Calculus- Early Transcendentals, 2021a

1.4. Additional Exercises 43
1.4 Additional Exercises
These problems require a comprehensive knowledge of the skills reviewed in this chapter. They are not
in any particular order. A proficiency in these skills will help you a long way as your learn the calculus
material in the following chapters.
Exercise 1.4.1 Rationalize the denominator for each of the following expressions. That is, re-write the
expression in such a way that no square roots appear in the denominator. Also, simplify your answers if
possible.
(a)
(b)
1
√
2
3h
√
x + h + 1 −
√
x + 1
Exercise 1.4.2 Solve the following equations.
(a) 2 − 5(x − 3)=4 − 10x
(b) 2x 2 − 5x = 3
(c) x 2 − x − 3 = 0
(d) x 2 + x + 3 = 0
(e) √ x 2 + 9 = 2x
Exercise 1.4.3 By means of counter-examples, show why it is wrong to say that the following equations
hold for all real numbers for which the expressions are defined.
(a) (x − 2) 2 = x 2 − 2 2
(b)
1
x + h = 1 x + 1 h
(c) √ x 2 + y 2 = x + y
Exercise 1.4.4 Find an equation of the line passing through the point (−2,5) and parallel to the line
x + 3y − 2 = 0.
Exercise 1.4.5 Solve x2 − 1
3x − 1 ≤ 1.
Exercise 1.4.6 Explain why the following expression never represents a real number (for any real number
x): √ x − 2 + √ 1 − x.