MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Fall 2006� � 18.01 Practice Questions for Exam 3 – Fall 2006 � 1 � �/2x dx 31. Evaluate a) b) cos x sin 2x dx 2 0 �1 + 3x�/3 1 2. Evaluate x dx directly from its definition as the limit of a sum. 0 n 1 Use upper sums (circumscribed rectangles). You can use the formula �i = n(n + 1). 2 1 3. A bank gives interest at the rate r, compounded continuously, so that an amount A0 deposited grows after t years to an amount A(t) = A0ert . You make a daily deposit at the constant annual rate k; in other words, over the time period �t you deposit k�t dollars. Set up a definite integral (give reasoning) which tells how much is in your account at the end of one year. (Do not evaluate the integral.) 4. Consider the function defined by F (x) = � x �3 + sin t dt. Without attempting to find 0 an explicit formula for F (x), a) (5) show that F (1) � 2; b) (5) determine whether F (x) is convex (“concave up”) or concave (“concave down”) on the interval 0 < x < 1; show work or give reasoning; c) (10) give in terms of values of F (x) the value of � 2 �3 + sin 2t dt. 1 x 5. If f(t) dt = e 2x cos x + c, find the value of the constant c and the function f(t). 0 6. A glass vase has the shape of the solid obtained by rotating about the y-axis the area in the first quadrant lying over the x-interval [0, a] and under the graph of y = �x. By slicing it horizontally, determine how much glass it contains. 7. A right circular cone has height 5 and base radius 1; it is over-filled with ice cream, in the usual way. Place the cone so its vertex is at the origin, and its axis lies along the positive y-axis, and take the cross-section containing the x-axis. The top of this cross-section is a piece of the parabola y = 6 − x2 . (The whole filled ice-cream cone is gotten by rotating this cross-section about the y-axis.) What is the volume of the ice cream? (Suggestion: use cylindrical shells.) 8. Rectangles are inscribed as shown in the quarter-circle of radius a, with the point x being chosen randomly on the interval [0, a]. Find the average value of their area. 9. Find the approximate value given for the integral below by the trapezoidal rule and also by Simpson’s rule, taking n = 2 (i.e., dividing the interval of integration into two equal subintervals): � �/2 sin6 x dx 0 Other possible problems: Volumes by vertical slicing 4B, Work problems (P.Set
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