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UM / Math / CCM 116 / What is the fundamental theorem of calculus?

What is the fundamental theorem of calculus?

What is the fundamental theorem of calculus?

Description

School: University of Michigan
Department: Math
Course: Calculus II
Professor: Gsi chen
Term: Fall 2016
Tags: Math, 116, final, exam, study, guide, Calculus, Calc, and II
Cost: 50
Name: Math 116 Final Exam Study Guide
Description: University of Michigan Math 116 - Calculus II Final exam study guide - includes material from exam 1, exam 2, and material covered after exam 2 that will be on the final exam
Uploaded: 12/18/2016
19 Pages 39 Views 1 Unlocks
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Math 116 Exam 1 Study GuideIf you want to learn more check out Giave a sample of format in writing a letter.
Don't forget about the age old question of What is stated in mischel’s findings on self­-control?

  • Fundamental Theorem of Calculus I
  • If f is continuous on [a, b] and F’(x) = f(x) then
  • Theorems about definite integrals
  • If f is continuous everywhere and a, b, and c are any number (not necessarily a < b < c), then…
  • 1.
  • 2.
  • Linearity Theorem

  • If f and g are continuous on [a, b] and c is equal to any real number, then…
  • 1.
  • 2.

  • Fundamental Theorem of Calculus II

  • If f is continuous on [a, b ] and F is an antiderivative of f: F’(x) = f(x), then
  •  
  • U - substitution
  • If an integral is in the form , you can put u = g(x), du = g’(x) dx, and modify the bounds appropriately: .

If you want to learn more check out What refers to the ability to influence a group toward the achievement of goals?
If you want to learn more check out Explain what tautology is.
Don't forget about the age old question of What are the models of psychopathology?
We also discuss several other topics like What happened during the radical reconstruction?

  • Integration by parts
  • ← explicit
  •  ← common form
  • For indefinite integration…
  • Picking u and dv:
  •  simpler than u
  •  easy to integrate
  • General priority of picking u (not always):
  • Log
  • Inverse trigonometry
  • Polynomial
  • Exponential
  • Trigonometry
  • “Lipet” Rule
  • Trapezoidal Rule=
  • Equivalent to getting a secant line
  • Secant line above the graph → overestimate concave up
  • Secant line below the graph → underestimate concave down

  • Midpoint Rule
  • Equivalent to a tangent line
  • Tangent line below the graph → underestimate concave up
  • Tangent line above graph- overestimate concave down

  • Area and Volume

  • Area under the curve: just the integral of the function
  • Area between two curves: one we call the ceiling, the other is the floor
  • Area =
  • Volume of a 3D region
  • Vertical strip revolving around a horizontal axis → washer method
  • Vertical strip revolving around a vertical axis → shell method
  • Horizontal strip revolving around a vertical axis → washer method
  • Horizontal strip revolving around a horizontal axis → shell method

When given the freedom to slice, consider:

  •  How the functions are given y = f(x), x = f(y)
  • Which variable the density function depends on
  • Objects with prescribed cross sections perpendicular to a given axis: find the area of cross section in terms of the independent variable, accumulate or integrate them.
  • Arc Length
  • A graph y = f(x) between x = a and x = b has the length:
  • Center of Mass
  • x / y / z coordinate of center of mass
  •  ; where 𝓵(x) is the object’s density
  • Mass with constant density:

  • 3D Objects: mass = volume * density
  • 2D Objects: mass = area * density

  • Mass with non-constant density

  • Divide the regions into small pieces in such a way that the density is approximately constant on each piece, then add together using an integral

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