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Sample-MT1

# Sample-MT1 - more accurate and why If one approximates(3 x...

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MATH 126 Midterm I Sample Problems Most of the following problems have appeared on my midterms or finals in the past. A. Find: i) D (arcsin( x 2 )); ii) sec(arcsin( 1– x 2 )); iii) e 5 x / 2 dx e 2 x + e 3 x " . B. Determine if the following limits exist, including ± : i) cos( x ) – cos(2 x ) cos(3 x ) – cos(4 x ) as x 0; ii) (cos( x )) 1/ x 2 as x 0; iii) 1 x " # \$ % & x as x C. Find antiderivatives for: i) cos 4 ( x ); ii) (sec( x )tan( x )) 3 ; iii) x 3 ( x 2 + 1) 1/3 ; iv) x + x 3 1 + x 4 ; v) x 2 + 2 x 3 + 2 x 2 + 2 x ; vi) 1 e 3 x –1 ; vii) x 5 1 + x 2 ; viii) 1– x 2 x 4 ix) cos 2 ( x ) – cos 3 ( x ) D. Determine if the following exist, and if so find the value: i) lim t "# dx x 4 + x 2 1 t \$ ; ii) lim t "# dx x 5/ 3 x 2 t \$ ; iii) lim t "# dx x x 8 –1 2 1/4 t \$ ; iv) lim t " 0 + ln 2 ( x ) dx t 1 # . E. If both the Trapezoid Rule and Simpson's Rule are used to approximate f ( x ) dx a b " with the same partition having a large number of equally spaced points, which rule is likely to be
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Unformatted text preview: more accurate and why? If one approximates (3 x 2 + 5 x " 7) dx a b # using the Trapezoid Rule with 10 8 intervals and with Simpson's Rule using 10 intervals, which is more accurate? Why? Write out the general expressions for the Trapizoid Rule and Simpson's Rule for f ( x ) dx a b " using the partition P = { a = x , x 1 , . . . , x 2 n = b } with all x i – x i –1 = ( b – a )/2 n . Answers: Ai) 2 x /(1 – x 4 ) 1/2 ; Aii) 1/ x ; Bi) 3/7; Bii) e –1/2 ; Biii) 0 Di) 1 – π /4; Dii) –3ln((2 2/3 – 1) 1/2 /2 1/3 ); Diii) π /24; Div) 2....
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