluate each definite integral
• 1)
• 2)

2
Part 1 Review

• FTC Part 1 stated that we can use antiderivatives
  to compute definite integrals
• This allowed us to compute the accumulation of a
  function over a given interval

3
FTC Part 2

• Assume that f(x) is continuous on an open
  interval and let a be in that interval. Then the
  area function
• is an antiderivative of f(x), or

4
Notes

• FTC Part 2 is used for integrals that we are
  unable to integrate
• Because derivatives and anti derivatives are
  inverses, we can at least find the derivatives of
  these integrals
• If the variable limit is more than x, use chain
  rule

5
Ex1

• Find a formula for the area function

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Ex 2 Antiderivative as Integral

• Let F(x) be the particular antiderivative of
  f(x) sin(x2) satisfying
• Express F(x) as an integral

7
Ex 3 Differentiating an Integral

• Find the derivative of
• and calculate A(2), A(3), and A(2)

8
Ex 4 FTC2 and Chain Rule

• Find the derivative of

9
Closure

• Calculate the derivative
• HW p.320 1-45 odds
• 5.1-5.4 Quiz Fri

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5.4 Practice / Quiz ReviewMon Jan 25

• Do Now
• Find formulas for the functions represented by
  the integrals
• 1)
• 2)

11
HW Review
12
Quiz Review

• 5.1 Approximating Areas
• Left/Right/Midpoint Approximations
• 5.2 Theorems/Properties of Definite Integrals
• 5.3 Definite Integrals / FTC Pt 1
• Absolute value
• 5.4 FTC Pt 2

13
Closure

• Describe the Fundamental Theorem of Calculus
  (both parts) and their uses
• 5.1-5.4 Quiz tomorrow!