Test 4 Review. dy du 9 5. sin5 zdz. dt. 5 Ê. x 2 È 1, 3. 2cos( x) dx is less than using Simpson's. ,1 t 5 t 2. ft () t2 4.
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1 Name: Class: Date: ID: A Test Review Short Aswer. Fid the geeral solutio of the differetial equatio below ad check the result by differetiatio. dy du 9 u. Use the error formula to estimate the error i approximatig the itegral ( 9x ) dx with usig Trapezoidal Rule.. Fid the smallest such that the error estimate i the approximatio of the defiite itegral cos(x) dx is less tha. usig Simpso's Rule.. Use the Trapezoid Rule to approximate the value of the defiite itegral x dx wth. Roud your aswer to four decimal places.. Fid the idefiite itegral of the followig fuctio ad check the result by differetiatio. ( s) s ds 6. Evaluate the followig defiite itegral. z cos 7z ˆ dz Use a graphig utility to check your aswer Fid the idefiite itegral x x dx. 8. Fid the idefiite itegral of the followig fuctio. sizdz 9. Fid the idefiite itegral t t ˆ dt.. Fid the average value of fx ( ) È iterval ÎÍ,. x ˆ o the x. Fid the average value of the fuctio over the give iterval ad all values t i the iterval for which the fuctio equals its average value. ft () t,t t Use a graphig utility to verify your results.. Fid the idefiite itegral ( z) dz. È. Determie all values of x i the iterval ÎÍ, for x ˆ which the fuctio fx ( ) equals its x average value 6.. Fid the area of the regio bouded by the graphs of the equatios y x x, x, y. Roud your aswer to the earest whole umber.
2 Name: ID: A. Determie the area of the give regio. y x six, x 9. Evaluate the defiite itegral of a fuctio u Use a graphig utility to verify your results. du.. Evaluate the itegral. x ˆ dx give, x dx 7, 6. Evaluate the defiite itegral of the trig fuctio. ( six cosx)dx Use a graphig utility to verify your results. 7. Evaluate the defiite itegral of the algebraic fuctio. x dx Use a graphig utility to verify your results. 8. Evaluate the defiite itegral of the fuctio. x ( t cost) dt Use a graphig utility to verify your results. x dx xdx dx. 7, 7,. Sketch the regio whose area is give by the defiite itegral ad the use a geometric formula to evaluate the itegral. a a a z dz. Write the followig limit as a defiite itegral o the iterval [, ], where c i is ay poit i the ith subiterval. lim x i 7ci ˆx i
3 Name: ID: A. Write the followig limit as a defiite itegral o È the iterval ÎÍ, 8 where c i is ay poit i the i th subiterval. 7. Sketch the regio whose area is give by the defiite itegral ad the use a geometric formula to evaluate the itegral. lim x i c i c i x i uˆ du ˆ 6. Write the limit lim x i, as a defiite i c i È itegral o the iterval ÎÍ 8,, where c i is ay poit i the i th subiterval.. The graph of the fuctio fx ( ) 6 x is give below. Which of the followig defiite itegrals yields the area of the shaded regio? 8. Evaluate the followig defiite itegral by the limit defiitio. s ds 9. Use the summatio formulas to rewrite the k k expressio ( ) without the summatio otatio. k 7. Fid the limit. lim i i ˆ 8 ˆ. Use the limit process to fid the area of the regio betwee the graph of the fuctio y 6 x ad È x-axis over the iterval ÎÍ,.. Fid the sum give below. 6. Evaluate the followig defiite itegral by the limit defiitio. 9s ˆ ds 8 k kk ( ). Use sigma otatio to write the sum.
4 Name: ID: A. Solve the differetial equatio. dp dx x, P( ) 8. Fid the limit of s( ) as. s ( ) È ( ) 7 ÎÍ. The diagram below shows upper ad lower sums for the fuctio usig subitervals. Use upper ad lower sums to approximate the area of the regio usig the subitervals. 6. The maker of a automobile advertises that it takes secods to accelerate from kilometers per hour to 8 kilometers per hour. Assumig costat acceleratio, compute the acceleratio i meters per secod per secod. Roud your aswer to three decimal places. 7. A ball is throw vertically upwards from a height of ft with a iitial velocity of ft per secod. How high will the ball go? Note that the acceleratio of the ball is give by a(t) feet per secod per secod. 8. Fid the idefiite itegral sis 7cossds. 9. Fid the idefiite itegral ad check the result by differetiatio.. Fid the idefiite itegral u ˆ u du.. Fid the idefiite itegral s ˆ s ds. z z 9 z dz. Fid the idefiite itegral x 8 dx.
5 Test Review Aswer Sectio SHORT ANSWER. ANS: 9 Yu ( ) u C PTS: DIF: Medium REF:..7 OBJ: Calculate the geeral solutio of a differetial equatio NOT: Sectio.. ANS: PTS: DIF: Easy REF:.6.a OBJ: Estimate the error i usig the Trapezoidal Rule to approximate a defiite itegral NOT: Sectio.6. ANS: 9 PTS: DIF: Medium REF:.6.b OBJ: Idetify the smallest value of eeded to approximate a defiite itegral to withi a desired degree of accuracy NOT: Sectio.6. ANS: 7.6 PTS: DIF: Easy REF:.6. OBJ: Approximate a defiite itegral usig the Trapezoidal Rule NOT: Sectio.6. ANS: s s C PTS: DIF: Easy REF:..7 NOT: Sectio. 6. ANS: PTS: DIF: Medium REF:..8 OBJ: Evaluate the defiite itegral of a fuctio usig substitutio NOT: Sectio.
6 7. ANS: x ˆ 9 C PTS: DIF: Medium REF:.. usig substitutio NOT: Sectio. 8. ANS: cosz C PTS: DIF: Easy REF:..7 usig substitutio NOT: Sectio. 9. ANS: t ˆ 6 C PTS: DIF: Easy REF:.. usig substitutio NOT: Sectio.. ANS: PTS: DIF: Easy REF:..a OBJ: Calculate the average value of a fuctio over a give iterval NOT: Sectio.. ANS: The average is 9 ad the poit at which the fuctio is equal to its mea value is. PTS: DIF: Medium REF:.. OBJ: Calculate the average value of a fuctio over a give iterval ad idetify the poit at which it occurs NOT: Sectio.. ANS: ( z) 6 C PTS: DIF: Easy REF:.. usig substitutio NOT: Sectio.
7 . ANS: x PTS: DIF: Easy REF:..b OBJ: Idetify the poits where a fuctio equals its average value over a give iterval NOT: Sectio.. ANS: 7 PTS: DIF: Medium REF:.. OBJ: Calculate the area bouded by a fuctio MSC: Applicatio NOT: Sectio.. ANS:. PTS: DIF: Medium REF:..8 OBJ: Calculate the area bouded by a fuctio MSC: Applicatio NOT: Sectio. 6. ANS: 6 PTS: DIF: Easy REF:..7 OBJ: Evaluate the defiite itegral of a fuctio NOT: Sectio. 7. ANS: 98 PTS: DIF: Medium REF:.. OBJ: Evaluate the defiite itegral of a fuctio NOT: Sectio. 8. ANS: PTS: DIF: Easy REF:.. OBJ: Evaluate the defiite itegral of a fuctio NOT: Sectio. 9. ANS: PTS: DIF: Medium REF:..6 OBJ: Evaluate the defiite itegral of a fuctio NOT: Sectio.. ANS: 9 PTS: DIF: Easy REF:..8 OBJ: Evaluate the defiite itegral of a fuctio NOT: Sectio.
8 . ANS: a PTS: DIF: Easy REF:.. OBJ: Evaluate a defiite itegral geometrically NOT: Sectio.. ANS: ( 7x )dx PTS: DIF: Easy REF:..9 OBJ: Evaluate a defiite itegral by the limit defiitio NOT: Sectio.. ANS: 8 x x dx PTS: DIF: Easy REF:.. OBJ: Write a limit as a defiite itegral o a iterval NOT: Sectio.. ANS: 8 6 x dx PTS: DIF: Easy REF:.. OBJ: Write a limit as a defiite itegral o a iterval NOT: Sectio.. ANS: 6 x ˆ dx PTS: DIF: Easy REF:..7 OBJ: Write a defiite itegral for a bouded regio NOT: Sectio.
9 6. ANS: PTS: DIF: Easy REF:..8 OBJ: Evaluate a defiite itegral by the limit defiitio NOT: Sectio. 7. ANS: PTS: DIF: Easy REF:..9 OBJ: Evaluate a defiite itegral geometrically NOT: Sectio. 8. ANS: 7 PTS: DIF: Easy REF:.. OBJ: Evaluate a defiite itegral by the limit defiitio NOT: Sectio. 9. ANS: PTS: DIF: Medium REF:..7 OBJ: Rewrite a sum without summatio otatio NOT: Sectio.. ANS: PTS: DIF: Medium REF:.. OBJ: Evaluate a ifiite limit of a sum NOT: Sectio.. ANS: 6 PTS: DIF: Medium REF:..6 OBJ: Calculate the area bouded by a fuctio usig the limitig process MSC: Applicatio NOT: Sectio.. ANS: 67 PTS: DIF: Easy REF:.. OBJ: Calculate a sum give i sigma otatio NOT: Sectio.
10 . ANS: j j PTS: DIF: Easy REF:..8 OBJ: Write a sum i sigma otatio NOT: Sectio.. ANS: Px ( ) x x 9 PTS: DIF: Medium REF:..9 OBJ: Solve a differetial equatio NOT: Sectio.. ANS: ubouded PTS: DIF: Medium REF:..7 OBJ: Evaluate limits at ifiity NOT: Sectio. 6. ANS:.7 m/sec PTS: DIF: Medium REF:..8a OBJ: Calcuate the acceleratio MSC: Applicatio NOT: Sectio. 7. ANS:.6 ft PTS: DIF: Medium REF:..7 OBJ: Solve differetial equatios related to positio/velocity/acceleratio MSC: Applicatio NOT: Sectio. 8. ANS: cos s 7sis C PTS: DIF: Easy REF:.. NOT: Sectio. 9. ANS: z 6 z z C PTS: DIF: Medium REF:..8 NOT: Sectio.. ANS: x C PTS: DIF: Easy REF:.. NOT: Sectio. 6
11 . ANS: lower:.66 ; upper:.666 PTS: DIF: Medium REF:.. OBJ: Estimate the area of a regio usig upper ad lower sums NOT: Sectio.. ANS: u u C PTS: DIF: Easy REF:..7 NOT: Sectio.. ANS: s 6s s C PTS: DIF: Easy REF:.. NOT: Sectio. 7
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