Brown University Mathematics Department
Self-Graded Calculus Placement Exam

SECTION A

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Question #1: Let \(\displaystyle y = x^4 - 3x + x^{1 \over 2}\). Find \(\displaystyle {dy \over dx}\).

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

Question #2: Let \(\displaystyle y = {\sqrt{1 - {1 \over x}}}\). Find \(\displaystyle {dy \over dx}\).

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

Question #3: The maximum value of \(y = 3x^2 - x^3\) for \(0 \leq x \leq 4\) occurs when \(x\) is what number?

  1. 2
  2. \(\displaystyle {3 \over 2}\)
  3. 0
  4. there is no maximum
  5. None of the above answers

Question #4: Find the area under the graph of \(y = x^3\) from \(x = 0\) to \(x = 1\).

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

Question #5: Find all values of \(x\) for which \(y = x + e^{-x}\) is increasing.

  1. \(x > 0\)
  2. \(x < 0\)
  3. all \(x\)
  4. no \(x\)
  5. None of the above answers

Question #6: Which of these represents the graph of \(\displaystyle y=\frac{x}{x^2-1}\)?

Problem #6

Question #7: Suppose that \(y\) is a function of \(x\) so that \(x^3 - y^3 + x + y = 10\), and so that \(y = 1\) when \(x = 2\). Find \(\displaystyle {dy \over dx}\) when \(x = 2\).

  1. undefined
  2. \(\displaystyle {13 \over 2}\)
  3. \(\displaystyle {13 \over 3}\)
  4. \(-13\)
  5. None of the above answers

Question #8: Assume that a piece of metal expands when heated while retaining the shape of a square. If the length of the side of the square is increasing at a rate of \({1 \over 4}\) inch per second when the side is 10 inches long, find the rate at which the area is increasing at that moment.

  1. \(\displaystyle {1 \over 16}\) square inches per second
  2. \(\displaystyle {1 \over 8}\) square inches per second
  3. 5 square inches per second
  4. \(\displaystyle 5{1 \over 16}\) square inches per second
  5. None of the above answers