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Quiz 10 Dec. 26, 2007

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Calculus I Name:

TA/classroom: Student ID:

Quiz 10

Dec. 26, 2007

1. (5 pts) Use Riemann Sums to compute the given definite integral

1

0

x2dx

Set f (x) = x2, ∆x = 1N−0 and xi = 0 + i∆x = Ni. Then

AN =

N i=1

f (xi)∆x =

N i=1

( i N)2 1

N = 1 N3

N i=1

i2 = 1 N3

N (N + 1)(2N + 1) 6

1

0

x2dx = lim

N→∞

1 N3

N (N + 1)(2N + 1)

6 = 1

3

2. (5 pts) Use the Fundamental Theorem to compute the given definite integral

1 0

x2dx = 1

3x3|10 = 1 3

3. (10 pts) Given F (x) =x2 x

√t2+ 1 dt, use the Fundamental Theorem to compute F0(x)

F0(x) =√

x4+ 1(x2)0 −√

x2+ 1(x)0 = 2x√

x4+ 10−√

x2+ 1.

or, let G(x) =x 0

√t2+ 1 dt, then we have F (x) = G(x2)−G(x) and, by Fundamental Theorem of Calculus, G0(x) =√

x2 + 1.

Thus

F0(x) = (G(x2)− G(x))0 = G0(x2)(x2)0− G0(x) = 2x√

x4+ 1−√

x2+ 1.

. .

• Theorem 1.1 If n is any positive integer and c is any constant, then

n i=1

c = cn,

n i=1

i = n(n + 1)

2 ,

n i=1

i2 = n(n + 1)(2n + 1) 6

• Fundamental Theorem of Calculus Part I: If f is continuous on [a, b] and F (x) is any antiderivative of f , thenb

a f (x)dx = F (b)− F (a).

• Fundamental Theorem of Calculus, Part II: If f is continuous on [a, b] and F (x) =x

a

f (t)dt, then F0(x) = f (x), on [a, b].

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