Math1011_Prelim3A - Math1011 Fall 2000 Prelim III-A1...

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Math1011 Fall 2000 Prelim III-A1 Consider the function ƒ (x) = 1 / x 2 on the interval [1,2]. Write the Riemann sum for ƒ corresponding to the subdivision of [ 1, 2] into 5 equal subintervals and the choice of left-hand endpoint in each subinterval. (Do not evaluate the sum!) Without evaluating the sum above, or the definite integral 2 1 2 x dx , decide which of these two numbers is bigger. Explain.
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Math1011 Fall 2000 Prelim A1 Consider the function ƒ (x) = 1 / x 2 on the interval [1,2]. Write the Riemann sum for ƒ corresponding to the subdivision of [ 1, 2] into 5 equal subintervals and the choice of left-hand endpoint in each subinterval. (Do not evaluate the sum!) 2 22222 1 21 1 55 9 678 555 5 11 1 1 1 1 5 5 ( 1 )( 6 / 5 7 / 5 8 / 5 9 / 5 ) () Left-hand endpoints are 1, , , , and x ba n fx L = Δ= = = ⎡⎤ =+ + + + ⎣⎦ Without evaluating the sum above, or the definite integral 2 1 2 x dx , decide which of these two numbers is bigger. Explain. ƒ (x) is a decreasing function, so left hand endpoints will yield an overestimate So, L 5 is larger than 2 1 2 x dx 0 0.2 0.4 0.6 0.8 1 1.2 1 1.2 1.4 1.6 1.8 2 (Doc #011p.51.01t) Overestimate
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M ath1011 Spring 2007 Prelim A2 Use two rectangles and (i) Left-endpoint rule and (ii) Right- endpoint rule to establish the area under the graph of f(x) = x³ between x = 1 and x = 5. Clearly label which part is which. Write down (but do NOT evaluate) the following area as a limit using sigma notation: the area under the graph of g(x) = e x /x between x = 2 and x = 8. Note: do NOT just use the symbols “x i ” or “ x” in your final answer. Rather, find an expression for them depending only on i and n and use them properly in your answer.
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M ath1011 Spring 2007 Prelim A2 Use two rectangles and (i) Left-endpoint rule and (ii) Right- endpoint rule to establish the area under the graph of f(x) = x³ between x = 1 and x = 5. Clearly label which part is which. Left-endpoint Rule Right-endpoint Rule L(f,P)=2(1³+3³)=2(1+27)=56 U(f,P)=2(3³+5³)=2(27+125)=304 Write down (but do NOT evaluate) the following area as a limit using sigma notation: the area under the graph of g(x) = e x /x between x = 2 and x = 8. Note: do NOT just use the symbols “x i ” or “ x” in your final answer. Rather, find an expression for them depending only on i and n and use them properly in your answer. 6 2 6 * 1 66 1 6 2 1 () / lim ( ) lim (2 ) lim ( ) i n x nn e fx f + = →∞ = →∞ + = →∞ = Δ + 82 6 * 1 2( ) 2 lim ( ) ba xa −− = →∞ Δ === = + Δ=+ =+ Δ (Doc #011p.51.02t)
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M ath1011 Fall 2007 Prelim A3 2 4 Let ( ) ( 2) , and P be the partition of [0,4] that divides the interval into four equal pieces. fx x =− 4 Find the lower (Riemann) sum, L(f,P ) 4 Find the upper (Riemann) sum, U(f,P ) 4 44 0 Show that L(f,P ) ( ) (f,P ) by evaluating the integral using the method of your choice. fxdx U ≤≤
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M ath1011 Fall 2007 Prelim A3 2 4 Let ( ) ( 2) , and P be the partition of [0,4] that divides the interval into four equal pieces.
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This note was uploaded on 12/06/2010 for the course MATH 1110 taught by Professor Martin,c. during the Fall '06 term at Cornell University (Engineering School).

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Math1011_Prelim3A - Math1011 Fall 2000 Prelim III-A1...

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