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Section Solutions

Linear Approximations

  1. Linear approximation is \(y = -2x + 1 + \pi.\)

  2. Exact volume = 6795.2, approximate volume = 6785.83.

Limits and Infinity

  1. \(\dsp \frac{2}{3}\)

  2. \(\infty, -\infty\)

  3. 0

  4. \(1, -1\)

Graphing

  1. For each of the following, find the greatest lower bound and the least upper bound for the function on the given interval.

    1. \(8, 12\)

    2. \(-2.5, -2\)

    3. no greatest lower bound or least upper bound, although there is a local min at \((0,5/4)\)

    4. greatest lower bound is 0, no least upper bound

  2. For each of the following graph \(f,\) \(f',\) and \(f''\) on the same set of axes.

    1. no graphs, but think about the relationships between the three functions

  3. Sketch an accurate graph of the following functions...

    1. Type them into Google search or your favorite software to see graphs.

Rolle's Theorem, Mean Value Theorem, and L'Hôpital's Rule

  1. Since all trig functions are differentiable everywhere and \(f(-1)=0=f(4),\) \(f\) satisfies the hypothesis. \(\dsp c = -\frac{3}{4}, -\frac{1}{4}, \frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}, \frac{9}{4}, \frac{11}{4}, \frac{13}{4}, \frac{15}{4}\) satisfy the conclusion.

  2. Since all polys are differentiable everywhere, \(f\) satisfies the hypothesis. \(\dsp c = \frac{1 \pm \sqrt{7}}{3}\) both satisfy the conclusion.

  3. Evaluate each of the following limits using L'Hôpital's Rule if L'Hôpital's Rule applies.

    1. \(\dsp 2/5\)

    2. \(2/3\)

    3. \(0\)

    4. \(0\)

    5. \(0\)

    6. \(3 \ln(4)\)

    7. \(0\)

    8. \(1\)

    9. \(e\)

    10. \(1\)

    11. \(1\)

    12. \(e\)

    13. \(e^4\)

    14. \(0\)

    15. \(1\)

    16. \(e^{-1}\)

    17. \(1\)

Max/Min Problems

  1. A 4 x 4 square

  2. Radius = \(\sqrt{2/3} \ h\)

  3. \(\dsp (\sqrt[6]{2},-\frac{1}{\sqrt[3]{2}})\)

  4. approximately 523 miles/hour

  5. Radius = \(\sqrt[3]{750/\pi}\)

Related Rate Problems

  1. 8 ft/sec

  2. 500 mi/hr

  3. 20/9 square feet/sec

  4. \(\dsp \frac{3}{8\pi}\) inches per second

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