Q. 95 f(x)聽= 3 sin x(a) Find the zero... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 95 (page 466)

$f (x)$ = 3 sin x
(a) Find the zeros of f on the interval $[- 2 蟺, 4 蟺]$
(b) Graph $f (x)$ = 3 sin x on the interval $[- 2 蟺, 4 蟺]$
(c) Solve localid="1647352910053" $f (x) = \frac{3}{2}$ on the interval $[- 2 蟺, 4 蟺]$ . What points
are on the graph of f ? Label these points on the graph
drawn in part (b).
(d) Use the graph drawn in part (b) along with the results of
part (c) to determine the values of x such that $f (x) > \frac{3}{2}, x 鈭� [- 2 蟺, 4 蟺]$

Short Answer

Expert verified

(a) The values of x are $- 2 蟺, - 蟺, 0, 蟺, 2 蟺, 3 蟺, 4 蟺 in the interval [- 2 蟺, 4 蟺]$ .

(b)

(c) The value $3 \sin x = \frac{3}{2} s a t i s f i e s t h e v a l u e o f x a r e x = - \frac{11 蟺}{6}, - \frac{7 蟺}{6}, \frac{蟺}{6}, \frac{5 蟺}{6}, \frac{13 蟺}{6}, \frac{17 蟺}{6}$

(d) $f (x) > \frac{3}{2}; - 2 蟺鈮� x 鈮� 4 蟺 \{x | - \frac{11 蟺}{6} < x < - \frac{7 蟺}{6}, \frac{蟺}{6} < x < \frac{5 蟺}{6}, \frac{13 蟺}{6} < x < \frac{17 蟺}{6}\}$

Step by step solution

Part (a) Step 1. Given Information

$y = 3 \sin x$

Part (a) Step 2. Calculation

$3 \sin x = 0 \sin x = 0; - 2 蟺鈮� x 鈮� 4 蟺$

x	0	$\frac{蟺}{6}$	$\frac{蟺}{4}$	$\frac{蟺}{3}$	$\frac{蟺}{2}$	$\frac{2 蟺}{3}$	$\frac{3 蟺}{4}$	$\frac{5 蟺}{6}$
sinx	0	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$

x	$蟺$	$\frac{7 蟺}{6}$	$\frac{5 蟺}{4}$	$\frac{4 蟺}{3}$	$\frac{3 蟺}{2}$	$\frac{5 蟺}{3}$	$\frac{7 蟺}{4}$	$\frac{11 蟺}{6}$	$2 蟺$
sinx	0	$- \frac{1}{2}$	$- \frac{1}{\sqrt{2}}$	$- \frac{\sqrt{3}}{2}$	$- 1$	$- \frac{\sqrt{3}}{2}$	$- \frac{1}{\sqrt{2}}$	$- \frac{1}{2}$	0

Therefore,

Sinx=0 has values $- 2 蟺, - 蟺, 0, 蟺, 2 蟺, 3 蟺, 4 蟺 in the interval [- 2 蟺, 4 蟺]$ .

Part (b) Step 1. Calculation

Part (c) Step 1. Calculation

$C o n s i d e r, f (x) = \frac{3}{2}; - 2 蟺鈮� x 鈮� 4 蟺 3 sinx = \frac{3}{2} sinx = \frac{1}{2}$

x	0	$\frac{蟺}{6}$	$\frac{蟺}{4}$	$\frac{蟺}{3}$	$\frac{蟺}{2}$	$\frac{2 蟺}{3}$	$\frac{3 蟺}{4}$	$\frac{5 蟺}{6}$
sinx	0	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$

x	$蟺$	$\frac{7 蟺}{6}$	$\frac{5 蟺}{4}$	$\frac{4 蟺}{3}$	$\frac{3 蟺}{2}$	$\frac{5 蟺}{3}$	$\frac{7 蟺}{4}$	$\frac{11 蟺}{6}$	$2 蟺$
sinx	0	$- \frac{1}{2}$	$- \frac{1}{\sqrt{2}}$	$- \frac{\sqrt{3}}{2}$	$- 1$	$- \frac{\sqrt{3}}{2}$	$- \frac{1}{\sqrt{2}}$	$- \frac{1}{2}$	$0$

therefore,

$x = - \frac{11 蟺}{6}, - \frac{7 蟺}{6}, \frac{蟺}{6}, \frac{5 蟺}{6}, \frac{13 蟺}{6}, \frac{17 蟺}{6}$

Part (d) Step 1. Calculation

$C o n s i d e r, f (x) > \frac{3}{2} 3 \sin x > \frac{3}{2} \sin x > \frac{1}{2} F r o m p r e v i o u s t a b l e w e s e e \sin x > \frac{1}{2} f o r \frac{蟺}{6} < x < \frac{5 蟺}{6} a l s o - \frac{11 蟺}{6} < x < - \frac{7 蟺}{6} a n d \frac{13 蟺}{6} < x < \frac{17 蟺}{6} i n t h e i n t e r v a l [- 2 蟺, 0] a n d [2 蟺, 4 蟺]$

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91影视

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part (a) Step 2. Calculation

Part (b) Step 1. Calculation

Part (c) Step 1. Calculation

Part (d) Step 1. Calculation

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