TR3.5

math 401 ret 21

Exr0n 2021-10-02 Sat 10:58

Table of Contents

#source openstax calculus volume 1 section 2.4 exercises

1 131

\[ x \le 0 \implies \boxed{\text{infinite}} \]

2 132

\[ \boxed{\text{no discontinuities}} \]

3 140

\[ \boxed{\text{Infinite discontinuity }} \left(\frac{-1}{0}\right) \]

4 141

\[ \boxed{\text{Continuous}} \left(\frac{\cancel{(2u-1)}(3u+2)}{\cancel{2u-1}}\right) \]

5 145

\[ 3x+2 = 2x-3 \implies \boxed{x = -5} \]

6 150

\[ \boxed{\text{The function is not continuous at }x = 2} \]

7 152

7.1 a

\[\cos t = t^3\]

7.2 b

Let \(f(x) = \cos x\) and \(g(x) = x^3\). For \(a = 0\) and \(b = \frac{\pi}{2}\): $$

\begin{aligned} f(a) &= 1\\ g(a) &= 0\\ f(b) &= 0\\ g(b) &= \frac{\pi^3}{8} > 1\\ \end{aligned}

$$ Because these functions each traverse \(0 \le y \le 1\) over the interval \(0 \le x \le \frac{\pi}{2}\) in opposite directions and are continuous over that range, they must cross somewhere in that range.

7.3 c

\[ \boxed{x = 0.8655 \pm 0.005} \]

8 164

\[\boxed{\text{It's true.}}\]