Calculus Help

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) = 3×2 (b) f (x) = 3×2 − 11 (c) f (x) = 3×2 + 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 − 2×3 − 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.
∫

3×2+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) = 3×2 over [0, a] with 4 partitions.

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