Chapter 1.1, Problem 79E

(a)

To determine

To find: The value of $f(\pi /6)$ .

Answer to Problem 79E

The value of $f(\pi /6)$ is $\frac{1}{2}$ .

Explanation of Solution

Given information: The function is $f(x)=\sin x$

Calculation:

The given function is $f(x)=\sin x$

Calculate $f$ at $x=\frac{\pi }{6}$ :

  $\begin{array}{c}f\left(\frac{\pi }{6}\right)=\sin \frac{\pi }{6}\\ =\frac{1}{2}\end{array}$

Therefore, the required value of $f(\pi /6)$ is $\frac{1}{2}$ .

(b)

To determine

To find: The values of a and b.

Answer to Problem 79E

Possible values for a: Any number that is rounded up from $0.3047$

Possible values for b: Any number that is rounded up from $0.7754$ .

Explanation of Solution

Given information: The function is $f(x)=\sin x$

Calculation:

The given function is $f(x)=\sin x$

The graph $f(x)$ , $y=0.3$ ,

  $y=0.7$ in the same coordinate plane. Here use the viewing window

  $[0.25,0.85],[0,1]$

  

The intersections at about $(0.3047,0.3)$ and $(0.7754,0.7)$ .

  $0.3<f(x)<0.7$ is true, the interval $(a,b)$ so that the inequality is satisfied.

For the lower bound an approximate value at $0.3047$ and therefore, to guarantee the inequality, must round it up. Possible values for a are $0.305,0.31$ and so forth.

For the lower bound an approximate value at $0.7754$ and therefore, to guarantee the inequality, must round it up. Possible values for b are $0.775,0.77$ and so forth.

Possible values for a: Any number that is rounded up from $0.3047$

Possible values for b: Any number that is rounded up from $0.7754$ .

(c)

To determine

To find: The values of a and b.

Answer to Problem 79E

Possible values for a: Any number that is rounded up from $0.512]$

Possible values for b: Any number that is rounded up from $0.5352$ .

Explanation of Solution

Given information: The function is $f(x)=\sin x$

Calculation:

The given function is $f(x)=\sin x$

The graph $f(x)$ , $y=0.49$ ,

  $y=0.51$ in the same coordinate plane. Here use the viewing window

  $[0.45,0.55],[0.45,0.55]$

  

The intersections at about $(0.5121,0.49)$ and $(0.5352,0.51)$

  $0.49<f(x)<0.51$ is true, the interval $(a,b)$ so that the inequality is satisfied.

For the lower bound an approximate value at $0.5121$ and therefore, to guarantee the inequality, must round it up. Possible values for a are $0.513,0.52$ and so forth.

For the lower bound an approximate value at $0.5352$ and therefore, to guarantee the inequality, must round it up. Possible values for b are $0.535,0.53$ and so forth.

Possible values for a: Any number that is rounded up from $0.512]$

Possible values for b: Any number that is rounded up from $0.5352$ .