Calculus Help

Name: The Definite Integral Section:

5.2 The Definite Integral Vocabulary Examples

Definite Integral For a function f defined on the interval [a, b],∫ b a f (x) =

Average Value (of a function) For a function f on the interval [a, b]

fave =

Properties of Definite Integrals

∫ a a f (x)d x =

∫ b a f (x)d x =

∫ b a

( f (x) + g(x)) d x = ∫ b a

( f (x) − g(x)) d x =

∫ b a c · f (x)d x =

∫ b a f (x)d x =

1. Given that ∫ 1 0 x =

1 2 ,

∫ 1 0 x

2 = 13 , and ∫ 1 0 x

3 = 14 , determine the following definite integrals.

(a) ∫ 1 0 (1 + x + x

2 + x3)d x (b) ∫ 1 0 (1 − x + x

2 − x3)d x (c) ∫ 1 0 (6x −

4 3 x

2)d x

(d) ∫ 1 0 (1 − x)

2d x (e) ∫ 1 0 (1 − 2x)

3d x (f) ∫ 1 0 (7 − 5x

3)d x

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Name: The Definite Integral Section:

For each definite integral ∫ b a f (x)d x, sketch a graph each function f over the interval [a, b] and then

use that graph to evaluate the definite integral.

(a) ∫ 4 0 (4 − |x − 4|)d x (b)

∫ 2 −2( √

4 − x2)d x

∫ 0 −5( √

25 − x2)d x ∫ 4 0

( 2 3|x − 3| − 1

) d x

2. The graph of f is shown below. Evaluate ∫ 8 2 f (x)d x

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9

10

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Name: The Fundamental Theorem of Calculus Section:

5.3 The Fundamental Theorem of Calculus Vocabulary Examples

Mean Value Theorem of Integrals

If f is continuous over the interval [a, b], then there exists

at least one value c�[a, b] such that

f (c) =

Furthermore, it can also be said that∫ b a f (x)d x =

The Fundamental Theorem of Calculus

If f is continuous over the interval [a, b] and

F(x) = ∫ x a f (t)dt, then

F′(x) =

Furthermore, if F(x) is any anti-derivate of f (x), then∫ b a f (x)d x =

1. Restate the second part of the Fundamental Theorem of Calculus in your own words.

Evaluate the following integrals.

2. ∫ x 0 3dt 3.

∫ x 0 tdt 4.

∫ x 0

1 3 t

2dt

5. ∫ x 0 dt 6.

∫ x 0 (2 − 6t

2)dt 7. ∫ x 0 (1 + cos t) dt

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Name: The Fundamental Theorem of Calculus Section:

Evaluate the following integrals.

8. ∫ 2 0 3d x 9.

∫ 4 1 x −1d x 10.

∫ 2 −2 e

xd x

11. ∫ π/2 π/3 (sin x)d x 12.

∫ 2 −2 7x

3d x 13. ∫ 2 1

( 1+x2−3x3

x5

) d x

14. The figure below shows the graphs of f (x) = 12x − 6 − 4x2 and g(x) = x3 − 3x2 + 2x + 2. Determine the area of the shaded region.

1 2 3

1

2

3

4

15. Determine the the area bound by f (x) = √

x and g(x) = x2 on the interval [0, 1]

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Name: Differentiation Rules Practice 2 Section:

A.4 Differentiation Rules Practice 2

1. Determine the equation of the line tangent to the function f (x) = x 2−4

(x−2)2 at x = 3

2. The height of a particle can be modeled by the function h(t) = 4.2t − 0.4t4

(a) Sketch a reasonable graph of the function. Be sure to consider all critical points.

(b) Based on your sketch, what is the derivative of h at the projectile’s highest point? (c) In this real-world context, what does h′(t) represent? (d) Determine the time at which the projectile reaches it’s greatest height.

(e) Determine the maximum height of the projectile.

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Name: Differentiation Rules Practice 2 Section:

3. Determine the equation of a line that is tangent to the function f (x) = 25 − x2 and also contains the point (13, 0).

4. The population, in millions, of arctic flounder in the Atlantic Ocean is modeled by the function P(t) = 8t+3

0.2t2+1

(a) Determine the initial population of arctic flounder.

(b) Determine P′(10). What does this mean in the context of the arctic flounder?

5. Given the function f (x) = −x 2

4 + 8.5x − 60.69, solve the following equations.

(a) f (x) = 0 (b) f ′(x) = 2 (c) f ′(x) = −2 (d) f ′(x) = 0

(e) How does our answer to part (d) relate to the original function f ?

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