Chapter 5.8

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. For each function \(f\) and each \(x_0\) that is given below, determine the difference quotient \(\frac{1}{h}\Delta_{x_0}f(h)\) for each real number \(h\).
    1. \(f(x) = -2x^2+x+3\), \(x_0 = 2\)
    2. \(f(x) = \frac{x^2 -3}{x}\), \(x_0 = 5\)
    3. \(f(x) = 3\cos(x)\), \(x_0 = \frac{\pi}{4}\)
    4. \(f(x) = \sqrt{x}\), \(x_0=9\)
    5. \(f(x)= \exp(x)\), \(x_0=1\)
    6. \(f(x)= 4\), \(x_0=1\)
  2. For each function \(f\) and \(x_0\) given below, set up a limit to determine \(f^\prime(x_0)\).
    1. \(f(x) = -2x^2+x+3\), \(x_0 = 2\)
    2. \(f(x) = \frac{x^2 -5}{x}\), \(x_0 = 5\)
    3. \(f(x) = 4\cos(x)\), \(x_0 = \frac{\pi}{4}\)
    4. \(f(x) = \sqrt{x}\), \(x_0=9\)
    5. \(f(x)= \exp(x)\), \(x_0=1\)
    6. \(f(x)= 4\), \(x_0=1\)
  3. For each function \(f\) that is given below, use limits to calculate \(f^\prime(x)\) for every \(x\) where this limit exists.
    1. \(f(x) = 4\)
    2. \(f(x) = 5x\)
    3. \(f(x) = -2x^2+x+3\)
    4. \(f(x) = \frac{x^2 -3}{x}\)
    5. \(f(x)=\sqrt{x}\)
  4. Take \(f\) to be the function that is given by \[f(x) = \cos(5x^2+1).\] Show that \(f\) is differentiable for each \(x\) in \(\mathbb R\) and determine \(f^\prime(x)\).
  5. Take \(f\) to be the function that is given by \[f(x) = \mathrm{e}^{x^2+4x}.\] Show that \(f\) is differentiable for each \(x\) in \(\mathbb R\) and determine \(f^\prime(x)\).

Practice Problems

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