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
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 Reiman
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 (
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
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 i
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
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
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 Poi
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
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
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