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UCR / Mathematics / MATH 009B / What is the content of the initial value theorem?

What is the content of the initial value theorem?

What is the content of the initial value theorem?

Description

School: University of California Riverside
Department: Mathematics
Course: First-Year Calculus
Professor: Thunwa theerakarn
Term: Fall 2018
Tags: Calculus
Cost: 50
Name: Chapter 5.1 to 6.3
Description: Study guide for midterm for math 9b
Uploaded: 10/29/2018
2 Pages 77 Views 1 Unlocks
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tstyn001 (Rating: )



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"}.lst-kix_vehpn6d87zq0-8 > li:before{content:"" counter(lst-ctn-kix_vehpn6d87zq0-8,lower-roman) ". "}.lst-kix_cpduhxrsj0v-8 > li{counter-increment:lst-ctn-kix_cpduhxrsj0v-8}.lst-kix_vehpn6d87zq0-7 > li:before{content:"" counter(lst-ctn-kix_vehpn6d87zq0-7,lower-latin) ". "}ol.lst-kix_n6gzn5j4l1fy-8{list-style-type:none}ol.lst-kix_vehpn6d87zq0-4{list-style-type:none}ol.lst-kix_n6gzn5j4l1fy-7{list-style-type:none}ol.lst-kix_vehpn6d87zq0-5{list-style-type:none}ol.lst-kix_vehpn6d87zq0-2{list-style-type:none}ol.lst-kix_vehpn6d87zq0-3{list-style-type:none}.lst-kix_vehpn6d87zq0-4 > li:before{content:"(" counter(lst-ctn-kix_vehpn6d87zq0-4,lower-latin) ") "}ol.lst-kix_n6gzn5j4l1fy-4{list-style-type:none}ol.lst-kix_vehpn6d87zq0-8{list-style-type:none}ol.lst-kix_n6gzn5j4l1fy-3{list-style-type:none}ol.lst-kix_n6gzn5j4l1fy-6{list-style-type:none}ol.lst-kix_vehpn6d87zq0-6{list-style-type:none}ol.lst-kix_n6gzn5j4l1fy-5{list-style-type:none}.lst-kix_vehpn6d87zq0-5 > li:before{content:"(" counter(lst-ctn-kix_vehpn6d87zq0-5,lower-roman) ") "}ol.lst-kix_vehpn6d87zq0-7{list-style-type:none}.lst-kix_vehpn6d87zq0-8 > li{counter-increment:lst-ctn-kix_vehpn6d87zq0-8}ol.lst-kix_vehpn6d87zq0-0{list-style-type:none}ol.lst-kix_vehpn6d87zq0-1{list-style-type:none}.lst-kix_at3gng7kxydv-2 > li{counter-increment:lst-ctn-kix_at3gng7kxydv-2}.lst-kix_eit46awnacp3-5 > li:before{content:"- "}ol.lst-kix_n6gzn5j4l1fy-4.start{counter-reset:lst-ctn-kix_n6gzn5j4l1fy-4 0}.lst-kix_eit46awnacp3-6 > li:before{content:"- "}.lst-kix_utaorsrqarc-6 > li:before{content:"- "}.lst-kix_utaorsrqarc-7 > li:before{content:"- "}.lst-kix_eit46awnacp3-7 > li:before{content:"- "}.lst-kix_n6gzn5j4l1fy-6 > li{counter-increment:lst-ctn-kix_n6gzn5j4l1fy-6}.lst-kix_cpduhxrsj0v-4 > li{counter-increment:lst-ctn-kix_cpduhxrsj0v-4}.lst-kix_utaorsrqarc-5 > li:before{content:"- "}.lst-kix_eit46awnacp3-8 > li:before{content:"- "}.lst-kix_at3gng7kxydv-0 > li{counter-increment:lst-ctn-kix_at3gng7kxydv-0}.lst-kix_utaorsrqarc-8 > li:before{content:"- "}.lst-kix_utaorsrqarc-0 > li:before{content:"- "}.lst-kix_s2lzqhn0v1kp-7 > li:before{content:"○ "}.lst-kix_s2lzqhn0v1kp-8 > li:before{content:"■ "}.lst-kix_utaorsrqarc-1 > li:before{content:"- "}ol.lst-kix_vehpn6d87zq0-3.start{counter-reset:lst-ctn-kix_vehpn6d87zq0-3 0}.lst-kix_s2lzqhn0v1kp-6 > li:before{content:"● "}.lst-kix_utaorsrqarc-2 > li:before{content:"- "}.lst-kix_utaorsrqarc-3 > li:before{content:"- "}.lst-kix_s2lzqhn0v1kp-3 > li:before{content:"● "}.lst-kix_s2lzqhn0v1kp-4 > li:before{content:"○ "}ol.lst-kix_cpduhxrsj0v-5.start{counter-reset:lst-ctn-kix_cpduhxrsj0v-5 0}.lst-kix_utaorsrqarc-4 > li:before{content:"- "}.lst-kix_s2lzqhn0v1kp-5 > li:before{content:"■ "}.lst-kix_2qy1kbsxslq6-0 > li:before{content:"- "}.lst-kix_2qy1kbsxslq6-1 > li:before{content:"- "}.lst-kix_2qy1kbsxslq6-3 > li:before{content:"- "}.lst-kix_2qy1kbsxslq6-2 > li:before{content:"- "}.lst-kix_n6gzn5j4l1fy-2 > li{counter-increment:lst-ctn-kix_n6gzn5j4l1fy-2}.lst-kix_2qy1kbsxslq6-5 > li:before{content:"- "}ol.lst-kix_vehpn6d87zq0-2.start{counter-reset:lst-ctn-kix_vehpn6d87zq0-2 0}.lst-kix_2qy1kbsxslq6-4 > li:before{content:"- "}ol.lst-kix_cpduhxrsj0v-4.start{counter-reset:lst-ctn-kix_cpduhxrsj0v-4 0}.lst-kix_at3gng7kxydv-6 > li{counter-increment:lst-ctn-kix_at3gng7kxydv-6}.lst-kix_2qy1kbsxslq6-6 > li:before{content:"- "}.lst-kix_2qy1kbsxslq6-7 > li:before{content:"- "}.lst-kix_vehpn6d87zq0-4 > li{counter-increment:lst-ctn-kix_vehpn6d87zq0-4}.lst-kix_2qy1kbsxslq6-8 > li:before{content:"- "}.lst-kix_vehpn6d87zq0-1 > li{counter-increment:lst-ctn-kix_vehpn6d87zq0-1}ol.lst-kix_vehpn6d87zq0-1.start{counter-reset:lst-ctn-kix_vehpn6d87zq0-1 0}.lst-kix_vehpn6d87zq0-3 > li{counter-increment:lst-ctn-kix_vehpn6d87zq0-3}ol.lst-kix_n6gzn5j4l1fy-2.start{counter-reset:lst-ctn-kix_n6gzn5j4l1fy-2 0}ol.lst-kix_cpduhxrsj0v-3.start{counter-reset:lst-ctn-kix_cpduhxrsj0v-3 0}ul.lst-kix_t48q2cxz4x5b-4{list-style-type:none}ul.lst-kix_t48q2cxz4x5b-5{list-style-type:none}ul.lst-kix_t48q2cxz4x5b-6{list-style-type:none}ul.lst-kix_t48q2cxz4x5b-7{list-style-type:none}.lst-kix_vehpn6d87zq0-2 > li{counter-increment:lst-ctn-kix_vehpn6d87zq0-2}ul.lst-kix_t48q2cxz4x5b-8{list-style-type:none}ol.lst-kix_n6gzn5j4l1fy-5.start{counter-reset:lst-ctn-kix_n6gzn5j4l1fy-5 0}ol.lst-kix_cpduhxrsj0v-0.start{counter-reset:lst-ctn-kix_cpduhxrsj0v-0 0}.lst-kix_n6gzn5j4l1fy-0 > li:before{content:"" counter(lst-ctn-kix_n6gzn5j4l1fy-0,decimal) ". "}.lst-kix_s2lzqhn0v1kp-0 > li:before{content:"● "}.lst-kix_s2lzqhn0v1kp-2 > li:before{content:"■ "}.lst-kix_cpduhxrsj0v-2 > li{counter-increment:lst-ctn-kix_cpduhxrsj0v-2}ul.lst-kix_t48q2cxz4x5b-0{list-style-type:none}ul.lst-kix_t48q2cxz4x5b-1{list-style-type:none}.lst-kix_n6gzn5j4l1fy-2 > li:before{content:"" counter(lst-ctn-kix_n6gzn5j4l1fy-2,lower-roman) ". "}ul.lst-kix_t48q2cxz4x5b-2{list-style-type:none}ul.lst-kix_t48q2cxz4x5b-3{list-style-type:none}ol.lst-kix_at3gng7kxydv-3.start{counter-reset:lst-ctn-kix_at3gng7kxydv-3 0}ol.lst-kix_at3gng7kxydv-6.start{counter-reset:lst-ctn-kix_at3gng7kxydv-6 0}ol.lst-kix_cpduhxrsj0v-1.start{counter-reset:lst-ctn-kix_cpduhxrsj0v-1 0}ul.lst-kix_8j1vjl8355n6-7{list-style-type:none}ul.lst-kix_8j1vjl8355n6-8{list-style-type:none}ul.lst-kix_8j1vjl8355n6-5{list-style-type:none}ul.lst-kix_8j1vjl8355n6-6{list-style-type:none}.lst-kix_eit46awnacp3-3 > li:before{content:"- "}.lst-kix_at3gng7kxydv-5 > li{counter-increment:lst-ctn-kix_at3gng7kxydv-5}ol.lst-kix_n6gzn5j4l1fy-0{list-style-type:none}.lst-kix_n6gzn5j4l1fy-8 > li:before{content:"" counter(lst-ctn-kix_n6gzn5j4l1fy-8,lower-roman) ". "}ol.lst-kix_n6gzn5j4l1fy-2{list-style-type:none}.lst-kix_eit46awnacp3-1 > li:before{content:"- "}ol.lst-kix_n6gzn5j4l1fy-1{list-style-type:none}ol.lst-kix_n6gzn5j4l1fy-3.start{counter-reset:lst-ctn-kix_n6gzn5j4l1fy-3 0}.lst-kix_t48q2cxz4x5b-8 > li:before{content:"■ "}.lst-kix_n6gzn5j4l1fy-4 > li:before{content:"" counter(lst-ctn-kix_n6gzn5j4l1fy-4,lower-latin) ". "}.lst-kix_n6gzn5j4l1fy-6 > li:before{content:"" counter(lst-ctn-kix_n6gzn5j4l1fy-6,decimal) ". "}.lst-kix_at3gng7kxydv-4 > li{counter-increment:lst-ctn-kix_at3gng7kxydv-4}ol.lst-kix_vehpn6d87zq0-0.start{counter-reset:lst-ctn-kix_vehpn6d87zq0-0 0}.lst-kix_cpduhxrsj0v-2 > li:before{content:"" counter(lst-ctn-kix_cpduhxrsj0v-2,lower-roman) ". "}.lst-kix_t48q2cxz4x5b-0 > li:before{content:"● "}.lst-kix_t48q2cxz4x5b-4 > li:before{content:"○ "}.lst-kix_cpduhxrsj0v-6 > li:before{content:"" counter(lst-ctn-kix_cpduhxrsj0v-6,decimal) ". "}.lst-kix_cpduhxrsj0v-0 > li:before{content:"" counter(lst-ctn-kix_cpduhxrsj0v-0,decimal) ". "}.lst-kix_cpduhxrsj0v-8 > li:before{content:"" counter(lst-ctn-kix_cpduhxrsj0v-8,lower-roman) ". "}ol.lst-kix_at3gng7kxydv-5.start{counter-reset:lst-ctn-kix_at3gng7kxydv-5 0}.lst-kix_t48q2cxz4x5b-6 > li:before{content:"● "}ul.lst-kix_2qy1kbsxslq6-0{list-style-type:none}ul.lst-kix_2qy1kbsxslq6-1{list-style-type:none}.lst-kix_cpduhxrsj0v-3 > li{counter-increment:lst-ctn-kix_cpduhxrsj0v-3}ul.lst-kix_2qy1kbsxslq6-2{list-style-type:none}ul.lst-kix_2qy1kbsxslq6-3{list-style-type:none}ul.lst-kix_2qy1kbsxslq6-4{list-style-type:none}ul.lst-kix_2qy1kbsxslq6-5{list-style-type:none}ul.lst-kix_2qy1kbsxslq6-6{list-style-type:none}ul.lst-kix_2qy1kbsxslq6-7{list-style-type:none}ul.lst-kix_2qy1kbsxslq6-8{list-style-type:none}ul.lst-kix_8j1vjl8355n6-3{list-style-type:none}ul.lst-kix_8j1vjl8355n6-4{list-style-type:none}ul.lst-kix_8j1vjl8355n6-1{list-style-type:none}ul.lst-kix_8j1vjl8355n6-2{list-style-type:none}.lst-kix_cpduhxrsj0v-4 > li:before{content:"" counter(lst-ctn-kix_cpduhxrsj0v-4,lower-latin) ". "}ul.lst-kix_8j1vjl8355n6-0{list-style-type:none}.lst-kix_t48q2cxz4x5b-2 > li:before{content:"■ "}.lst-kix_at3gng7kxydv-8 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-8,lower-roman) ". "}.lst-kix_at3gng7kxydv-7 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-7,lower-latin) ". "}.lst-kix_at3gng7kxydv-6 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-6,decimal) ". "}.lst-kix_cpduhxrsj0v-7 > li{counter-increment:lst-ctn-kix_cpduhxrsj0v-7}ol.lst-kix_at3gng7kxydv-1.start{counter-reset:lst-ctn-kix_at3gng7kxydv-1 0}.lst-kix_at3gng7kxydv-5 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-5,lower-roman) ". "}.lst-kix_at3gng7kxydv-2 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-2,lower-roman) ". "}.lst-kix_at3gng7kxydv-0 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-0,decimal) ". "}.lst-kix_at3gng7kxydv-4 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-4,lower-latin) ". "}.lst-kix_at3gng7kxydv-3 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-3,decimal) ". "}ol.lst-kix_at3gng7kxydv-7.start{counter-reset:lst-ctn-kix_at3gng7kxydv-7 0}ol.lst-kix_vehpn6d87zq0-5.start{counter-reset:lst-ctn-kix_vehpn6d87zq0-5 0}.lst-kix_at3gng7kxydv-1 > li:before{content:"" counter(lst-ctn-kix_at3gng7kxydv-1,lower-latin) ". 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"}.lst-kix_n6gzn5j4l1fy-1 > li:before{content:"" counter(lst-ctn-kix_n6gzn5j4l1fy-1,lower-latin) ". 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Calculus - Math 9B - MidtermIf you want to learn more check out What are the characteristics of crystalline candy?

Study Guide

(5.1 - 6.3)

5.1

Formulas:

  • antiderivative of f(x) is F(x), such that F’(x) = f(x) integration or antiderivative of f➝ഽf(x)dx
  • Constant “c” is added to the solution of ഽf(x)dx

Eg: ഽf(x)dx = f(x) + CDon't forget about the age old question of Name the 5 classes of lipids.
Don't forget about the age old question of What is the charactertistic of prosocial behavior?

Example of integration (antiderivatives) with trigs:

  1. ഽsin x dx = -cos (x) + c
  2. ഽcos (x) dx = sin (x) + c
  3. ഽsec² (x) dx = tan (x) + c
  4. ഽ csc² (x) = -cot (x) + c
  5. ഽ sec (x) tan (x) dx = sec (x) + c
  6. ഽcsc (x) cot (x) = -csc (x) + c

We also discuss several other topics like What is social gospel?

Initial value theorem:

  • Derivative of position = velocity s’ (x) = v (x)
  • Derivative of velocity = acceleration v’ (x) = a (x)

        Example:

  • The acceleration due to gravity of a falling object is -32ft/s². At the time f=3, a falling object had a velocity of -10ft/s. Find equation of the object’s velocity
  • We know: derivative velocity = acceleration

         V’= -32                                        we also know v(3)=-10If you want to learn more check out What is the history fo the exile in 597 b.c.?

We want to find v: ഽv’= ഽ-32                                        now we find “c”

                         v= -32ഽ+c

                      -10 = -32(3) + c

                      -10 = -96 + c                                Equation: -32t + 86 = ↓We also discuss several other topics like What does horticulture mean?

                      +96   +96

                        86 = C

  • Understanding motion, plug in t=# to find it’s sign (upward or downward)

Example 2: find f(t)

                f’’(t) = cos(t) , f’(0)=3 and f(0)=5

  • Derive each function and plug in the input and output

5.2

Definite Integrals

  • Finding total signed area from [a,b] using aഽb+(x)dx
  • a and b are bounds of integration

Note: integrate f(x) before solving the bounds

Important properties of definite integral

 switching bound change problem to negative*

5.3

Riemann Sums (Do not Integrate for Riemann Sum)

3 types

  1. Left hand rule: use the left n number to solve for the area

  1. Right hand endpoint: use the # on the right side of intervals

  1. Midpoint: use the # number between each interval

Summation Notation:

 

  • Add each number from “a” up until “b”

Example:

5.4

Fundamental Theorem of Calculus

States:

Let f be continuous on [a,b] and let f(x) = aഽx(t)d(t). Then F is a differentiable function on (a,b) and F’(x)=f(x)

States 2:

Let f be continuous on [a,b] and let F be any antiderivative ct f. Then aഽbf(x)dx = F(b) - F(a)

Ex: 0ഽ4 (4x-x²) = F(4) - F(0) = (2(4)² - ⅓ 4^3) = 32/3

  1. Integrate 4x - x2
  2. Plug in b in integrated f(x) and solve.

FTC and Chain rule

Derive 2ഽx2 (n(t)d+ ➝ dy/dx 2ഽx2 ln(t)dt         f(x)=2²             f’(x)=2x

                        

                        = ln(x²)2x

  1. Derive “b” which is x2
  2. Replace “t” with “b”
  3. Multiply together

Area between Curves

*usually bounds aren’t given    

*integration f(x) and g(x) before using bounds

Find bound by putting f(x) = g(x) to find two bounds [a,b]

Example:

Find bound

Mean Value Theorem and Average Value

*Mean value = f(c)

To find:

*Use c in f(x) function

*Average value=

Integrate f(x)

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