[mean value theorem][section 13]
(1.) f(x) = 1 + 2x2 ;
(-1,1)
f'(x) = 4x take the derivative
f(-1) = 1 + 2(-1)2 f(1) = 1 + 2(1)2
[replace x with -1 and 1 in f]
f(-1) = 3 f(1) = 3 multiply, add
m = (y2 - y1)/(x2 - x1) use the slope formula
m = (3 - 3)/(1 - -1) make substitutions
m = 0 subtract, add, divide
4x = 0 set 4x equal to 0
___
___
4
4 divide each side by 4
x = 0 divide and cancel
result: c = 0
(2.) f(x) = 1/x ; (1,4)
here is the problem
f(1) = 1/1 ;
f(4) = 1/4 replace x with 1
and 4
f(1) = 1 ;
f(4) = 1/4 divide
m = (y2 - y1)/(x2 - x1) use the slope
formula
m = [(1/4) - 1]/(4 - 1) make substitutions
m = (-3/4)/(3) subtract
m = -1/4 divide
f'(x) = -1/x2 take the derivative of x
-1/x2 = -1/4 set the derivative equal to -1/4
1/x2 = 1/4 multiply each side by -1
x2 = 4 take reciprocals
x = 2 take square roots
result:
c = 2
_
(3.) f(x) = √x ;
(1,4) here is the
problem
_ _
f(1) = √1
; f(4) = √4
replace x with 1 & 4
f(1) = 1 ;
f(4) = 2 take square roots
m =
(y2 - y1)/(x2 - x1) use the slope
formula
m = (2 - 1)/(4 - 1) make substitutions
m = 1/3 subtract
f'(x) = (1/2)x-1/2 take the derivative of f
1/3 = (1/2)x-1/2 replace f'(x) with 1/3
2 = 3x-1/2 multiply each side by 6, cancel
___ ______
3
3 divide each side by 3
2/3 = x-1/2 cancel
3/2 = x1/2 take the reciprocal of each side
9/4 = x square each side
result: c = 9/4
_
(4.) f(x) = \3/x ;
(-8,-1) here is the
problem
__ __
f(-8) = \3/-8 ;
f(-1) = \3/-1
[replace x with -8][replace x with -1]
f(-8) = -2 f(-1) = -1 take cube roots
m = (y2 - y1)/(x2 - x1) use the slope formula
m = (-1 - -2)/(-1 - -8) make substitutions
m = 1/7 add
f'(x) = (1/3)x-2/3 take
the derivative of f
1/7 =(1/3)x-2/3 replace f' with 1/7
3 = 7x-2/3 multiply each side by 21, cancel
__ ______
7
7 divide each side
by 7
3/7 = x-2/3 cancel
7/3 = x2/3 take reciprocals of each side
x2/3
= 7/3 just rearrange like this
x2 = (7/3)3 cube each side
______ ______
x =
√(7/3)3 x
= -√(7/3)3 take the sq rt of ea side
_______
result: c = -√(7/3)3
(5.) f(x) = x3 ;
(1,2) here is the problem
f(1) = 13 f(2) = 23 replace x with 1 and with 2
f(1) = 1 f(2) = 8 cube 1 and cube 2
m = (y2 - y1)/(x2 - x1) use the slope
formula
m = (8 - 1)/(2 - 1) make substitutions
m = 7 subtract, divide
f'(x) = 3x2 take the
derivative of f
7 = 3x2 replace f'
with 7
__ ___
3 3 divide each side by 3
7/3 = x2 cancel
___
x = √7/3 take square roots
____
result: c = √7/3
(6.) f(x) = x3 ; (-2,2)
(7.) f(x) = (x + 3)(x - 1)(x - 5) ; (-3,1)
f(x) = (x + 3)(x2 - 6x + 5) foil multiply combine like terms
f(x) = x3 - 6x2 +
5x + 3x2 - 18x + 15 multiply
f(x) = x3 - 3x2 - 13x + 15 combine like terms
f(-3) = (-3 + 3)(-3 - 1)(-3 - 5) f(1) =
(1+3)(1-1)(1-5)
[replace x with -3][replace x with 1]
f(-3) = 0 f(1) = 0 simplify
m = (y2 - y1)/(x2 - x1) use the slope
formula
m = (0 - 0)/(1 - -3) make substitutions
m = 0 subtract add divide
f'(x) = 3x2 - 6x - 13 take
the derivative of f
3x2 - 6x - 13 = 0 set the
derivative equal to 0
+ 16 +16
add 16 to each side
____________________
3x2 - 6x + 3 = 16 add
3(x2 - 2x + 1) = 16
factor
3(x - 1)2 = 16
factor
(x - 1)2 = 16/3 divide
each side by 3, cancel
_ _
x - 1 = 4/√3 x - 1 = -4/√3 take square roots
+
1 +1 +1
+1 add 1 to each side
____________ ______________
_ _
x = 1 + (4/√3) x
= 1 - (4/√3)
_
result: c = 1 - (4/√3)
(8.) f(x) = x3 - 3x2
+ 1 ; (0,3)
f(0) = 03
- 3(0)2 + 1 f(3) = 33
- 3(3)2 + 1
[replace x with 0][replace x with 3]
f(0) = 1 f(3) = 1 simplify
m = (y2 - y1)/(x2 - x1) use the slope
formula
m = (1 - 1)/(3 - 0) make substitutions
m = 0 subtract and divide
f'(x) = 3x2 - 6x take
the derivative of f
3x2 - 6x = 0 set the
derivative equal to 0
3x(x - 2) = 0 factor
x - 2 = 0 set this factor equal to 0
+
2 +2 add 2 to each side
___________
x =
2 add
result: c = 2
(9.) f(x) = cos x ; (0, pi/2)
f(0) =
cos 0 f(pi/2) = cos (pi/2)
[replace x with 0][replace x with pi/2]
f(0) = 1 f(pi/2) = 0 use the unit circle
m = (y2 - y1)/(x2 - x1) use the slope formula
m = (0 - 1)/[(pi/2) - 0] make substitutions
m = -2/pi subtract, and take the reciprocal of
the bottom
f'(x) = -sin x take the derivative
of f
-sin x = -2/pi set the derivative equal to -2/pi
sin x = 2/pi multiply each side by -1
x = arcsin (2/pi) take the arcsin of each side
result: c = arcsin (2/pi)
(10.) f(x) = sin 2x ; (0,
pi/12)
f(0) = sin 2(0) f(pi/12) = sin 2(pi/12)
[replace x with 0][replace x with pi/12]
f(0) = sin 0 f(pi/12) = sin
(pi/6) multiply, reduce
f(0) = 0 f(pi/12) = 1/2 use the unit circle
m = (y2 - y1)/(x2 - x1) use the slope
formula
m = [(1/2) - 0]/[(pi/12) - 0] make substitutions
m = (1/2)/(pi/12) subtract
m = (1/2)*(12/pi) multiply by the
reciprocal of the bottom
m = 6/pi multiply and
reduce
f'(x) = 2 cos 2x take the
derivative of f
2 cos 2x = 6/pi set the
derivative equal to 6/pi
cos 2x = (1/2)(6/pi) multiply each side by 1/2, cancel
cos 2x = 3/pi divide
2x =
arccos (3/pi) take the
arccos of each side
x = (1/2)arccos (3/pi) multiply each side by 1/2, cancel
result: c = (1/2)arccos (3/pi)