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by: Ren K.

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# MTH 132 Calculus Week 3 notes MTH 132

Marketplace > Michigan State University > Mathematics > MTH 132 > MTH 132 Calculus Week 3 notes
Ren K.
MSU
GPA 4.0

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These are all the lecture notes from week 3 of calculus.
COURSE
Calculus 1
PROF.
Z. Zhou
TYPE
Class Notes
PAGES
7
WORDS
CONCEPTS
Z.Zhou, Zhou, MTH, 132, Engineering, Calculus, Math, Calc
KARMA
25 ?

## Popular in Mathematics

This 7 page Class Notes was uploaded by Ren K. on Sunday September 18, 2016. The Class Notes belongs to MTH 132 at Michigan State University taught by Z. Zhou in Fall 2016. Since its upload, it has received 3 views. For similar materials see Calculus 1 in Mathematics at Michigan State University.

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Date Created: 09/18/16
MTH 132 ­ Lecture 5 ­ Continuity    Recap of Continuity  ● We studied the limit x → a f(x) and the limit x→a±.  ● Definition: f(x) is continuous at x = a, if and only if lim x→a  f(x) = f(a)  1. Limit exists.  2. f(a) = defined.  3. f(x) = f(a)    ● At a = 1 limit = 1 ≠ f(1) not continuous at 1.  ● At a = 2 limit = 0  ≠ f(2) not continuous at 0.  ● At a = 3 limit = DNE  ≠ f(3) not continuous at f(3)      ● Example:  ○ xsin1/x  ○ f(x) isn’t continuous at x = 0 since f(0) is not defined.  ○ Limit x → 0 f(x)  ○ − |x| ≤ f(x) ≤ |x|  ○ Both ­|x| and |x| = 0, meaning that 0 is a removable singularity.  ○ f(x) is continuous everywhere if g(x) {xsin1/x   x ≠0}  ○                                                          {0            x =0}      Between [a,b]  ● Definition:​ if f(x) is defined in (a,b) we say that it is continuous if f(x) is continuous at  every point of (a,b).  ● If continuous at [a,b] a​_____________​   1. If continuous at m(a,b)   2. Limit x→a f(x) = f(a)  3. Limit x→b f(x) = f(b)  ● Example:  ○ f(x) = 1/x  ○ f(x) is continuous on (­infinity, 0) U (0, infinity)  ■ AUB = A union B  ■ A∩B = A intersection B or the common part of a and b  ○ Limit x → 0+ f(x) = 1 / 0+ = infinity  ○ Limit x → 0­ f(x)  = 1/ 0­ = ­infinity    ● Example:   ○ f(x) = 1/x^2  ○ f(x) is continuous on (­infinity,0)U(0,infinity)  ○ Limit x→ 0 f(x) 1/0+ = infinity    ● Example:  ○ f(x) = (x+2)/(x­2)  ○ f(x) is continuous on all but (­infinity,2)U(2,infinity)  ○ Limit x→2+ = 4/0+ = infinity  ○ Limit x→2­ = 4/0­ = ­infinity  ● Example:  3 ○ f(x) = (x − 4x)/(x   2)   ○ f(x) is continuous on (­infinity,2)U(2,infinity)  ○ Limit = f(x) x→2 = 8  ○ X = 2 is a removable singularity because we can substitute the value of that point  by its limit.    Intermediate value theorem  ● If f(x) is continuous on m[a,b] and L is between f(a) and f(b) then there is at least one  point such that f(c) = L.  ● Continuity = necessary.     ● Example:  ○ f(x) = x + 8x − 10  =  0 has a solution in the interval [0,2]  ○ Has to be one point less than 0, and one point greater than 0.   ○ Since f(x) is continuous is between [f(0),f(2)] by the intermediate value theorem,  there exists cε[0,2] that f(c) = 0.  ● Proof:  ○ f(x) = g(x)  ○ h(x) = f(x) ­ g(x)  ○ h(0) = 0 ­ cos(0) = ­1  ○ h(1000) = 1000 ­ cos(1000) = answer doesn’t matter, if it's a positive value that  means that point c exists.                MTH 132 ­ Lecture 6 ­ Derivatives    Derivatives  ● Given any function f(x) defined in an interval containing (a), the derivative of f at a,  denoted by f’(a) = limit h approaches 0 (f(a+h)­f(a))/h  ● Note that f’(a) = slope of the tangent line at (a, f(a))  ● The derivative of constants is always 0.  ● The derivative of simple equations mx will always be x.  ○ M = slope ex: 4x, 2333232x etc      Example:  ● f(x) = 3x+2  ● f’(3) = 3  ● Why?   ○ 3x +2 = straight line and its slope is 3; that’s the solution.     Example:  ● f(x) = 2x^2 + 5  ● f’(2)= ?  ● [ ​ (2+h​ (2)​ ] / h  ● ( 2(2+h)(2+h) + 5 )​ ​ 2*4 +5)  ● 8 + 8h + 2h^2 +5​ ­​ 13  ● ( 2h^2 + 8h ) / h  ● 2h + 8 limit h approaches 0 = 2(0)+8 = 8    Simplification of Cubed polynomials review  ● (a^3 ­ b^3) = (a­b)(a^2+ab+b^2)    Definition of Derivative functions  ● Definition: f(x) is defined on (a,b), we define the derivative function f’(x) on (a,b) as f’(x) =  [f(x+h) ­ f(x)] / h.  ● Simply alternate phrasing ­ it’s really the same equation, but the professor mentioned  that in some texts they will say this is the equation.         MTH 132 ­ Lecture 7 ­ Derivatives (The easy way)    Derivatives  ● The hard way ­ limit of x as it approaches a [f(x+h)­f(x)]/h  ● The easy way­  ○ f(x) = constant  f’(x) = 0  ○ f(x) = mx           f’(x) = m  ○ f(x) = x^2          f’(x) = 2x  ■ Because 2*x^(2­1) = 2x  ■ Formula power*x^(power ­ 1) = derivative version  ○ f(x)= 1/x          f’(x) = 1/(x^2)  ■ Because 1/x = x^­1   ■ = x^(­1­1)   ■ = x^­2 = 1/x^2  ○ f(x) √  x        f’(x)√= 1/(2 x )  ■ Because  √ = x^½  ■ x^(½) ­ 1 = x^­½  ■ x^­½ = 1/2√x  Theorem  ● If f(x) is differentiable at a, meaning that f’(a) is possible then f(x) must be continuous at  point a.    Exceptions  ● An equation can be continuous at all points, but cannot be differentiated.  ● Example:  ○ f(x) = |x|   ■ X if x≥ 0  ■ ­x if x≤ 0  ○ Because the slope alternates, you cannot derive it.   ○ H tends to 0 must be the same from both sides.

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