Derivatives
Find the derivative: f(x) = 4x5 -3x2 +7
20x4 - 6x
Find the critical numbers:
f(x) = x2 + 6x
x = -3
What is f′′(x) called?
The second derivative
Find the critical numbers:
f(x) = x2 - 6x + 5
x = 3
What equation do you usually solve to find critical numbers?
f'(x) = 0
Find the derivative: f(x) = 6x3 - 8x + 2
18x2 - 8
f(x) = x2 - 10x + 3
x = 5
Find f′′(x):
f(x)=3x4−5x2+7x
f′′(x)=36x2−10
Find the critical numbers:
f(x) = x2 + 8x + 12
x = -4
Find the critical number(s).
f(x)=x2+2x
x = -1
Find the derivative: f(x) = 5x3 - 2x2 + 8x - 1
15x2 - 4x + 8
Find the critical numbers:
f(x) = x3 - 12x
x = -2, 2
Find f′′(x):
f(x)=5x3−2x3+8x−1
f''(x)= 30x - 4
A student finds f′(x)=2x−8 and says the critical number is x=8. Are they correct and if not what is the correct answer?
No, the student is incorrect. x=4
Find the critical number(s).
f(x) = x3−3x
x = -1, 1
Find the derivative: f(x) = 7x4 - 3x3 + 5x2 - 9
28x3 - 9x2 + 10x
Find the critical numbers:
f(x) = 3x2 + 12x
x = -2
Find f′′′(x):
f(x)=2x5 − 4x3 + x
f′′′(x)=120x2 − 24
A student finds f′(x)=3x2 and says there are two critical numbers, x=0 and x = -0. What is wrong?
0 and −0 are the same number, so there is only one critical number.
f(x) = x2 - 4x
x = 2
Find the derivative: f(x) = 3x5 - 4x4 + 7x2 - 6x + 10
15x4 - 16x3 + 14x - 6
Find the critical numbers:
f(x) = 2x2 - 8x
x = 2
f(x)=x6−3x4+2x2 , Find f'''(x)
f′′′(x)=120x3−72x
What is the first step in finding critical numbers?
Find the derivative
Find the critical numbers:
f(x) = x2
x= 0