Math 220 Sample Exam 4 1) (Section 3.7) Find the dimensions of the rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and graph of y=8-x3 2) (Section 3.7) A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle of the movie screen is as large as possible? 3) (Section 3.7) A rectangular piece of paper is 12 inches high and six inches wide. The lower right-hand corner is folded over so as to reach the leftmost edge of the paper. Find the minimum length of the resulting crease. 4) (Section 2.8) Assume that y is a function of x . Find y' = dy/dx for 5) (Section 2.8) Find all points (x, y) on the graph of x2/3 + y2/3 = 8 (See diagram.) where lines tangent to the graph at (x, y) have slope -1 6) (Section 3.8) A 200 foot tree is falling in the forest; the sun is directly overhead. At the o moment when the tree makes an angle of 30 with the horizontal, its shadow is lengthening at the rate of 50 feet/sec. How fast is the angle changing at that moment ? 7) (Section 3.8) A trough is 10m long and its ends have the shape of equilateral triangles (i.e. all three sides have equal length) that are 2m across, as shown in the diagram below. If the trough is being filled with water at the rate of 12m3 / min , how fast is the water level rising when the water is 60cm deep? 2m 10m 8) (Section 3.8) Two parallel sides of a rectangle are being lengthened at the rate of 2in / sec while the other two sides are shortened in such a way that the figure remains a rectangle with constant area A of 50 square inches. What is the rate of change of the perimeter P when the length of an increasing side is 5in? 9) (Section 4.1) Find the general anti-derivative:  x  2 x 3/ 4 dx x 5/ 4 4  1  x2 dx 10) (Section 4.1) Find all functions that satisfy f '''( x)  sin x  e x 11) (Section 4.1) Find a function f(x) such that the point (-1, 1) is on the graph of y=f(x), the slope of the tangent line at (-1, 1) is 2 and f ''( x)  6 x  4 12) (Section 4.2) Translate this sentence into summation notation and then use the summation rules to compute the sum: the sum of the squares of the first 24 positive integers. 2 13) (Section 4.3) Approximate the area under the curve y  f ( x)  4  x on the interval [-1, 1] using 16 rectangles and right endpoints as evaluation points. 14) (Section 4.3) Use Riemann sums and a limit to compute the exact area under the curve y  f ( x)  4 x  2 on the interval [1, 3]. 15) (Section 4.4) Evaluate the integral by computing the limit of Riemann sums: 5  (1  2 x 3 ) dx 0 16) (Section 4.4) Evaluate each integral by interpreting it in terms of areas 3 3 a)  (2  x)dx b)  9  x 2 dx 1 17) (Section 4.4) If 0 1 4 4 3 0 0 3 1  f (t )dt  2,  f (t )dt  6, and  f (t )dt  1, find  f (t )dt 18) (Section 4.6) Evaluate each integral a) cos(1 / x)  x 2 dx b)  3 x2 dx x3 19) (Section 4.6) Evaluate each integral 4 a)  0 x dx 1  2x a b) x a 2  x 2 dx 0 9 20) (Section 4.6) If f is continuous and  0 3 f ( x)dx  4, find  xf ( x 0 2 ) dx