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Practice set

For every answer state the domain, assumptions and a check. Attempt first; the key is for diagnosis.

  1. Find the inverse and domain of ln(x+4). Solve exp(−2t)=1/5.
  2. Compute the limit of (x²−16)/(x−4) at 4. Explain whether the original function is continuous there.
  3. Differentiate x² exp(−x) and ln(2+x²). Identify each rule.
  4. Minimize x+25/x on [1,10], then on [6,10].
  5. Evaluate ∫₀² (3x²+1)dx and d/dx ∫₀^(3x) exp(t)dt.
  6. Evaluate ∫₀¹ 2x exp(x²)dx and ∫₀¹ x exp(−x)dx.
  7. Normalize c x on [0,4]; find the mean and P(X≤2).
  8. Compare ∫₁∞ x^(−2)dx with ∫₁∞ x^(−1)dx. Why does a finite upper cutoff not settle convergence?
  9. Bound quadratic Taylor error for exp(x) at 0.2 using exp(x)<2 on [0,0.2].
  10. How large must n be to guarantee trapezoidal error at most 0.0001 for x² on [0,1]? Give a bisection iteration count guaranteeing midpoint error at most 0.001 from a bracket of width 2.