questions in calculus

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PRACTICE FINAL FALL 2008 1. Evaluate the following integrals: (a) e (b) x (c)  (d)  (x (e)  x− (f)  sin (g)  tan cos xdx 2 e x dx dx 25 x 2 − 4 2 − 2x + 4 + 1)( x − 1) 2 dx  (h) x  xe x 2 x cos 3 xdx 5 xdx −x dx 0 x (i) F(x)=  sin( t 2 )dt for 0  x  3 0 Use the Trapezoidal Rule with four equal subdivisions of the closed interval [0, 1] to approximate F (1) . The base of a solid S is the shaded region in the first quadrant enclosed by the coordinate axes and the graph of y = 1 − sin x , as shown in the figure below. For each x , the cross section of S perpendicular to the x -axis at the point (x,0) is an isosceles right triangle whose hypotenuse lies in the xy -plane. 2. (a) Find the area of the triangle as a function of x . (b) Find the volume of S.  Let f be the function defined by f ( x) =  3. n =1 xnnn for all values of x for 3n n! which the series converges. (a) (b) (c) Find the radius of convergence of this series. Find the first three terms of this series to find an approximation of f (−1) . Estimate the amount of error involved in the approximation in part (b). Justify your answer.  4. Let a k = (−1) k +1 k  sin kxdx . 0 (a) Evaluate a k . (b) Show that the infinite series  a k =1  (c) Show that 1   a k  k =1 3 . 2 k converges. 5. At any time t  0 , in days, the rate of growth of a bacteria population is given by y ' = ky , where k is a constant and y is the number of bacteria present. The initial population is 1,000 and the population triples during the first 5 days. (a) (b) (c) Write an expression for y at any time t  0 . By what factor will the population have increased in the first 10 days? At what time t , in days, will the population have increased by a factor of 6? Let R be the shaded region between the graphs of y = 6. 3x 3 and y = 2 x x +1 from x = 1 to x = 3 , as shown in the figure below. (a) (b) Find the area of R . Find the volume of the solid generated by revolving R about the y axis.  x  Let f ( x) = 14x 2 and g ( x) = k 2 sin   for k  0 .  2k  7. (a) (b) (c) (d) Find the average value of f on [1, 4]. For what value of k will the average value of g on [0, k] be equal to the average value of f on [1, 4]. Find the value of c that satisfies the Mean Value Theorem for part (a). Draw a diagram to represent parts (a) and (c) geometrically. Let R be the region bounded by the graph of y = ln x , the line x = e , and the x − axis. 8. (a) (b) 9. Find the volume of the region generated by revolving R about the x − axis. Find the volume of the region generated by revolving R about the y − axis. Let R be the region of the first quadrant bounded by the x -axis and the curve y = kx − x 2 , where k  0 . (a) (b) (c) 10. In terms of k , find the volume produced when R is revolved about the x -axis. In terms of k , find the volume produced when R is revolved about the y -axis. Find the value of k for which the volumes found in parts (a) and (b) are equal. A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table below gives the measurements of the diameter of the blood vessel at selected points along the length of the blood vessel, where x represents the distance from one end of the blood vessel and B (x ) is a twice-differentiable function that represents the diameter at that point. (a) (b) Write an integral expression in terms of B (x ) that represents the average radius, in mm, of the blood vessel between x = 0 and x = 360 . Approximate the value of your answer from part (a) using the data from the table and a midpoint Reimann sum with three subintervals of equal length. Show the computations that lead to your answer. (c) (d) 2  B( x)  Using correct units, explain the meaning of     dx in terms of 2   125 the blood vessel. Explain why there must be at least one value x , for 0  x  360 , such that B ' ' ( x) = 0 . 275 11. Consider the differential equation (a) dy = x 4 ( y − 2) . dx One the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. (a) (b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xy -plane. Describe all the points in the xy -plane for which the slopes are negative. (c) Find the particular solution y = f (x) to the given differential equation with the initial condition f (0) = 0 . 12. Find the area of the region outside the cardiod r = 1+ cos and inside the circle r = 3 sin  . 13. In each of the following cases, decide whether the infinite series converges. Justify your answers. (a) (b)  k k =1 sin k 2 + k   1 1 +   k k =1  k (c)  1  k ln k =1 2 k 14. Let R be the region bounded by the curves f ( x) = (a) (b) 15. 16. 4 and g ( x) = ( x − 3) 2 . x Find the area of R . Find the volume of the solid generated by revolving R about the x axis. Find the horizontal and vertical tangents to the cardiod r = 1− cos , 0    2 . 1 Let f be the function defined by f ( x) = . 1 − 2x (a) (b) (c) 17. Write the first four terms and the general term of the Taylor series expansion of f (x ) about x = 0 . What is the interval of convergence for the series found in part (a)? Show your method. 1 Find the value of f at x = − . How many terms of the series are 4  1 adequate for approximating f  −  with an error not exceeding one  4 per cent? Justify your answer. An automobile’s gasoline tank is in the shape of a right-circular cylinder of radius 8 in. with a horizontal axis. Find the total force on one end when the gasoline is 12 in. deep and  oz/in3 is the weight density of gasoline. 18. Find the centroid of the upper half of the circle of radius r . 19. Find the center of mass of a thin plate of constant density  covering the region bounded above by the parabola y = 4 − x 2 and below by the x − axis. 20. Find the center of mass of a thin, flat plate covering the region enclosed by the parabola y 2 = x and the line x = 2 y if the density function is  ( y ) = 1 + y .
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