Sumário

Preface
1Functions
2Linear Functions
3Polynomial and Rational Functions
4Exponential and Logarithmic Functions
5Trigonometric Functions
6Periodic Functions
7Trigonometric Identities and Equations
8Further Applications of Trigonometry
9Systems of Equations and Inequalities
10Analytic Geometry
11Sequences, Probability and Counting Theory
12Introduction to Calculus
ABasic Functions and Identities
Answer Key
Index

Review Exercises

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use Figure 1.

Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point.

Figure 1

$\lim_{x \to {−1}^{+}} f (x) \lim_{x \to {−1}^{+}} f (x)$

$\lim_{x \to {−1}^{-}} f (x) \lim_{x \to {−1}^{-}} f (x)$

$\lim_{x \to - 1} f (x) \lim_{x \to - 1} f (x)$

$\lim_{x \to 3} f (x) \lim_{x \to 3} f (x)$

At what values of $x x$ is the function discontinuous? What condition of continuity is violated?

Using Table 1, estimate $\lim_{x \to 0} f (x) . \lim_{x \to 0} f (x) .$

$x x$	$F (x) F (x)$
−0.1	2.875
−0.01	2.92
−0.001	2.998
0	Undefined
0.001	2.9987
0.01	2.865
0.1	2.78145
0.15	2.678

Table 1

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 $x x$ approaches $a . a .$ If the function has limit as $x x$ approaches $a, a,$ state it. If not, discuss why there is no limit.

$f (x) = {\begin{array}{l} | x | - 1, & i f & x \neq 1 \\ x^{3}, & i f & x = 1 \end{array} a = 1 f (x) = {\begin{array}{l} | x | - 1, & i f & x \neq 1 \\ x^{3}, & i f & x = 1 \end{array} a = 1$

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

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

Finding Limits: Properties of Limits

For the following exercises, find the limits if $\lim_{x \to c} f (x) = −3 \lim_{x \to c} f (x) = −3$ and $\lim_{x \to c} g (x) = 5. \lim_{x \to c} g (x) = 5.$

10.

$\lim_{x \to c} (f (x) + g (x)) \lim_{x \to c} (f (x) + g (x))$

11.

$\lim_{x \to c} \frac{f (x)}{g (x)} \lim_{x \to c} \frac{f (x)}{g (x)}$

12.

$\lim_{x \to c} (f (x) \cdot g (x)) \lim_{x \to c} (f (x) \cdot g (x))$

13.

$\lim_{x \to 0^{+}} f (x), f (x) = {\begin{matrix} 3 x^{2} + 2 x + 1 \\ 5 x + 3 \end{matrix} \begin{matrix} x > 0 \\ x < 0 \end{matrix} \lim_{x \to 0^{+}} f (x), f (x) = {\begin{matrix} 3 x^{2} + 2 x + 1 \\ 5 x + 3 \end{matrix} \begin{matrix} x > 0 \\ x < 0 \end{matrix}$

14.

$\lim_{x \to 0^{-}} f (x), f (x) = {\begin{matrix} 3 x^{2} + 2 x + 1 \\ 5 x + 3 \end{matrix} \begin{matrix} x > 0 \\ x < 0 \end{matrix} \lim_{x \to 0^{-}} f (x), f (x) = {\begin{matrix} 3 x^{2} + 2 x + 1 \\ 5 x + 3 \end{matrix} \begin{matrix} x > 0 \\ x < 0 \end{matrix}$

15.

$\lim_{x \to 3^{+}} (3 x - [x]) \lim_{x \to 3^{+}} (3 x - [x])$

For the following exercises, evaluate the limits using algebraic techniques.

16.

$\lim_{h \to 0} (\frac{{(h + 6)}^{2} - 36}{h}) \lim_{h \to 0} (\frac{{(h + 6)}^{2} - 36}{h})$

17.

$\lim_{x \to 25} (\frac{x^{2} - 625}{\sqrt{x} - 5}) \lim_{x \to 25} (\frac{x^{2} - 625}{\sqrt{x} - 5})$

18.

$\lim_{x \to 1} (\frac{- x^{2} - 9 x}{x}) \lim_{x \to 1} (\frac{- x^{2} - 9 x}{x})$

19.

$\begin{matrix} \lim \\ ^{x \to 4} \end{matrix} \frac{7 - \sqrt{12 x + 1}}{x - 4} \begin{matrix} \lim \\ x \to 4 \end{matrix} \frac{7 - \sqrt{12 x + 1}}{x - 4}$

20.

$\lim_{x \to - 3} (\frac{\frac{1}{3} + \frac{1}{x}}{3 + x}) \lim_{x \to - 3} (\frac{\frac{1}{3} + \frac{1}{x}}{3 + x})$

Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at $x = a . x = a .$ If not, describe the behavior of the graph of the function at $x = a . x = a .$

21.

$f (x) = \frac{- 2}{x - 4}; a = 4 f (x) = \frac{- 2}{x - 4};$

22.

$f (x) = \frac{- 2}{{(x - 4)}^{2}}; a = 4 f (x) = \frac{- 2}{{(x - 4)}^{2}};$

23.

$f (x) = \frac{- x}{x^{2} - x - 6}; a = 3 f (x) = \frac{- x}{x^{2} - x - 6};$

24.

$f (x) = \frac{6 x^{2} + 23 x + 20}{4 x^{2} - 25}; a = - \frac{5}{2} f (x) = \frac{6 x^{2} + 23 x + 20}{4 x^{2} - 25};$

25.

$f (x) = \frac{\sqrt{x} - 3}{9 - x}; a = 9 f (x) = \frac{\sqrt{x} - 3}{9 - x};$

For the following exercises, determine where the given function $f (x) f (x)$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

26.

$f (x) = x^{2} - 2 x - 15 f (x) = x^{2} - 2 x - 15$

27.

$f (x) = \frac{x^{2} - 2 x - 15}{x - 5} f (x) = \frac{x^{2} - 2 x - 15}{x - 5}$

28.

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

29.

$f (x) = \frac{x^{3} - 125}{2 x^{2} - 12 x + 10} f (x) = \frac{x^{3} - 125}{2 x^{2} - 12 x + 10}$

30.

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

31.

$f (x) = \frac{x + 2}{x^{2} - 3 x - 10} f (x) = \frac{x + 2}{x^{2} - 3 x - 10}$

32.

$f (x) = \frac{x + 2}{x^{3} + 8} f (x) = \frac{x + 2}{x^{3} + 8}$

Derivatives

For the following exercises, find the average rate of change $\frac{f (x + h) - f (x)}{h} . \frac{f (x + h) - f (x)}{h} .$

33.

$f (x) = 3 x + 2 f (x) = 3 x + 2$

34.

$f (x) = 5 f (x) = 5$

35.

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

36.

$f (x) = \ln (x) f (x) = \ln (x)$

37.

$f (x) = e^{2 x} f (x) = e^{2 x}$

For the following exercises, find the derivative of the function.

38.

$f (x) = 4 x - 6 f (x) = 4 x - 6$

39.

$f (x) = 5 x^{2} - 3 x f (x) = 5 x^{2} - 3 x$

40.

Find the equation of the tangent line to the graph of $f (x) f (x)$ at the indicated $x x$ value.

$f (x) = - x^{3} + 4 x f (x) = - x^{3} + 4 x$ ; $x = 2. x = 2.$

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.

41.

$f (x) = \frac{x}{| x |} f (x) = \frac{x}{| x |}$

42.

Given that the volume of a right circular cone is $V = \frac{1}{3} π r^{2} h V = \frac{1}{3} π r^{2} h$ and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of $π π$