Practice Test

For the following exercises, use the graph of $f$ in Figure 1.

Graph of a piecewise function with two segments. The first segment goes from negative infinity to (-1, 0), an open point, and the second segment goes from (-1, 3), an open point, to positive infinity.

Figure 1

$f (1)$

$\lim_{x \to {−1}^{+}} f (x)$

$\lim_{x \to {−1}^{-}} f (x)$

$\lim_{x \to −1} f (x)$

$\lim_{x \to −2} f (x)$

At what values of $x$ is $f$ discontinuous? What property of continuity is violated?

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a .$ If the function has a limit as $x$ approaches $a,$ state it. If not, discuss why there is no limit

$f (x) = {\begin{array}{l} \frac{1}{x} - 3, i f & x \leq 2 \\ x^{3} + 1, i f & x > 2 \end{array} a = 2$

$f (x) = {\begin{array}{l} x^{3} + 1, & i f & x < 1 \\ 3 x^{2} - 1, & i f & x = 1 \\ - \sqrt{x + 3} + 4, & i f & x > 1 \end{array} a = 1$

For the following exercises, evaluate each limit using algebraic techniques.

$\lim_{x \to −5} (\frac{\frac{1}{5} + \frac{1}{x}}{10 + 2 x})$

10.

$\lim_{h \to 0} (\frac{\sqrt{h^{2} + 25} - 5}{h^{2}})$

11.

$\lim_{h \to 0} (\frac{1}{h} - \frac{1}{h^{2} + h})$

For the following exercises, determine whether or not the given function $f$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

12.

$f (x) = \sqrt{x^{2} - 4}$

13.

$f (x) = \frac{x^{3} - 4 x^{2} - 9 x + 36}{x^{3} - 3 x^{2} + 2 x - 6}$

For the following exercises, use the definition of a derivative to find the derivative of the given function at $x = a .$

14.

$f (x) = \frac{3}{5 + 2 x}$

15.

$f (x) = \frac{3}{\sqrt{x}}$

16.

$f (x) = 2 x^{2} + 9 x$

17.

For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable.

Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.

Figure 2

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

18.

$f (x) = | x - 2 | - | x + 2 |$

19.

$f (x) = \frac{2}{1 + e^{\frac{2}{x}}}$

For the following exercises, explain the notation in words when the height of a projectile in feet, $s,$ is a function of time $t$ in seconds after launch and is given by the function $s (t) .$

20.

$s (0)$

21.

$s (2)$

22.

$s' (2)$

23.

$\frac{s (2) - s (1)}{2 - 1}$

24.

$s (t) = 0$

For the following exercises, use technology to evaluate the limit.

25.

$\lim_{x \to 0} \frac{\sin (x)}{3 x}$

26.

$\lim_{x \to 0} \frac{\tan^{2} (x)}{2 x}$

27.

$\lim_{x \to 0} \frac{\sin (x) (1 - \cos (x))}{2 x^{2}}$

28.

Evaluate the limit by hand.

$\begin{matrix} \lim \\ ^{x \to 1} \end{matrix} f (x), where f (x) = {\begin{matrix} 4 x - 7 & x \neq 1 \\ x^{2} - 4 & x = 1 \end{matrix}$

At what value(s) of $x$ is the function below discontinuous?

$f (x) = {\begin{matrix} 4 x - 7 x \neq 1 \\ x^{2} - 4 x = 1 \end{matrix}$

For the following exercises, consider the function whose graph appears in Figure 3.

Figure 3

29.

Find the average rate of change of the function from $x = 1 to x = 3.$

30.

Find all values of $x$ at which $f' (x) = 0.$

31.

Find all values of $x$ at which $f' (x)$ does not exist.

32.

Find an equation of the tangent line to the graph of $f$ the indicated point: $f (x) = 3 x^{2} - 2 x - 6, x = - 2$

For the following exercises, use the function $f (x) = x {(1 - x)}^{\frac{2}{5}}$ .

33.

Graph the function $f (x) = x {(1 - x)}^{\frac{2}{5}}$ by entering $f (x) = x {({(1 - x)}^{2})}^{\frac{1}{5}}$ and then by entering $f (x) = x {({(1 - x)}^{\frac{1}{5}})}^{2}$ .

34.

Explore the behavior of the graph of $f (x)$ around $x = 1$ by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at $x = 1.$

For the following exercises, find the derivative of each of the functions using the definition: $\lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

35.

$f (x) = 2 x - 8$

36.

$f (x) = 4 x^{2} - 7$

37.

$f (x) = x - \frac{1}{2} x^{2}$

38.

$f (x) = \frac{1}{x + 2}$

39.

$f (x) = \frac{3}{x - 1}$

40.

$f (x) = - x^{3} + 1$

41.

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

42.

$f (x) = \sqrt{x - 1}$