h(t)=cot t
a.) [pi/4,3pi/4] b.) [pi/6,pi/2]
Find the average rate of change of the function over the give n interval and intervals. Please show step by step instructions.
Thanks
Of course in this case you can write:
The average rate of change = ( h2 - h1 ) / ( t2 - t1 )
t1 = π / 4
t2 = 3 π / 4
h1 = cot ( π / 4 ) = 1
h2 = cot ( 3 π / 4 ) = - 1
( h2 - h1 ) / ( t2 - t1 ) =
( - 1 - 1 ) / ( 3 π / 4 - π / 4 ) =
- 2 / ( 2 π / 4 ) =
( - 2 / 1 ) / ( 2 π / 4 ) =
- 2 ∙ 4 / 1 ∙ 2 π =
- 2 ∙ 4 / 2 π =
- 4 / π
b.)
t1 = π / 6
t2 = π / 2
h1 = cot ( π / 6 ) = √3
h2 = cot ( π / 2 ) = 0
( h2 - h1 ) / ( t2 - t1 ) =
( 0 - √3 ) / ( π / 6 - π / 2 ) =
- √3 / ( π / 6 - 3 π / 6 ) =
- √3 / ( - 2 π / 6 ) =
( - √3 / 1 ) / ( - 2 π / 6 ) =
- √3 ∙ 6 / 1 ∙ ( - 2 π ) =
- 6 ∙ √3 / - 2 π =
- 2 ∙ 3 ∙ √3 / - 2 ∙ π =
3 √3 / π
Bosnian, part b should be negative. Believe you solved as y2-y1/x1-x2 instead of y2-y1/x2-x1which would flip it to positive. Otherwise correct
To find the average rate of change of a function, we need to determine the change in the function values over the given interval and divide it by the length of the interval.
Let's start with option a) [π/4, 3π/4]:
1. Substitute the upper and lower limits of the interval (t values) into the given function to find the corresponding function values.
h(π/4) = cot(π/4)
h(3π/4) = cot(3π/4)
2. Calculate the change in function values by subtracting the initial value from the final value.
Change in function values = h(3π/4) - h(π/4)
3. Find the length of the interval by subtracting the lower limit from the upper limit.
Interval length = 3π/4 - π/4
4. Divide the change in function values by the interval length.
Average rate of change = (h(3π/4) - h(π/4)) / (3π/4 - π/4)
Now, let's move on to option b) [π/6, π/2]:
1. Substitute the upper and lower limits of the interval (t values) into the given function to find the corresponding function values.
h(π/6) = cot(π/6)
h(π/2) = cot(π/2)
2. Calculate the change in function values by subtracting the initial value from the final value.
Change in function values = h(π/2) - h(π/6)
3. Find the length of the interval by subtracting the lower limit from the upper limit.
Interval length = π/2 - π/6
4. Divide the change in function values by the interval length.
Average rate of change = (h(π/2) - h(π/6)) / (π/2 - π/6)
Following these steps, you can find the average rate of change for both options a) and b) by substituting the given values into the formula and performing the calculations.
The average rate of change = ( y2 - y1 ) / ( x2 - x1 )
a.)
x1 = π / 4
x2 = 3 π / 4
y1 = cot ( π / 4 ) = 1
y2 = cot ( 3 π / 4 ) = - 1
( y2 - y1 ) / ( x2 - x1 ) =
( - 1 - 1 ) / ( 3 π / 4 - π / 4 ) =
- 2 / ( 2 π / 4 ) =
( - 2 / 1 ) / ( 2 π / 4 ) =
- 2 ∙ 4 / 1 ∙ 2 π =
- 4 / π
b.)
x1 = π / 6
x2 = π / 2
y1 = cot ( π / 6 ) = √3
y2 = cot ( π / 2 ) = 0
( y2 - y1 ) / ( x2 - x1 ) =
( 0 - √3 ) / ( π / 6 - π / 2 ) =
- √3 / ( π / 6 - 3 π / 6 ) =
- √3 / ( - 2 π / 6 ) =
( - √3 / 1 ) / ( - 2 π / 6 ) =
- √3 ∙ 6 / 1 ∙ ( - 2 π ) =
- 6 ∙ √3 / - 2 π =
- 2 ∙ 3 ∙ √3 / - 2 ∙ π =
3 √3 / π