Mean Value Theorem
Mean Value Theorem
x
Suppose f(x)= 3
Mean Value Theorem
x
Suppose f(x)= 3
x
y
0
0
Mean Value Theorem
x
Suppose f(x)= 3
x
y
0
0
1
1
Mean Value Theorem
x
Suppose f(x)= 3
x
y
0
1
2
0
1
8
Mean Value Theorem
x
Suppose f(x)= 3
x
y
0
1
2
-1
0
1
8
-1
Mean Value Theorem
x
Suppose f(x)= 3
x
y
0
1
2
-1
-2
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
x
y
0
1
2
-1
-2
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
x
y
0
1
2
-1
-2
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
x
y
0
1
2
-1
-2
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
y
0
1
2
-1
-2
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
2
-1
-2
y
0
1
8
-1
-8 (-2,-8)
(2,8)
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
8(8)
2(2)
x
0
1
2
-1
-2
y
0
1
8
-1
-8 (-2,-8)
(2,8)
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
8(8) 16  4
2(2) 4
x
0
1
2
-1
-2
y
0
1
8
-1
-8 (-2,-8)
(2,8)
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
8(8) 16  4
2(2) 4
Find derivative and set = 4
x
0
1
2
-1
-2
y
0
1
8
-1
-8 (-2,-8)
(2,8)
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x) 3x2
y
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x) 3x2  4
y
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x) 3x2  4 so
x2  4
3
y
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x)  3x2  4
x2  4  x=  2 3
3
3
y
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x)  3x2  4
x2  4  x=  2 3  1.15
3
3
y
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x)  3x2  4
x2  4  x=  2 3  1.15
3
3
y
0
1
8
-1
-8
Draw tangent
lines
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x)  3x2  4
x2  4  x=  2 3  1.15
3
3
y
0
1
8
-1
-8
Draw tangent lines
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x)  3x2  4
x2  4  x=  2 3  1.15
3
3
y
0
1
8
-1
-8
Draw tangent lines
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x)  3x2  4
x2  4  x=  2 3  1.15
3
3
y
0
1
8
-1
-8
Mean Value Theorem
x
Suppose f(x)= 3
F(x) is continuous on [-2,2]
Draw line from -2 to 2
Find slope of line:
x
0
1
8(8) 16  4
2
2(2) 4
-1
Find derivative and set = 4 -2
f '(x)  3x2  4
x2  4  x=  2 3  1.15
3
3
y
0
1
8
-1
-8
c=  2 3
3