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If x2 + y2 = 25, what is the value of y'' at the point (4,3)? -25/27 25/27 7/27 3/4 -7/27 If a function f is continuous for all x and if f has a relative maximum at (-1,4) and a relative minimum at (3,-2), which of the following statements must be true? The graph of f has a point of inflection somewhere between x=-1 and x=3 The graph of f has a horizontal tangent line at x = 3. The graph of f intersects both axes. The graph of f has a horizontal asymptote. f'(-1) = 0 If a, b, c, d and e are real numbers and a ≠ 0, then the polynomial equation ax7 + bx5 + cx3 + dx + e = 0 has at least one real root an odd number of nonreal roots. no real roots. no positive real roots. only one real root What is the average ( mean) value of 3t3 - t2 over the interval -1 ≤ t ≤ 2? 11/4 7/2 33/4 8 16 At t = 0 a particle starts at rest and moves along a line in such a way that at time t its acceleration is 24t2 feet per second. through how many feet does the particle move during the first 2 seconds? 32 48 96 64 192 If y = tant, t= v -(1/v), and v=lnx, what is the value of dy/dx at x = e 1/e 0 1 2/e sec2e The derivative of f(x) = (x4/3) - (x5/5) attains its maximum value at x = -1 0 4/3 1 5/3 What is the area of the region completely bounded by the curve y = -x2 + x + 6 and the line y = 4? 3/2 7/3 9/2 33/2 31/6 ∫ 0 1 ¼ e ¼(e-1) x3ex4 dx = e e - 1 4(e - 1) The region enclosed by the graph of y = x2, the line x = 2, and the x - axis. The volume of the solid generated is (32/5)π 8π 4π (16/3)π (8/3)π The base of a solid is the region enclosed by the graph of y = e-x, the coordinate axes, and the line x = 3. If all plane cross sections perpendicular to the x - axis are squares, then its volume is ½(1 - e-6 ) e-6 ½(e-6 ) e-3 1 - e-3 The diagram shows part of the graph of y = 1/x . The area of the shaded region is 2 units. a = (round to the second decimal place) Let f(x) = ex cos x. Find the gradient of the normal to the curve of f at x = π. (round to the third decimal place) A function f has its first derivative given by f′(x) = (x – 3)3. f ''(3) = f '(3) = Consider the function ƒ(x) = 3x2 – 5x + k. The equation of the tangent to the graph of ƒ at x = p is y = 7x – 9. Find the value of p and k p = k = The function f is such that f '' (x) = 2x – 2.When the graph of f is drawn, it has a minimum point at (3, –7). f(-1) = f '(-1) = f(0) = The graph of y = √x between x = 0 and x = a is rotated 360° about the x-axis. The volume of the solid formed is 32π. Find the value of a. a = The region enclosed between the curves y = √x(ex) and y =e√x is rotated through 2π about the x-axis. Find the volume of the solid obtained. (round off to whole number) ∫ ∫ 1 e 3 0 (lnx)3 2x + 3 x 1 dx = dx = ln √P, P = |