If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then m(b − a) ≤ b f(x) dx a ≤ M(b − a). Use this property to estimate the value of the integral. π/12 5 tan(4x) dx π/16

The correct question is:

If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then

m(b − a) ≤ (integral from a to b)f(x)dx ≤ M(b − a).

Use this property to estimate the value of the (integral from π/16 to π/12) 5 tan(4x)dx.

The estimated value of the integral is 0.447030768

Step-by-step explanation:

Given f(x) = 5tan(4x)

f(x) is an increasing function on the interval (-infinity, infinity)

This implies that x must be in the interval (-π/8, π/8)(1)

But we are given the interval

[π/16, π/12] (2)

Since this interval is contained in (1), we have π/16 to be the minimum, and π/12 to be the maximum.

Now, the minimum value is

f(π/16) = 5tan(4π/16)

= 5tan(π/4)

= 5 × 1

m = 5

Maximum value is

f(π/12) = 5tan(4π/12)

= 5tan(π/3)

= 5 × √3

M = 5√3

Now, because

m(b - a) ≤ (integral from a to b) f(x)dx ≤ M(b - a)

We have

5(π/12 - π/16) ≤ (Integral from (π/16 to π/12) 5tan(4x)dx ≤ 5√3(π/12 - π/16)

5π/48 ≤ (Integral from (π/16 to π/12) 5tan(4x)dx ≤ 5√3π/48

Let's take the midpoint of this interval to be the approximate value of the integral.

(Integral from (π/16 to π/12) 5tan(4x)dx is approximately

(5π/48 + 5√3π/48)/2

= 0.447030768

square root of 36/6 = 2.449489743

square root of 16/6 = 1.632993162

square root of 6 = 2.449489743

the answer is square root of 16/6 because the square root of 36/6 is the same answer as square root of 6 = 2.449489743

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕