Brown University Mathematics Department
Self-Graded Calculus Placement Exam

SECTION D

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Question #17: The series \(1 + x + x^2 + \cdots\) converges for which \(x\)?

  1. all \(x\)
  2. only for \(x = 0\)
  3. \(-1 < x < 1\)
  4. \(x \geq 0\)
  5. None of the above answers

Question #18: For which numbers \(p\) does the series \(\displaystyle {1 \over 1^p} + {1 \over 2^p} + {1 \over 3^p} + \cdots\) converge?

  1. \( p > 1\)
  2. \(p \geq 1\)
  3. no \(p\)
  4. \(p \geq 0\)
  5. None of the above answers

Question #19: Find the Taylor series at \(x = 0\) for \(\displaystyle y = {1 \over {1 + x}}\).

  1. \(1 + x + x^2 + x^3 + \cdots\)
  2. \(1 - x + x^2 - x^3 + \cdots\)
  3. \(1 - x + (2!) x^2 - (3!) x^3 + \cdots\)
  4. \(\displaystyle 1 + x + {x^2 \over 2!} + {x^2 \over 3!} + \cdots\)
  5. None of the above answers

Question #20: The series \(\displaystyle 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^ 6 \over 6!} + \cdots\) is the Taylor series for which function?

  1. \(\sin x\)
  2. \(e^{-x}\)
  3. \(\displaystyle {{e^x + e^{-x}} \over {2}}\)
  4. \(\cos x\)
  5. None of the above answers

Question #21: The parametric curve defined by \(x = 2 \cos t\) and \(y = 1 + \sin t\) describes what kind of geometric figure?

  1. circle
  2. ellipse
  3. parabola
  4. hyperbola
  5. None of the above answers

Question #22: A particle travels in the \(xy\)-plane so that it has position \(x = 1 + t^2, y = 1 - t^3\) at time \(t\). Find the length of the velocity vector at \(t = 1\).

  1. 2
  2. 0
  3. \(\sqrt{13}\)
  4. \(\sqrt{2}\)
  5. None of the above answers

Question #23: Suppose that \(\displaystyle {dy \over dx} + y = 1\) and that \(y = 0\) when \(x = 0\). Find the value of \(y\) when \(x = 1\).

  1. 0
  2. 1
  3. \(\displaystyle {{e - 1} \over e}\)
  4. \(1 - e\)
  5. None of the above answers

Question #24: Find the general solution to \(\displaystyle {{d^2 y} \over {d x^2}} + y = e^x\).

  1. \(e^x\)
  2. \(\displaystyle {1 \over 2} e^x\)
  3. \(\displaystyle {1 \over 2} e^x - {1 \over 2} \cos x + c \sin x\), where \(\displaystyle c\) is a constant.
  4. \({1 \over 2} e^x + c_1 \cos x + c_2 \sin x\), where \(c_1, c_2\) are constants.
  5. None of the above answers