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# Week 2 Calc II Notes MATH 142 001

UTK

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This 7 page Class Notes was uploaded by Jared Hopland on Friday January 22, 2016. The Class Notes belongs to MATH 142 001 at UTK taught by Dr. Stephenson in Spring 2016. Since its upload, it has received 30 views. For similar materials see Calculus II in Applied Mathematics at UTK.

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Date Created: 01/22/16

Week 2 Notes Calc II N Summation Notation; ∑ f (x) i=i Used for adding a lot of numbers or series of numbers with a pattern. Ex: 1+2+3+4+…+100 (easy to see pattern) Ex: .5+2+4.5+8+…+128 (not easy to see clear pattern) N ∑ f (x) i=i is a better way of writing these long patterned lists. 16i2 12 22 32 162 Ex: ∑ = + + +…+ =748 (this is the same i=12 2 2 2 2 pattern from above that didn’t seem to have a clear pattern) Properties of Summation Notation n n ai+¿∑ bi=∑ (i +i ) i=1 i=1 n ∑ ¿ i=1 n n i=1kai=k i=1i n ∑ k=k n i=1 Power Sums n 2 n(n+1) n n ∑ i= 2 = 2 +2 i=1 n 2 3 2 ∑ i =n(n+1)(n+1)= n(n+1) = n + n +n i=1 6 6 3 2 6 n n (n+1)2 n4 n3 n 2 ∑ i = = + + i=1 4 4 2 4 Estimating Area “Under” a Curve Finding the area between the curve/function line and x-axis in interval [a,b] Image 1 Image 2 Estimating comes into play on more complicated graphs that we don’t necessarily have an equation for the area of the shape the graph creates. (e.g. polynomials, exponential functions, trigonometry functions, anything with a curve, etc…) Image 3 As a method of estimating we can fill in the area we don’t know with rectangles. (the more rectangles the more accurate the estimation) Image 4 Image 5 Because the more rectangles we have means the more accurate the estimation is, the limit # of rectangles approaches infinity is the actual area. ( R→ ∞¿ Areaundercu)ve Process of Finding the Area of the Rectangles 1.) Start with a function, f(x), and graph it 2.) Choose a base number of rectangles you want to use that would fit well with the function you have. (e.g. fx=x +1 you could choose 4 rectangles or some number with a whole number for the square root [4,9,16,25]) N=number of rectangles or number of subintervals 3.) Find the ∆ x that works with the with the amount of rectangles you chose in the above step on interval [a,b] b−a ∆ x= N f(x=x +1 Ex: on [1,3], N you chose is N=4, 3−1 1 ∆ x= = 4 2 4.) Find area of rectangles LN For (Approximation using the left endpoint) Image 6 ∆ x∗f(a+0∆x )+∆ x∗f (a+1∆ x)+∆ x∗f a+2∆ x +…+∆x∗f (a+(N−1)∆x) N−1 Or ∑ ∆ x∗f (a+i∆ x) i=0 For RN (Approximation using the right endpoint) Image 7 ∆ x∗f(a+1∆x )+∆ x∗f a+2∆ x +∆ x∗f (a+3∆ x +…+∆ x∗f (a+N ∆ x) N Or ∑ ∆x∗f (a+i ∆ x) i=1 L =R N N as N →∞ (R f ,P ,Cfor interval[a ,b]) Riemann Sum a=x <x <x <…<x =b Partition P: 0 1 2 N C ∈[x , x ] Sample Points(C): i i−1 i (∆x) ∆ x =x −x Subintervals : i i i−1 N R(f ,P ,= ∑ f( i∆ xi i=1 Norm ¿P|∨¿ : The maximum intervals of ∆ i If the limit of Riemann sums exists it is equal to the b f(xdx= lim R( f ,P ,C) definite integral ∫a ¿P|∨→0 We say that f is integrable over [a,b] if the limit exists The limit won’t exist if f(x) isn’t a continuous function “Signed” Area If f(x) goes below the x-axis then the area between the curve and the x-axis is considered to be negative. Image 8 Area of fx on a,b = (∑ of areaabovethe xaxis)−(∑ of areabelowthe xaxis) Definite Integral b ∫ f x dx=signed areaof f x oninterval[a,b] a Properties of Definite Integrals b ∫ k=k(b−a) a b b b ∫ f (x)±g(x)dx= ∫ f(x)dx− ∫ (x)dx a a a b b ∫ k∗f x dx=k ∫ f(x)dx a a a ∫ f(x)dx=0 a b a ∫ f(x)dx=− ∫ f(x)dx a b (a and b are flipped) b c c If a≤b≤c then ∫ f x dx+ ∫ f(x)dx= ∫ f x dx a b a Image 9 Fundamental Theorem of Calculus (FTC) f x dx=¿F (b)−F(a) b If F = f ,th∫n ¿ a Explanation: A function, f, is continuous on [a,b] and F is the anti-derivative of f on [a,b] b F =f ∴ ∫ f x =F b −F(a) a Further explanation see image 10 8 ∫ 3x +4x+1 Ex: 3 2 3 2 F ( ) ∫x +4 x+1=x +2x +x (8) [¿¿3+2 (8)+(8)]−[(3)+2 (3)+(3)] F b −F a =¿ F (b−F a = [12+128+8 −] [+18+3 ] F (b−F a = [48 ] [ ]=600 Keep in mind the function needs to be continuous on [a,b] 1 x dx Ex: −1 Does not exist (DNE) because the x−1 function isn’t continuous at x=0 Fundamental Theorem of Calculus II (FTC II) If a function, f, is continuous on [a,b] then the area x function, A(x), would be as follows: A (x)=∫a ft)dt A (x)=f (x) This also means d x ∫ f (t)dt= f (x) dx a b f (x)a means f(x) evaluated on [a,b] x x Ex: A(x)=∫0os t)dt=sin?(t)¿0=(sin(x−0 =)in ?(x) 3 x d ∫√ 1+t dt= d −∫√ 1+t dt =− 1√x 4 Ex: dx x dx ( 3 ) x2 d ∫ cos1+t 1)dt Ex: dx 0 x 10 A (x)=∫cos? (1+t )dt 0 ' 10 A x)=cos?(1+x ) 10 2 cos 1+t )2t=¿ A(x ) x ∫ ¿ 0 d A(x2=A (x2∗2x=cos 1+x 20∗2x dx (Chain Rule)

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