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Test III Calculus I Place your name in the upper right hand corner. Show your work on a separate sheet of paper. Staple your work sheet(s) to the test when you hand it in. You may use your calculator (but not to do calculus), the trig. identities, and the area via summation sheets. Please enjoy yourself, and remember, Calculus is Latin for more fun than a human being should be allowed to have. _____________ 1) Use the summation technique to find the area under the curve y = x 2 between x = a / 2 and x = a 3) A box with a square base and an open top is to have a volume of V . Find the dimensions of the box that minimizes the material being used. 4) A box with a cylindrical base and an open top is to have a volume of V . Find the dimensions of the box that minimizes the material being used. 5) Find the area of the largest rectangle that can be inscribed in the circle, x 2 + y 2 = r 2 . 6) Compute the following derivatives with respect to x. (e ( ) ) x x ____________ a) y(x) = ln (e ) __________ e) y(x) = ____________ b) y(x) = sin(x 2 ) __________ f) y(x) = tan(tan !1 (x)) 2 ____________ c) y(x) = ee e ____________ d) y(x) = ln x e x ! e! x 2 7) Compute the following integrals __________ a) "x __________ c) "x 3 x2 ! 3 dx ! 3x + 1 dx ! n __________ b) ! ln(x) dx __________ d) #e x!" dx 7) Draw a curve with the following properties; on (!",a) y’ > 0 and y’’< 0, on (a,b) y’ < 0, and y’’ < 0, on (b,c ) y’> 0, and y’’ > 0, on (c, d ) y’ > 0 and y’’< 0 on (d,!) y’=0. Assume that a < b < c < d. Also note that the derivatives need not be smooth at x=a, x=b, x=c, x=d. 8) If f(x) is a differentiable function such that ! x 0 f (t)dt = ( f (x)) for all x, find f. 3