# SOLUTION: MATH 2413 HCCS The Fundamental Theorems of Calculus in Mathematics Question

SOLUTION: MATH 2413 HCCS The Fundamental Theorems of Calculus in Mathematics Question.

Homework Assignment 14 (Written)

Chapter 6

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1. (10 points) Several statements about the definite integral of a continuous function f (x) over [a, b] are written below. Determine those statements that are true by writing “TRUE” and those that are false by writing “FALSE.” (No work

need be included with this question.)

Z b

I. One of the Fundamental Theorems of Calculus states that

f (x) dx = F (b)−F (a) where F 0 = f .

.

a

Z b

Z a

f (x) dx = −

II.

a

f (x) dx.

.

b

Z

III.

f (x) dx denotes all possible anti-derivatives of f (x).

Z b

IV.

.

f 0 (sin x) · cos x dx = f (sin(b)) − f (sin(a)).

.

a

V. One of the Fundamental Theorems of Calculus states that

1

d

dx

Z x

a

f (t) dt = f (x).

.

2. (10 points) A portion of the graph y = f (x) is shown in the image below, and the region bounded by this graph

and the x-axis is shaded.

Figure 1: Graph of y = f (x) over [0, 4].

Z 4

If we are told that

Z 4

f (x) dx = 16/3,

0

Z 2

f (x) dx = 37/12 and

3

Z 4

f (x) dx =

0

Z 3

f (x) dx =

2

2

f (x) dx evaluate the definite integral

2

3. (10 points) Two graphs are shown below (along with the shaded regions bounded by them and the x-axis).

A) (5 points) Calculate the deifnite integral of the graphed function without usig the Fundamental Theorem(s) of Calculus.

(Be sure to explain how you calculated this number.)

Figure 2: Graph of f (x) = 3 +

Z 7

3+

√

10x − x2 − 21.

p

10x − x2 − 21 dx =

3

B) (5 points) Calculate the deifnite integral of the graphed function without usig the Fundamental Theorem(s) of Calculus.

(Be sure to explain how you calculated this number.)

Figure 3: Graph of g(x) = |4 + 4x|.

Z 3

|4 + 4x| dx =

−2

3

4.

(10 points)

A graphZ of a function y = f (t) is shown below. Use this graph to answer the following questions

x

f (t) dt.

about the function F (x) =

−4

Figure 4: Graph of y = f (t)

A) (3 points) Compute the values of F (1) and F 0 (1).

B) (7 points) Find the maximum and minimum values of F (x) on the interval [−4, 2].

4

5. (10 points) The functions F (x) and G(x) are given by

Z x

F (x) =

e

arctan t

Z x2

dt

and G(x) =

0

earctan t dt

0

2

Note: Like the function e−t , the integrands used to define both F (x) and G(x) are not possible to anti-differentiate in terms

of our old, familiar functions, so don’t waste your precious time trying.

A) (2 points)

We

can express the function G(x) as a composition of F (x) with another function; that is, we can write

G(x) = F h(x) . Identify the “inside function” h(x).

B) (4 points) Find F 0 (x) and G0 (x).

C) (6 points) Check that L’Hopital’s rule applies to the limits

lim

x→0

F (x)

x

and lim

x→0

and use this rule to evaluate these limits.

5

G(x)

x2

6. (40 points) Complete the following cross-word puzzle with English words (this means writing numbers out as words),

showing your work for clues 4, 7, 8, 12, 13, and 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Across

1. (5 points) A function f (x) with derivative f 0 (x) = ex (5 − x)3 has a local

Z 1

4. (5 points)

ln 7 · 7x dx + 1

at x = 5.

0

6. (2 points) An anti-derivative for the exponential, ex , is the

Z 8

8. (5 points)

−1

Z

9. (5 points)

√

Z e32

1 + x dx +

.

x−1 dx

1

(1 + x2 )−1 dx =

(x) + C.

11. (0 points) When computing an indefinite integral, don’t forget the

!

12. (2 points) One half of Clue 8 minus the value 15 · F (π/4) where F (x) is the function used in Clue 9.

4

14. (points) The score you will earn on your final; a.k.a. the minimum value of f (x) = 89ex + 11.

Down

2. (2 points) If a particle is traveling along a horizontal access with velocity s0 (t) = v(t) = 30t + 5 ft/sec then its

Z 0

√

√

3. (2 points) The definite integral

2 + cos x dx equals the

area under the graph 2 + cos x over [0, 2π].

2π

5. (2 points) The technique of integration that “undoes the Chain Rule” (and was also used in the first integral of Clue 8).

7. (3 points) The area under the curve of y = sec2 x over [0, π/4].

10. (5 points) If f 0 (x) > 0 over an interval, then the y-values of f (x) “

Z b

Z b

13. (2 points) The value of b that satisfies

|t| dt = 2 ·

|t| dt = 81

−9

.”

0

6

is 30 ft/sec2 .

Use this page, if needed, to include (clearly labelled) work from the previous question.

7

7. (10 points) This problem consists of two parts.

A) (7 points) Evaluate the following indefinite integral

Z

√

x2

5

ln(x + 1)

+

+

dx.

3

2

1+x

x+1

1−x

B) (3 points) Based on your work in part A), find a function y(x) that satisfies the differential equation

y 0 (x) = √

x2

ln(x + 1)

5

+

+

3

2

1+x

x+1

1−x

and initial condition y(0) = 5.

8

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SOLUTION: MATH 2413 HCCS The Fundamental Theorems of Calculus in Mathematics Question