1225FormA

Math 1225

Common Final Exam

Fall 2021

Form A

Instructions :

• Fill in A, B or C in the Test Version section.

• Enter your NAME, ID Number, CRN (under Class ID) and write A, B, or C (under Test ID) on the op-scan sheet.

• Darken the appropriate circles below your ID number and Class ID (CRN). Use a number 2 pencil. Machine grading may ignore faintly marked circles.

• For each problem, mark only ONE answer choice.

• Mark your answers to the test questions in rows 1–14 of the op-scan sheet. Your score on this test will be the number of correct answers.

• You have one hour to complete this portion of the exam. Turn in the op-scan sheet with your answers, this exam, and all scrap paper at the end of this part of the final exam.

• Read each question carefully .

Exam Policies : You may not use a book, notes, formula sheet, calculator, or a computer.

Name (printed):

Student ID #:

Honor Pledge: I have neither given nor received unauthorized assistance on this exam.

Signature:

Page 1 of 7 — Form A

1. Which of the following derivatives is correct ?

d dx d dx d dx d dx

arctan( x ) = 2 x = 2 x log 2 ( x ) =

1 √ 1 − x 2

(A)

(B)

1 x ln(2)

(C)

sin 2 ( x ) = 2 cos( x )

(D)

2. The graph of y = v ( t ), the velocity of a particle at time t , is given below.

y

2 4 6 8 10

y = v ( t )

x

1

2

3

4

5

6

7

− 8 − 6 − 4 − 2

On which of the following intervals is the particle slowing down throughout the entire interval?

(A) (1 , 2) (B) (2 , 3) (C) (3 , 4) (D) (5 , 6)

Page 2 of 7 — Form A

3. Let f and g be two differentiable functions for all real values of x . The following table shows some values for f , f ′ , g , and g ′ .

f ′ ( x ) g ( x ) g ′ ( x )

x f ( x )

− 1 2 0 0 2 1

− 4 3

8 5

− 5

2

− 3 1 10

Evaluate the following at x = 2:

d dx

g ( x ) f ( x − 3)

5 4

(A)

(B) 3 (C) 7 (D) 12

4. A cylindrical tank with radius 10 feet is being filled with water at the rate of 300 ft 3 / min. How fast is the water level rising? Recall: The volume of a cylinder is given by V = πr 2 h , where r is the radius and h is the height of the cylinder.

3 π

(A)

ft / min

15 π 3 4 π

(B)

ft / min

(C)

ft / min

(D) Without more information about dr dt

, this problem cannot be solved.

5. Which of the following definite integrals is equal to the following expression?

n X i =1

+ 2

2

1 n

i n

lim n →∞

(A) Z

(B) Z

(C) Z

(D) Z

1

2

1

3

x 2 dx

x 2 dx

( x + 2) 2 dx

( x + 2) 2 dx

0

0

0

2

Page 3 of 7 — Form A

6. Suppose that

• f is a continuous, real-valued function on the closed interval [ a, b ] and • f has an absolute minimum at x = c , where a < c < b .

Consider the following statements about f :

Statement I: f ( c ) ≤ f ( x ) for all x in [ a, b ]. Statement II: f has an absolute maximum on [ a, b ].

Which of the above statements MUST be TRUE ?

(A) Statement I only (B) Statement II only

(C) both Statement I and Statement II (D) neither Statement I nor Statement II

7. Suppose f is continuous on ( −∞ , ∞ ) and the graph of its derivative, y = f ′ ( x ), is shown below.

y

3

y = f ′ ( x )

2

1

x

1

2

3

4

5

6

Which of the following MUST be FALSE about the function f ?

(A) f has a critical number at x = 3. (B) f has an inflection point at x = 3. (C) f has a critical number at x = 4 . 5. (D) f is concave up on (1 , 3).

Page 4 of 7 — Form A

8. A bowl of hot soup is set to cool. Let y = R ( t ) represent the rate at which the temperature of the soup is changing, in degrees Fahrenheit per minute, for a period of 10 minutes, 0 ≤ t ≤ 10. Values of R are given in the table below:

t (minutes) 0 3 8 10 R ( t ) ( ◦ F/minute) − 5 − 4 − 3 − 1

Assume that the temperature of the soup is 110 ◦ F at t = 0. Estimate the temperature of the soup at t = 10 minutes using L 3 , a Riemann sum with three sub-intervals whose sample points are left endpoints.

(A) − 36 ◦ F (B) − 41 ◦ F (C) 69 ◦ F (D) 74 ◦ F

9. The graph of y = f ( x ) is given below.

y

x

x 4

x 1

a

x 2 x 3

b

y = f ( x )

How many of the x -values labeled on the graph above ( x 1 , x 2 , x 3 , and x 4 ) satisfy the conclusion of the Mean Value Theorem for f on the interval [ a, b ]?

(A) 1

(B) 2

(C) 3

(D) 4

Page 5 of 7 — Form A

10. Let f be a function such that x 2 − 1 ≤ f ( x ) ≤ 1 − x 2 for all x in ( − 1 , 1) and assume that f is defined everywhere. Which of the following statements MUST be TRUE ?

(A) lim x → 0

f ( x ) does not exist.

(B) f (1) = 0. (C) lim x → 0

f ( x ) = 1.

(D) lim

f ( x ) = 0.

x →− 1 +

11. Consider the graph of y = f ( x ) given below:

y

3

x

− 1 2

y = L ( x )

y = f ( x )

Suppose that L is the linearization of f at x = 3. Which of the following MUST be FALSE ?

(A) If we use x 1 = 3 as the initial approximation of Newton’s Method, then L ( x 2 ) = 0 (where x 2 is the second approximation of Newton’s method). (B) L also represents the linearization of f at a = − 1 2 .

(C) f ′ (3) = L ( x ) . (D) The tangent line to the graph of y = f ( x ) at x = 3 is y = L ( x ). 12. Given f ( x ) = Z 3 x 2 sin 4 ( t ) dt , find f ′ ( x ). (A) f ′ ( x ) = − 2 x sin 4 ( x 2 ) (B) f ′ ( x ) = − sin 4 ( x 2 ) d dx

(C) f ′ ( x ) = − 4 sin 3 ( x 2 ) cos( x 2 ) (D) f ′ ( x ) = − 8 x sin 3 ( x 2 ) cos( x 2 )

Page 6 of 7 — Form A

13. Evaluate the following limit for fixed x > 0:

lim h → 0 p

( x + h ) 2 + 2( x + h ) − √ x 2 + 2 x h

(A) The limit does not exist.

1 2 √ x 2 + 2 x

(B)

(C) √ 2 x + 2

x + 1 √ x 2 + 2 x

(D)

14. Consider the graph of y = f ( x ) below.

y

6

5

y = f ( x )

4

3

2

1

x

1 2 3 4 5 6

Choose the largest value of δ > 0 such that the following statement is true.

If 0 < | x − 4 | < δ , then | f ( x ) − 2 | < 1.

(A) No such value of δ exists. (B) 0.25

(C) 0.5 (D) 1.5

Page 7 of 7 — Form A