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Station A: Riemann SumsThe table give the values of a function from a survey. Use the values to estimate 010f(t) dt for the following Riemann sums of five equal subintervals.tf(t)0214.528312.5418524.5632740.5850960.51072For each approximation be sure the state whether the value is too big or too small. Explain clearly or draw a diagram.L5R52. fx=.25x2-3x 0≤x≤6with 3 subintervals M33. y=34x 0≤x≤8 with 4 subintervalsT4Station B: Indefinite/Definite Integrals1. .532-1x dx2. 132x-65 dx3. 2sec2x4. 0π1+cosx dx5. 4secxtanx dx6. 491-uu du 7. 0πsinx dx8. 3t2-t+tt3 dt 9. 05x32 dxStation C: U-substitution1. 0136(2x+1)3 dx2. -112xsin1-x2 dx3. 3x2(2x3+2)8 dx4. 4xx2+5 dxStation D: FTC1. Given fx=0x(t3-t)5 dt find f'(x) 2. Given hx=2xtan3u du find h'(x)3. Given gx= x6ln1+t2 dt find g'(x)4. Given fx=6x2cot3t dt find f'(x)5.If hx= sinxcosxt2 dt find h'(0)6. If gx=xxsinr2dr find g'(1)Station E: Initial ConditionDirections: Construct a function f(x) given the following conditions.1. f'x=sinx, and y=0 when x=12. dydx=3x2+2x, and y=2 when x=13. f'x=2-x+cosx, and y=-2 when x=04. d2ydx=x2, and f'0=3 when f1=5Station F: Properties of Integrals1. Suppose f and g are continuous functions and that,12fxdx=-415fxdx=615gxdx=8Calculate the following:22gxdx51gxdx123fxdx25fxdx2. Suppose h is a continuous function and that,-11hrdr=0 -13hrdr=6Calculate the following:13hrdr-31hrdrStation G: GraphicallyLet gx= 0xftdt, where f(t) is the function whose graph is shown. Find g(0)d. Find g(0)Find g(3)e. Find g’(0)Find g(-3)f. Find g’(3)On what intervals is g(x) concave up?On what intervals is g(x) decreasing?Where does g(x) have an absolute minimum on the interval 0, 5? Be sure to justify your answer!Station H: MiscellaneousFind an expression in terms of c and d for the value of the definite integral: cd4x3+ x-3 dxFind an expression in terms of c and d for the value of the definite integral: cdx+ 5 dxEvaluate 05(1+ 25-x2 ) dx by finding the area under the curve.4. Evaluate -70(1+ 49-x2 ) dx by finding the area under the curve.5. Find a function fx such that ax2f(t)t3dt=3x for x>0 and some number a by taking the derivative of each side and solving for fx. 6. Find a function fx such that ax5f(t)t2dt=2x for x>0 and some number a by taking the derivative of each side and solving for fx. ................
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