Fall 2009 Math 151LIFE Review for Final Examcourtesy: David J. Manuel(covering 1.1 - 6.4)1. Computeddx ˆsin x0p4 + 5t4dt!2. Compute limn→∞nXi=13n3in+ 423. A conical tank shown be low is losing water ata rate of one cubic foot per minute. How fastis the height of the water changing when theheight is 6 feet?4. Suppose you have 600 grams of a radioac tivesubstance which decays e xponentially. After5 years, 2/3 of the substance remains. Whenwill only 10% of the original substance re-main?5. A rectang ular beam is cut from a cylindricallog of radius 10cm. The strength o f the beamis proportional to the width and the square ofthe depth of a cross-sectio n. Find the dimen-sions of the strongest beam.6. Let a be the vector 3i+2j and b be the vector−i + 4j. Find a unit vector in the direction of2a + 3b.7. A triangle has vertices A(−3, −2), B(1, 2),and C(0, 4). Find the angle of the triangle atvertex A.8. Find parametric equations of the line tangentto the graph of r(t) = (t3−t)i +6tt + 1j atthe point where t = 1.9. Find a tangent vector of unit length to thecurve r(t) = (t sin t)i + (t cos t)j at the pointwhere t = π.10. Find the slope-intercept equation of the linetangent to the curve x = tet, y =1√tat thepoint (e, 1).11. Compute limx→2√x + 2 −√2xx2− 2x12. Find the values of a and c which make thefunction below continuous a nd differentiableat x = 3:f(x) =ax2− 9x + c if x < 32ax2+ a2x − 5 if x ≥ 313. Compute limx→−∞√x2+ 44x + 1.14. Compute limx→∞tan12x13x.15. Given f(x) =√x2+ 9, findlimh→0f(4 + h) − f (4)h.16. Find the point(s) on the curve xy2= 1 wherethe tangent line also passes thro ugh the point(0, 1).17. The function f (x) = e2x+ 2x3+ 1 is one-to-one. If g = f−1, what is g′(2)?18. Simplify tan(sin−1x)19. Given f′(x) =3√1 − x2+x3+3xand f12=0,find f (x).20. Given f (3) = 6, f′(3) = 0, and f′′(3) = −4.State everything you k now about the graphof f at x = 3.21. Find the linear approximation to f(x) =3√xat x =278and use it to approximate3√3.22. Approximate the area under the graph off(x) = 16 − x2between x = −2 and x = 4using a partition P = {−2, 0, 4} and takingx∗i= midpoint.23. Find the derivative of g(x) = ln(x2e4x) + e2+x2e+ πx124. Compute limx→0ln(1 + x) − xx2.25. Find f′′if f (x) = e(−1/2)x2.26. Compute limx→2−4x2+ xx2− 4.27. Findddxxcos2(4x).28. Determine the interval(s) on which f (x) =x5−15x4+6 is decreasing and concave
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