Sample Math 5 Midterm Exam Spring, 04 The midterm examination is on Wednesday, March at 5:45PM 7:45PM The midterm examination will be in Budig 0 Look for your instructor who will direct you where to sit Only simple graphing calculators (TI-84 plus and below) are allowed for the common exams You can do all your work in exam booklets and circle one answer for each problem Having done with all the problems, you will need to complete the bubble sheet with a # pencil You will need to write your instructor s name on margins of the bubble sheet You are considered responsible to bring pens/pencils and a calculator to the common exams Pens or pencils will not be provided for you, and interchanging calculators will be prohibited during the exams The common midterm will cover Chapters and 3 (excluding Section 37) The problems below, all multiple-choice with exactly one correct answer, are intended to be reasonably representative of what might appear on the actual exam, which will have 5 problems The domain of function f(x) = x+3 x x 3 is (, + ) (, 3 ) (3, + ) (C) (, ) (, 3 ) (3, + ) (D) (, ) (, + ) Let f(x) = x and g(x) = 3x + 5 Then, (f g)(x) is (3x + 5) 3 x + 5 (C) x (D) 3x + 5 3 Let f(x) = x + and g(x) = x Then, f(g()) is + 5 (C) 3 (D) 3
4 Find lim x 3 x 9 x 3 limit does not exist (C) 6 (D) 3 5 Find lim +x x 0 x limit does not exist (C) 0 (D) 3x 6 Find lim +x+4 x x 3x+ 4 3 (C) limit does not exist (D) 0 x 7 Find the limit lim +3 x x+ 0 (C) (D) 3 8 Let f(x) = x 4x When simplified, the difference quotient f(x+h) f(x) h becomes x + h 4 (C) 0 (D) x 4 9 Let f(x) = f(x+h) f(x) When simplified, the difference quotient x h x(x + h) becomes (C) h (D) 0 0 The slope of the line passing through the points (-,3) and (3,5) is (C) 4 (D) 4 What is the profit in terms of the cost C(x) to produce x units and the unit price p(x) at which x units will sell? x(p(x) C(x)) p(x) xc(x) (C) p(x) + xc(x) (D) xp(x) C(x)
Given the demand equation 4x+p 36 = 0 and the supply equation x p+0 = 0, where p is the unit price and x represents the quantity, find the equilibrium quantity and the equilibrium price x = and p = 4 x = 3 and p = 6 (C) x = 4 and p = 8 (D) x = 8 and p = 4 3 It is known that lim f(x) = 3, lim f(x) = 3, and f() = Which of the following x + x statements is False? f(x) is discontinuous at x = The graph of f(x) is broken at x = (C) lim x f(x) exists (D) f(x) is differentiable at x = (E) f(x) is defined at x = 4 It is known that f(x) is continuous in (, ) and f( ) =, f(0) =, and f() = 4 Which of the following statements is True? f(x) must have a zero in (-,0) f(x) must have a zero in (-,-) (C) f(x) must have a zero in (,4) (D) f(x) must have a zero in (0,) is true 5 An equation for the line tangent to the curve y = x 3 x + 5 at the point (-, ) is y + = (x )(x + ) y = (x )(x + ) (C) y = x (D) y = 5x + (E) y = 0x + 6 A ball is thrown straightly up into the air so that its height (in feet) after t seconds is given by s(t) = 6t + 64t The average velocity of the ball over the interval [, 05] is 4956 ft/sec 56 ft/sec (C) 3 ft/sec (D) 48 ft/sec 3
7 Assume that the distance function s(t) is given as in Problem 6 The velocity of the ball at time t = is 50 ft/sec ft/sec (C) 3 ft/sec (D) 48 ft/sec 8 Find an equation of the tangent line at the point (, 3) of the graph of y = x(x+) 5 y 3 = (x + ) 4 (6x + ) y + 3 = (x + ) 4 (6x + ) (C) y = x 80 9 For f(x) = (D) y = x + 80 + x, evaluate f (4) 64 6 (C) 4 (D) 0 An equation for the line tangent to the curve y = (x + x + )(x 3 x + ) at the point (, 3) is y = x + y = 3x (C) y = 6x 3 (D) y = 7x 4 Let y = u and u = 7x x Find dy dx (7 4x)(7x x ) / (7 4x)(7x x ) / (C) (7 4x) (D) (7x x ) / Suppose that F (x) = f(x + ) and f () = 3 Find F () 3 4 (C) 5 (D) 6 3 Suppose h = f g Find h (0) given that f(0) = 6, f (5) =, g(0) = 5, and g (0) = 3 6 8 (C) 0 (D) 6 4
4 What is the marginal profit in terms of the cost C(x) to produce x units and the unit price p(x) at which x units will sell? p (x) C (x) p (x) + xp(x) C(x) (C) x(p(x) C(x)) (D) p(x) + xp (x) C (x) 5 Find dy dx in terms of x and y when x and y are related by the equation x y y 3 = xy 3y + x x x 3y (C) xy (D) x + y 3y x x + y 6 Find dy dx at point (, 5) when x and y are related by the equation x y = 3 4 5 5 5 (C) 5 (D) 0 7 Find dy dx in terms of x and y when x and y are related by the equation x +y +x y = x( + y ) y( + x ) y( + x ) x( + y ) (C) x( + x ) y( + y ) (D) x y 8 The second derivative of function f(x) = (x + ) 5 is 0(x + ) 3 0x(x + ) 4 (C) 0(x + ) 3 (9x + ) (D) 0(x + ) 3 (7x + 4) 9 The third derivative of f(x) = x is (C) (D) 6 x x 3 x 3 x 4 30 The distance s (in feet) covered by a car t seconds after starting from rest is given by s = t 3 + 8t + 0t Find the car s acceleration at time t t 3 + 8t + 0t 3t + 6t + 0 (C) 6t + 6 (D) 6t + 36 5