Chapter 5.8 Practice

Questions

  1. For each function \(f\) and \(x_0\) given below, set up a limit to determine \(f^\prime(x_0)\):

    1. \(f(x) = 2x^2-10x+1\), \(x_0 = 2\)
    2. \(f(x) = \frac{x^2 -x}{x+1}\), \(x_0 = 2\)
    3. \(f(x) = 4\sin(x)\), \(x_0 = \frac{\pi}{6}\)
    4. \(f(x) = 3\sqrt{x}\), \(x_0=4\)
    5. \(f(x) = \sqrt{x-1}\), \(x_0 = 10\)

  2. For each function \(f\) that is given below, use limits to calculate \(f^\prime(x)\) for every \(x\) where this limit exists:

    1. \(f(x) = 2x^2-10x+1\)
    2. \(f(x) = \frac{x^2 -x}{x+1}\)
    3. \(f(x) = 4\sin(x)\)
    4. \(f(x) = 3\sqrt{x}\)
    5. \(f(x) = \sqrt{x-1}\)
    6. \(f(x)=-5x^2-5x-3\)

  3. For each function \(f\) that is given below, determine where \(f\) is differentiable and determine \(f^\prime(x).\)

    1. \(f(x)=\tan(x+1)\)
    2. \(f(x)=\cos(-x^2+197)\)
    3. \(f(x)=\sqrt{6x^2+82}\)
    4. \(f(x)=\sin(7x^2+226)\)
    5. \(f(x)=e^{x^2+4x}\)

  4. Take \(f\) to be the function that is given by \[f(x)=\begin{cases}6x^2+bx+c&\text{if }x<19\\ 9x+2&\text{if }x\geq 19.\end{cases}\] Determine \(b\) and \(c\) so that \(f\) is differentiable.

  5. Take \(f\) to be the function that is given by \[f(x)=\begin{cases}-20x^2+bx+c&\text{if }x<-16\\ -4x-11&\text{if }x\geq -16.\end{cases}\] Determine \(b\) and \(c\) so that \(f\) is differentiable.

Answers

    1. \(\Delta_{2}f(h)=2h^2-2h\), so \(f'(2)=\displaystyle \lim_{h\to 0}\frac{1}{h}\Delta_{2}f(h)=-2\)
    2. \(\Delta_{2}f(h)=\frac{3h^2+7}{3h+9}\), so \(f'(2)=\displaystyle \lim_{h\to 0}\frac{1}{h}\Delta_{2}f(h)=\frac{7}{9}\)
    3. \(\Delta_{\frac{\pi}{6}}f(h)=2\left(1-\cos(h)\right)+2\sqrt{3}\sin(h)\), so \(f'( \tfrac{\pi}{6})=\displaystyle \lim_{h\to 0}\frac{1}{h}\Delta_{ \frac{\pi}{6}}f(h)=2\sqrt{3}\)
    4. \(\Delta_{4}f(h)=\frac{9h}{3\sqrt{4+h}+3\sqrt{4}}\), so \(f'( 4)=\displaystyle \lim_{h\to 0}\frac{1}{h}\Delta_{ 4}f(h)=\frac{3}{4}\)
    5. \(\Delta_{10}f(h)=\frac{h}{\sqrt{9+h}+\sqrt{9}}\), so \(f'( 10)=\displaystyle \lim_{h\to 0}\frac{1}{h}\Delta_{ 10}f(h)=\frac{1}{6}\)
    1. \(f'(x)=4x-10\)
    2. \(f'(x)=\frac{x^2+2x-1}{(x+1)^2}\)
    3. \(f'(x)=4\cos(x)\)
    4. \(f'(x)=\frac{3}{2\sqrt{x-1}}\)
    5. \(f'(x)=\frac{1}{2\sqrt{x-1}}\)
    6. \(f'(x)=-10x-5\)
    1. \(f'(x)=\sec^2(x+1)\) all \(x\not=\frac{\pi}{2}k-1\) where \(k\) is an integer
    2. \(f'(x)=2x\sin(-x^2+197)\) for all \(x\) in \(\mathbb{R}\)
    3. \(f'(x)=\frac{6x}{\sqrt{6x^2+82}}\) for all \(x\) in \(\mathbb{R}\)
    4. \(f'(x)=14x\cos(7x^2+226)\) for all \(x\) in \(\mathbb{R}\)
    5. \(f(x)=(2x+4)e^{x^2+4x}\) for all \(x\) in \(\mathbb{R}\)

  4. \(b=-219\) and \(c=2168\).

  5. \(b=-644\) and \(c=-5131\).

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