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Name: Antiderivatives Section:

4.10 Antiderivatives Vocabulary Examples

Antiderivative A function F, related to f , such that F′(x) = for all x

within the domain of f Indefinite Integral For a function f such that F′(x) = f (x),∫

f (x)d x = Power Rule for Integrals

∫ xnd x = for n , 1

Table of Integrals

∫ kd x =

∫ xnd x =

∫ 1 x d x =

∫ exd x =

∫ (cos x)d x =

∫ (sin x)d x =

∫ (sec2 x)d x =

∫ (csc x cot x)d x =

∫ (sec x tan x)d x =

∫ (csc2 x)d x =

∫ 1

√ 1−x2

d x = ∫

1 1+x2

d x =

1. Determine the derivative of each of the following functions.

(a) f (x) = 3x2 (b) f (x) = 3x2 − 11 (c) f (x) = 3x2 + 37

2. Determine a function F for which F′(x) = f (x) given the function: f (x) = 6x

3. Considering our answers to #1 and #2, what is something that must be true for all antiderivatives.

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Name: Antiderivatives Section:

Determine the antiderivatives of the following functions.

4. f (x) = 12 x 4 − 2x3 − 7x + 11 5. f (x) = 1

x2 6. f (x) = ex + e2x

Determine the antiderivatives of the following functions.

7. f (x) = x 1/3

x2/3 8. f (x) = 2 sin x + sin(2x) 9. f (x) = sin x cos x

Determine the following integrals

10. ∫

(−1)d x 11. ∫

3x2+2 x2

12. ∫

4 √

x + e−x + 4 √

x

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Name: Antiderivatives Section:

13. Determine the function f for which f ′(x) = x−3 and f (1) = 1.

14. Determine the function f for which f ′(x) = cos x + sec2 x and f

( π 4

) = 2 +

√ 2

2 .

15. The velocity of a particle can be given by the function v(t) = 14t − 3.2t2. Determine the function that models the position of the particle if it’s initial position at t = 0s is 0 m.

16. Determine the equation of the quadratic equation whose instantaneous rate of change at x = 2 is 12 and for which f (0) = 10 and f (2) = 4.

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Name: Approximating Areas Section:

5.1 Approximating Areas Vocabulary Examples

Sigma Notation A compact algebraic notation to represent a

5∑ i=1

i =

Sum Properties For sequences a1 and bi and constant c, n∑

i=1 c =

n∑ i=1

cai =

n∑ i=1

(ai + bi) = n∑

i=1 (ai − bi) =

n∑ i=1

ai = n∑

i=1 i =

n∑ i=1

i2 = n∑

i=1 i3 =

Reimann Sum The of the partitions under a curve.

Notation: n∑

i=1 for a closed interval where

∆x is the of each partition, or subinterval, on

the interval and x∗i is the x-value of any on the

partition. Area Under a Curve For a function f the area under curve f on the interval

[a, b] is given by the formula:

lim x→

n∑ i=1

1. Find the left sum and right sum of the following function on the interval [0, 5] and with 5 partitions.

1 2 3 4 5

1 2 3 4 5 6 7 8 9

10

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Name: Approximating Areas Section:

2. Find the left sum and right sum of the following function on the interval [0, 5] and with 5 partitions.

1 2 3 4 5

−3 −2 −1

1 2 3 4

3. Determine the left sum of f (x) = e x over [0, 1] with 4 partitions.

4. Determine the right sum of f (x) = x2 − x over [1, 4] with 6 partitions.

5. Determine the area under the following ”curve” over the interval [0, 10].

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9

10

6. Write a formula for the left sum of f (x) = 3x2 over [0, a] with 4 partitions.

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