Suppose f(x) is the derivative (rate of change) of F(x), and f(x) is above the x-axis from the interval a to b. f(x) is the slope of F(x), and since f(x) is always positive, F(x) is always increasing from a to b. The integral of f(x) is the net change in
Formal Statement If f is continuous on [a, b], then 1. If g(x) = 2.
u ( x)
a
f (t ) dt , then g(x) = f( u(x) ) u(x)
b
a
f ( x)dx = F (b) F (a ) , where F is any antiderivative of f, that is, F = f.
If the graph of the function f is continuous on the close
Exercise Indenite Integrals
1. ShowZthat
(a) (3 x2 )3 dx = 27x
(c)
Z
(e)
Z
(g)
(i)
(m)
(s)
x)(1
dx = a ln jxj
2x)(1
(2x + 3x )2 dx =
Z
Z
Z
Z
17
x +C
7
a2
x
(b)
a3
+C
2x2
(d)
cos x + sin x + C
1
+C
2(1 + x2 )
dx
p
=
1 + e2x
ln e
p
1
x
x+C
p
+
1+e
2x
+C
1
l
The Fundamental Theorem of Calculus (Part2)
f ( x)dx
a
b
= F(b) F(a) where F(x) = f (x)
Proof: In the beginning, we established that integration is the sum of the area under the curve. We used Reimann sums to define integrals:
a
b
f ( x)dx
=
lim
n
f (
The Fundamental Theorem of Calculus (Part 1) If f is continuous on [a, b], then the function g defined by g ( x) = f (t )dt
a x
axb
Is continuous on [a, b] and differentiable on (a, b), and g(x) = f (x). Proof: g ( x + h) g ( x ) =
x+h x
a
f (t )dt f (t )
AP Calculus Practice Multiple Choice and FRQ Part 1: Calculator Inactive
y
1.
x a c b
f
The function f, whose graph consists of two line segments, is shown above. Which of the following are true for f on the open interval ( a, b ) ?
I. II. III.
(A) (B) (C
Historical Background Historical The true mathematician who found the Fundamental Theorem of Calculus is disputed. Fundamental While some believe Isaac Newton and Gottfried While Leibniz each worked independently on the Leibniz development of calculus con
The Fundamental Theorem of Calculus (Part1)
If f is continuous on an interval I, then f has an antiderivative on I. In particular, if a is any number in I, then the function F defined by F ( x) = f (t )dt
a x
is an antiderivative of f on I; that is, F(x)
Practical Application
CALCULUS IS EVERYWHERE Calculus is in Physics. derivative of speed = velocity derivative of velocity = acceleration Calculus is in Biology. Calculus is involved in biology, such as demonstrated by Poiseuilles Law and blood flow, etc.
CALCULUS IS EVERYWHERE Calculus is in Physics. derivative of speed = velocity derivative of velocity = acceleration Calculus is in Biology. Calculus is involved in biology, such as demonstrated by Poiseuilles Law and blood flow, etc. Calculus is in Econom
1
Suppose f(x) is the derivative (rate of change) of F(x), and f(x) is above the x-axis from the interval a to b.
THE FUNDAMENTAL THEOREM OF CALCULUS PART II
LAYMANS STATEMENT
2
f(x) is the slope (the derivative) of F(x), and since f(x) is always positive
Solution Indenite Integrals
Question 1
(a)
Z
Z
23
(3 x ) dx =
27x2 + 9x4
27
9
9x3 + x5
5
= 27x
x6 dx = 27x
27
x3
x5
+9
3
5
x7
+C
7
17
x +C
7
(b)
Z
1
(c)
Z
1
x2
Z
q
p
x xdx =
Z
=
a a2 a3
+
+
x x2 x3
p
1
x2
1
x 3= 4
x
dx = a ln jxj + a
xx1=2 dx =
Z
1
4
dx =