MATH115 GEOMETRY AND CALCULUS
Quiz 2
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1.
Lines and planes Which of the following statements is TRUE?
A) There is a unique curve passing through any two points in ,
B) There is a unique plane passing through any two points in ,
C) Any plane in has a unique normal vector,
D) For each pair of non-zero vectors in , there exists such that is orthogonal to ,
E) Vectors in are orthogonal if and only if for some .
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2.
Thinking about gradients The following is a sketch of the derivative of a function (not of itself!).
Which of the following best describes the point ?
A) A local minimum, B) A local maximum, C) A saddle point,
D) It’s not a stationary point, E) It’s impossible to tell.
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3.
Turning points How many turning points does the polynomial have?
A) 1, B) 2, C) 3, D) 4, E) 5.
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4.
Arc length Let , . What is the length of the curve between the points and ?
A) , B) 4, C) , D) , E) 1.
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5.
Tangent line What is the equation of the tangent line to the parametrized curve , at the point ?
A) , B) , C) , D) , E) .
Assessed Exercises
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1.
(2 marks) Let be the plane curve given by . Show that the image of is contained in an ellipse, and determine the equation of the ellipse. Is the image the whole ellipse?
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2.
A metal bar has one end fixed at the origin, and rotates anti-clockwise at a constant rate of radians per second; at time it points along the -axis. A beetle, starting at the origin, walks along the bar at a constant speed of metres per second.
a) (1 mark) What is its position at time ?
b) (2 marks) What is its acceleration at time ?
c) (2 marks) The beetle will stop walking when the magnitude of its acceleration is m s. At what distance along the bar will it stop?
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3.
(3 marks) Calculate the length of the curve , between the points and .
Bonus question. Let be the surface in defined by . By introducing the auxiliary variable , or otherwise, show that the minimum distance of a point on from the origin is .