Practice Test 3B - Calculus for Business and Life Sciences | MATH 160, Exams of Mathematics

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Math 160. Practice test 3B.

Name:

f(x)^ A^ B

C D^ E

F (^) G H

1. Which of the above is f ′(x)? Which of the above is f ′′(x)?

f(x)^ A^ B

C D^ E

F G (^) H

2. Which of the above is f ′(x)? Which of the above is f ′′(x)?

Problems 3-7 use these pictures of f (x):

(A) (B)

3. On which intervals is f ′(x) > 0?
4. Where is f ′(x) = 0?
5. On which intervals is f ′′(x) > 0?
6. Where is f ′′(x) = 0?
7. Where is f ′(x) increasing?

Problems 8-18 refer to these functions:

(A) f (x) = x^ + 2 2 x + 1

(B) f (x) =^2 x^ + 5 x^2

8. What is the domain of f (x)?
9. Give the equations for any vertical asymptotes of f (x).
10. Give the equations for any horizontal or oblique asymptotes of f (x).
11. Give the coordinates of all x and y intercepts of f (x)
12. Calculate f ′(x).
13. What are the partition numbers for f ′(x)?
14. Where is f (x) increasing? Where is f (x) decreasing?
15. Calculate f ′′(x).
16. What are the partition numbers for f ′′(x)?
17. Where is f (x) concave up? Where is f (x) concave down?

6. (A) x = 0 and x = 6; (B) x = 4. 5
7. (A) (− 3. 2 , 0) and (3, 6); (B) (−∞, 0) and (4. 5 , ∞)
8. (A) All real x except x = − 1 /2 so (−∞, − 1 /2) ∪ (− 1 / 2 , ∞); (B) all real x except x = 0. (so (−∞, 0) ∪ (0, ∞)).
9. (A) x = − 12 ; (B) x = 0
10. (A) y = 12 ; (B) y = 0
11. (A) (− 2 , 0) and (0, 2); (B) (− 5 / 2 , 0)
12. (A)

f ′(x) = (2x^ + 1)^ −^ (x^ + 2)(2) (2x + 1)^2

= −^3

(2x + 1)^2

(B)

f ′(x) = − 2 x−^2 − 10 x−^3 =

− 2 x − 10 x^3

13. (A) x = − 1 /2; (B) x = 0 and x = − 5
14. (A) Decreasing (−∞, − 1 /2) and (− 1 / 2 , ∞); (B) increasing on the interval (− 5 , 0); decreasing on the intervals (−∞, −5) and (0, ∞).
15. (A) f ′′(x) = 12(2x + 1)−^3

(B)

f ′′(x) = 4x−^3 + 30x−^4 =

4 x + 30 x^4

16. (A) x = − 1 /2; (B) x = 0 and x = − 15 / 2
17. (A) Concave down for on the interval (−∞, − 1 /2) and concave up on (− 1 / 2 , ∞); (B) Concave up on the intervals (− 15 / 2 , 0) and (0, ∞). Concave down on the interval (−∞, − 15 /2)

18. (A) (B)

V = 16πh =⇒ dVdt = 16π dhdt =⇒ −1 = 16π dhdt =⇒ dhdt = − (^161) π ≈ −. 02 f t/hr

20. Let x be the distance along the ground and let y be the length of the kite string. Then

x^2 + 100^2 = y^2 =⇒ 2 x ·

dx dt = 2y^ ·^

dy dt

When y = 200, x^2 + 100^2 = 200^2 =⇒ x ≈ 173, so

2(173)(8) = 2(200) · dy dt

=⇒ dy dt

≈ 6. 92 f t

21. Let x be the distance car B has traveled and let y be the distance car A has traveled. Let D be the distance between them. Then

D^2 = x^2 + y^2 =⇒ 2 D ·

dD dt = 2x^ ·^

dx dt + 2y^ ·^

dy dt

After two hours, car A has traveled 90 miles and car B has traveled 130 miles. The total distance between them at that instant is: D =

902 + 130^2 ≈ 158. Plug in:

2 · 158 · dD dt

= 2 · 130 · 65 + 2 · 90 · 45 =⇒ dD dt

dV dt = 4πr

(^2) · dr dt When V = 4π,

4 π =^43 πr^3 =⇒ 3 = r^2 =⇒ r = 3

1 = 4π( 3

3)^2 ·

dr dt =⇒^

dr dt =^

4 π 32 /^3

≈. 038 f t/min

x^3

dy dx +^ y^ ·^3 x

(^2) + x (^2) · 2 y dy dx +^ y

(^2) · 2 x + x · 3 y 2 dy dx +^ y

=⇒ x^3 dy dx

• 2x^2 y dy dx

• 3xy^2 dy dx

= − 3 x^2 y − 2 xy^2 − y^3

=⇒ dy dx

(x^3 + 2x^2 y + 3xy^2 ) = − 3 x^2 y − 2 xy^2 − y^3

dy dx =^

− 3 x^2 y − 2 xy^2 − y^3 x^3 + 2x^2 y + 3xy^2

Plug in (1,1): dy dx (1,^ 1) =^

1 + 2 + 3 =^

6 =^ −^1

Equation of the tangent line:

y − 1 = −1(x − 1) =⇒ y = −x + 2