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1997test2

# 1997test2 - Physical Sciences Division University of...

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Physical Sciences Division University of Toronto at Scarborough MATA26Y February 4, 1998 TERM TEST II 1. Let I = Z π/ 2 0 cos( x 2 ) dx . (a) Find n > 0 so that | Simp( n ) - I | ≤ 10 - 5 . The error bound for Simpsons Rule for Z b a f ( x ) dx is M ( b - a ) 5 16 · 180 n 4 where M ≥ | f (4) ( x ) | for all x [ a,b ]. Compute Simp( n ). (b) How large would n have to be to achieve the same accuracy with the Trapezoidal Rule? error bound is M ( b - a ) 3 12 n 2 , M ≥ | f 00 ( x ) | , x [ a,b ] · . (c) How large would n have to be to achieve the same accuracy with the Right Hand Rule? error bound is M ( b - a ) 2 2 n , M ≥ | f 0 ( x ) | , x [ a,b ] · . 2. Use a Taylor polynomial to approximate ln1 . 1 towithin 10 - 5 . 3. Compute the following integrals: (a) Z (ln x ) 2 dx (b) Z 4 + x x dx (c) Z ln(1 + y ) 1 + y dy (d) Z xdx x 2 + 3 x + 2

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MATA26Y page 2 4. (a) Find the average value of the following functions: (i) f ( t ) = cos t t [0 , 2 π ] (ii) g ( t ) = | cos t | t [0 , 2 π ] (iii) k ( t ) = (cos t ) 2 t [0 , 2 π ] (b) Write the averages you have just found in ascending order. Explain clearly, using
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1997test2 - Physical Sciences Division University of...

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