MATH69.204* ANSWERS TO THE FIRST TEST Fall, 1998

• 1. [1 mark] If , where f(t) is an even function and g(t) is an odd function, then g(t) is equal to

• (a) , (b) , (c) , (d) , (e) 9.

• Answer: (a). It's the only odd function in the sum of the three given functions.

• 2. [1 mark] If f is an odd periodic function of period 4 and if for all t satisfying , then f(-135) is equal to
• (a) 9, (b) 3, (c) 0, (d) -3, (e) -9.

• Answer: (e). Because .

• 3. [1 mark] The circle in polar coordinates is

• (a) , (b) , (c) , (d) , (e) .

• Answer: (b). Putting and in , we get . Cancel r and rearrange the identity.

• 4. [1 mark] Let be the equation of a curve in polar coordinates. Then, using as a parameter, the parametric equation of this curve is

• (a) , (b) , , (c) , \
• (d) , , (e) , .

• Answer: (e). This follows immediately upon setting and .

• 5. [2 marks] Find the Fourier series for a period 2 function f(t)=1-t for 0<t<2.

• [1 mark for the correct formula and its correct application, 0.5 mark for the correct procedure and 0.5 mark for the correct final answer.]

• Answer. The period is 2L=2. so L=1. A sketch of the graph of f shows that f is an odd function and hence its Fourier series must be a sine series. Now

  

  for . So the Fourier series for f is

  

  (We have extracted a common factor outside the summation sign.)

• 6. [2 marks] Find the area bounded by one loop of the trefoil .

• [1 mark for the correct area formula; 0.5 mark for the correct limits of the integral and 0.5 mark for the correct answer.]

• Answer. As ranges between and , increases from 0 to 1 and then decreases back to 0. This makes up one loop. Thus the required loop area is

  

• 7. [2 marks] Find the arc length of the parametric curve , in the plane with .

• [1 mark for the correct formula; 0.5 mark for the correct procedure and 0.5 mark for the correct answer.]

• Answer. The required arc length is

  

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