Exercises - Differentiability and Continuity

  1. Given the following function, $$f(x) = \left\{ \begin{array}{ccc} -\sqrt{16-x^2} &,& x \lt 0\\\\ 3 &,& x = 0\\\\ x^2-4 &,& 0 \lt x \le 3\\\\ 3+\sqrt{7-x} &,& x \gt 3 \end{array} \right.$$

    1. Sketch the graph of $y=f\,(x)$

    2. For what values of $x$ does $f\,'(x)$ not exist? Explain.

  2. Determine where the derivatives of the following functions fail to exist, characterizing the behavior of the related graph (i.e., corner, cusp, vertical tangent, hole, gap, vertical asymptote, etc...) at each such $x$-value. Classify any discontinuities found as either removable or non-removable. Justify your claims with appropriate function values, derivative values, and limiting values.

    1. $\displaystyle{f\,(x) = |x+2|}$

    2. $f(x) = (x+1)^{2/3}$

    3. $g(x) = 1 - \sqrt[3]{x}$

    4. $\displaystyle{f\,(x) = \left\{ \begin{array}{rlc} x^2+1 &,& x \lt 1\\\\ 3 &,& x = 1\\\\ 2 \sqrt{2x-1} &,& x \gt 1 \end{array} \right.}$

    5. $\displaystyle{g(x) = \left\{ \begin{array}{rlc} x+1 &,& x \le 2\\\\ 1 &,& 2 \lt x \le 4\\\\ x^2 - 8x + 17 &,& x \gt 4 \end{array} \right.}$

    6. $\displaystyle{h(x) = \left\{ \begin{array}{rlc} 2-x &,& x \le 3\\\\ x^2 - 10 &,& x \gt 3 \end{array} \right.}$

    7. $\displaystyle{f\,(x) = \left\{ \begin{array}{rlc} \sqrt{16-x^2} &,& x \lt 0\\\\ x^3+4 &,& 0 \le x \le 1\\\\ -x^2+5 &,& x \gt 1 \end{array} \right.}$

    8. $\displaystyle{f\,(x) = \left\{ \begin{array}{rlc} \frac{9}{x^2-4} &,& x < -1\\\\ 3\ln(-x)-3 &,& -1 \le x \le 0\\\\ \frac{2x^2-x}{2x-1} &,& 0 < x < 1\\\\ \sin(x-1)+1 &,& 1 < x \le \pi + 1\\\\ 2(x-5)^{2/3} &,& \pi + 1 < x \le 6\\\\ 5-\sqrt{-x^2+4x+21} &,& 6 < x < 7\\\\ \sqrt[3]{x-7}+5 &,& x \ge 7 \end{array} \right.}$

    9. $\displaystyle{f\,(x) = \left\{ \begin{array}{rlc} x^2-5 &,& x \le -2\\\\ x+1 &,& -2 \lt x \lt 0\\\\ -1 &,& x = 0\\\\ \sqrt{4-x^2} - 1 &,& 0 \lt x \lt 2\\\\ 2 &,& x \ge 2 \end{array} \right.}$

    10. $\displaystyle{h(x) = \left\{ \begin{array}{ccc} x+9 &,& x \le -9\\\\ (x+1)^{1/3} &,& -9 \lt x \lt 0\\\\ \cos x &,& 0 \le x \lt \pi\\\\ -\tan \left( \frac{x}{4} \right) &,& \pi \lt x \le 3\pi\\\\ \frac{1}{x-12} &,& x \gt 3\pi \end{array} \right.}$

    11. $\displaystyle{g(x) = \left\{ \begin{array}{ccc} |x+7| &,& x \le -4\\\\ \sqrt{25-x^2} &,& -4 \lt x \lt 0\\\\ 5-x^2 &,& 0 \le x \lt 2\\\\ \frac{-x}{x-4} &,& x \gt 2 \end{array} \right.}$

    12. $\displaystyle{f\,(x) = \left\{ \begin{array}{ccc} x^3-3x &,& x \lt 1\\\\ -2 &,& x = 1\\\\ x^2 + 1 &,& x \gt 1 \end{array} \right.}$

    13. $\displaystyle{g(x) = \left\{ \begin{array}{ccc} \frac{-x}{x+4} &,& x \le -2\\\\ x+5 &,& -2 < x < 0\\\\ \sqrt{25-x^2} &,& 0 < x \le 4\\\\ (31-x)^{1/3} &,& x > 4 \end{array} \right.}$

    14. $\displaystyle{g(x) = \left\{ \begin{array}{ccc} x^2 - 1 &,& x \gt 3\\\\ 6x - 10 &,& x \le 3 \end{array} \right.}$

    15. $\displaystyle{g(x) = \left\{ \begin{array}{ccc} x/2 &,& x \le -2\\\\ x^2 &,& -2 \lt x \lt 1\\\\ -x+5 &,& x \ge 1 \end{array} \right.}$

    16. $\displaystyle{f(x) = \left\{ \begin{array}{ccc} (x+6)^{1/3} - 1 & \textrm{ if } & x < -5 \\\\ \sqrt{25-x^2} & \textrm{ if } & -5 \le x < -3 \\\\ e^x - x & \textrm{ if } & -3 < x \le 0 \\\\ \cos x & \textrm{ if } & 0< x < \frac{\pi}{2} \\\\ \sin 3x & \textrm{ if } & \frac{\pi}{2} \le x < \pi \\\\ 2|x-5| & \textrm{ if } & \pi \le x < 6 \end{array} \right.}$

    17. $\displaystyle{f\,(x) = \left\{ \begin{array}{ccc} -x -1 &,& x \lt -1\\\\ x^2-1 &,& -1 \le x \le 1\\\\ 2x-2 &,& x \gt 1 \end{array} \right.}$

    18. $\displaystyle{f\,(x) = \left\{ \begin{array}{ccc} 5-x^2 &,& x \lt 0\\\\ 3 &,& x=0\\\\ \sqrt{25-x^2} &,& 0 \lt x \lt 4\\\\ 1-4(x-5)^{1/3} &,& 4 \le x \lt 6\\\\ \frac{3-3x}{2x-7} &,& x \le 6 \end{array} \right.}$