What You
NEED
To Know In
Calculus
The fundamental theorem of calculus is used to evaluate
definite integrals using antiderivatives.
It is central to the evaluation of integrals, allowing you to
find the area under a curve or between two functions.
If f is continuous at every point of [a, b], and if F is any antiderivative of f on [a, b] then:
∫
a
b
f(x)dx = F(b) – F(a)
Example:
Find
∫
3
(x3 + 1)dx
-1
∫
3
(x3 + 1)dx
x4
=
4
-1
Antiderivative of
(x3 + 1)
+x
]
3
-1
Antiderivative
evaluated at 3
=
(81 / 4 + 3) – (1 / 4 – 1)
=
24
Antiderivative
evaluated at -1
This rule is used in calculus when you want to find the
derivative of a function that is a quotient.
Hi
Lo
LoDeHi-HiDeLo
Lo2
Example:
Find f’(x)=x2-1
2
x +1
Hi=x2-1
DeHi=2x
2
Lo=x +1 DeLo=2x
2
2
F’(x)= (x +1) (2x)-(x -1)(2x)
2
(x +1)2
The Chain Rule applies whenever there is a function that is formed from two
simpler function.
F(x)= g(h(x)) >>>>
F(x)= g’(h(x)) h’(x)
Example:
(sin(2x))’=2sin(x)cos(x)
(sin(2x))’=2(sin’(x)cos(x)+sin(x)cos’(x))
(sin(2x))’=2(cos2(X)-sin2(x))
(sin(2x))’=
2cos(2x)
f’’’(x)
To get the Jerk of an equation is to take the
derivative of that equation 3 separate times.
f’’’(x)
Example:
Find the Jerk of 4x5.
1st Der.: 20x4
2nd Der.: 80x3
3rd Der.: 240x2
Works Consulted
Finney, Ross, Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus.
New Jersey: Pretince Hall, 2003. Print.
Kahn, David S. Cracking the AP Calculus AB & BC Exams. New York:
Random House, 2009. Print.