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by: Gavin B

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# MATH 1225 - Test 4 Study Guide MATH 1225

Gavin B
Virginia Tech
GPA 4.0

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This is the study guide I used to study for test 4. The things you need to know how to do for test 4 are all right here. It is condensed into two pages. It may not include some extremely simple as...
COURSE
Calculus of a Single Variable
PROF.
Eun Chang
TYPE
Study Guide
PAGES
3
WORDS
CONCEPTS
Math, 1225, Calculus, Single Variable, VT, Virginia Tech, Study Guide, Test 4, Exam 4
KARMA
50 ?

## Popular in Mathematics (M)

This 3 page Study Guide was uploaded by Gavin B on Tuesday March 1, 2016. The Study Guide belongs to MATH 1225 at Virginia Polytechnic Institute and State University taught by Eun Chang in Fall 2015. Since its upload, it has received 76 views. For similar materials see Calculus of a Single Variable in Mathematics (M) at Virginia Polytechnic Institute and State University.

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Date Created: 03/01/16
VT  –  MATH  1225  -­‐  UNIT  4     TEST  4   STUDY  GUIDE                                           Antiderivatives.     Riemann  Sums:   When  f  is  increasing:  LHS  is  underestimate,  RHS  is  overestimate.   When  f  is  decreasing:  RHS  is  underestimate,  LHS  is  overestimate.    lim ???? = ???????????????? = lim ????   ▯→▯ ▯ ▯→▯ ▯   ▯▯▯  Δ???? = ▯  for  width  of  interval  [a,b].   ▯ ???? = ???? + ????∆????    ???? = ???? ???? ∆???? + ???? ???? ∆???? + ⋯+ ????(???? )∆????   ▯ ▯ ▯ ▯▯▯ ▯???? = ???? ???? ∆▯ + ???? ???? ∆???? + ⋯+▯????(???? )∆????   ▯    ???????????????? = ???? = lim ???? = lim▯???? ???? ∆???? + ???? ???? ∆???? ▯ ⋯+ ???? ???? ]∆???? = ▯im ▯ ▯▯▯ ????(???? )▯????   ▯→▯ ▯→▯ ▯→▯ ▯  ???????????????? = ???? = lim▯→▯= lim▯???? ???? ▯→▯+ ???? ???? ∆???? ▯ ⋯+ ???? ???? ▯ ▯▯▯ ]∆???? = lim▯→▯ ▯▯▯ ????(???? ▯▯▯ )∆????   VT  –  MATH  1225  -­‐  UNIT  4     ▯ ▯▯▯ ???? = ???? + ???? + ⋯+ ???? = ????????   ▯▯▯ ???????? ▯ ???????? + ????▯ + ⋯+ ????▯ = ???? ▯ ▯▯▯ ???? ▯   ▯ ▯ ▯ ▯▯▯(???? ▯ ???? ) ▯ ▯▯▯ ???? ▯ ▯▯▯ ???? ▯    ???? ???? ???????? = lim ▯ ????(???? )∆????   ▯ ▯→▯ ▯▯▯ ▯   Sum  Functions:     ???? = ▯(▯▯▯)   ▯▯▯ ▯ ▯ ▯ ▯(▯▯▯)(▯▯▯▯) ▯▯▯ ???? = ▯     Net  Area  =▯  ???? ▯ ????  where▯  ????  is  area  above  the ▯ x-­‐axis  and  ????  is  area  below  the  x-­‐axis.     Properties  of  Definite  Integral:    ???????????? = ???? − ???? ????   ▯   ???? ???? ± ???? ???? ???????? = ▯ ???? ???? ???????? ± ▯ ???? ???? ????????   ▯ ▯ ▯  ???????? ???? ???????? = ???? ▯ ???? ???? ????????   ▯ ▯  ???? ???? ???????? = − ▯ ???? ???? ????????   ▯ ▯   Comparison  Properties:   If  ???? ???? ≥ 0,∀???? ∈ [????,????]  then   ▯ ???? ???? ???????? ≥ 0   ▯ ▯ ▯ If  ???? ???? ≥ ????(????),∀???? ∈ [????,????]  then   ???? ???? ???????? ≥ ???? ???? ????????   ▯ ▯ ▯ If  ???? ≤ ???? ???? ≤ ????,∀???? ∈ [????,????]  then  ????(???? − ????) ≤ ???? ???? ???????? ≤ ????(???? − ????)   ▯       Fundamental  Theorem  of  Calculus  (FTC):   FTC  1:   •   If  f  is  continuous  on  [a,b]  and  differentiable  on  (a,b), ???? ???? ????????,???? ≤ ???? ≤ ????   ▯ •   F  ‘(x)=f(x)   o   The  derivative  of  F  is  f.  The  antiderivative  of  f  is  F.   FTC  2:   ▯ •   ▯ ???? ???? ???????? = ???? ???? − ????(????)   If  a  or  b  is  not  some  constant  for  example  (???? )  then  you  must  substitute  by  saying  “Let  u=???? ”.   ▯▯ ▯▯ ▯▯ Then   = ×  and  the  derivative  of  the  integral  is  just  the  inside  function  of  the  integral.   ▯▯ ▯▯ ▯▯ The  lower  limit  does  not  change  the  derivative  value  for  these  problems  (because  of  FTC  1).   •   Because  you  first  take  the  antiderivative,  plug  in  the  constant  lower  limit  value  which   gives  a  constant  and  then  when  you  take  the  derivative  of  that  constant  it  is  just  0  so  it   does  not  impact  it.     VT  –  MATH  1225  -­‐  UNIT  4   Table  of  Integrals:   ???????? ???? ???????? = ???? ???? ???? ????????   cos???????????? = sin???? + ????   cosh???????????? = sinh???? + ????   ???????????? = ???????? + ????   sin???????????? = −cos???? + ????   sinh???????????? = cosh???? + ????   ▯ ????▯▯▯ ▯ 1 ▯▯ ???? ???????? = + ????   sec ????  ???????? = tan???? + ????   ▯ ???????? = sin ???? + ????   ???? + 1 1 − ???? ???? ???????? = ???? + ????   csc????cot???????????? = −csc???? + ????   1 ▯▯ 1 + ???? ▯ ???????? = tan ???? + ????   ????▯ 1 ???? ???????? = + ????   sec????tan???????????? = sec???? + ????   ???????? ln???? ???? ???? − 1 = sec ▯▯ ???? + ????   1 ???? ▯▯???????? = ???????? = ln???? + ????   csc ????  ???????? = −cot???? + ????     ????   Definite  vs.  Indefinite  Integrals:   •   No  “+C”  for  definite  integrals  but  there  is  one  for  indefinite  integrals.   •   Definite  integrals  produce  a  number;  indefinite  integrals  produce  a  function.     U-­‐Substitution.     Theorem  for  Symmetry:   If  f  is  continuous  on  [-­‐a,a]  and  f  is  an:   ▯ ▯ •   Even  function  (  f(x)=f(-▯▯▯???? ???? ???????? = 2then ▯ ???? ???? ????????   •   Odd  function  (  f(-­‐x)=  -???? ???? ???????? = 0   then   ▯▯

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