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Find the average or mean slope of the function on this interval?
Consider the function f(x)=4x^3−4x on the interval [−3,3].
a) Find the average or mean slope of the function on this interval. Average Slope= what?
b) By the Mean Value Theorem, we know there exists at least one c in the open interval (−3,3) such that f′(c) is equal to this mean slope. Find all values of c that work and list them (separated by commas) in the box below.
List of values:
1 Answer
- AlanLv 76 years ago
(a)
average slope= ( f(3) - f(-3)) / ( 3 - (-3))=
average slope = ( 4*(3)^3 - 4*(3) - ( 4*(-3)^3 -4*(-3) ) ) / (6) =
average slope = ((4*27 - 12 ) -(4*(-27) +12) / 6
average slope = (( 108 -12)- (-108 +12) ) / 6
average slope = (96 - ( -96) ) / 6 =
average slope = 192/6 = 32
answer = 32
(b)
df(x) /dx = 12x^2 - 4
so when does df(x)/dx = 32
12x^2 -4 = 32
12x^2 = 36
x^2 = 36/12 = 3
x= +/- sqrt(3)
c = - sqrt(3), + sqrt(3)
List of values : -sqrt(3), + sqrt(3)
