Applications of Derivatives Test

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Applications of Derivatives Test - 4

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Weekly Quiz Competition

Question 1
1 / -0.25

The radius of air bubble is increasing at the rate of 0. 25 cm/s. At what rate the volume of the bubble is increasing when the radius is 1 cm.

A
4 πcm³ /s

B
22 πcm³ /s

C
2 πcm³ /s

D
πcm³ /s
Question 2
1 / -0.25

The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 5x² + 22x + 35^. Find the marginal revenue, when x = 7, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant

A
Rs 7

B
Rs 127

C
Rs 92

D
Rs 48

Solution
Question 3
1 / -0.25

Find the approximate change in total surface area of a cube of side x metre caused by increase in side by 1%.

A
12 m²

B
0.12x² m²

C
1.2x m²

D
12x m²
Question 4
1 / -0.25

The volume of cube is increasing at the constant rate of 3 cm³ /s. Find the rate of change of edge of the cube when its edge is 5 cm.

A
25 cm³ /sec

B
25 cm/s

C
1/25 cm/s

D
1/25 cm³ /s

Solution

Let V be the instantaneous volume of the cube.
dV/dt = 3 cm³ /s
Let x be the side of the cube.
V = x³
dV/dt = 3x² * (dx/dt)
3 = 3*(5² )*(dx/dt)
So, dx/dt = 1/25 cm/s
Question 5
1 / -0.25

The total cost associated with the production of x units of a product is given by C(x) = 5x² + 14x + 6. Find marginal cost when 5 units are produced

A
Rs.64

B
Rs.70

C
Rs.50

D
Rs.(10x + 14)

Solution

C(x) = 5x² + 14x + 6
Marginal Cost M(x) = C ’(x)
M(x) = 10x +14
So, M(5) = 50 + 14
= 64 Rs
Question 6
1 / -0.25

At a distance of 120ft from the foot of a tower the elevation of its top is 60o. If the possible error in measuring the distance and elevation are 1 inch and 1 minute, find approximate error in calculated height.

A
0.342

B
0.284

C
0.154

D
0.986

Solution

Tan(z) = ^h ⁄ _x
Question 7
1 / -0.25

A stone is dropped into a quiet lake and waves move in circles at a speed of 2cm per second. At the instant, when the radius of the circular wave is 12 cm, how fast is the enclosed area changing ?

A
Decreasing at the rate of 48 πcm² / sec

B
Increasing at the rate of 24 πcm² / sec

C
Increasing at the rate of 48 πcm² / sec

D
Decreasing at the rate of 24 πcm² / sec

Solution

Rate of increase of radius dr/dt = 2 cm/s
Area of circle A = πr²
dA/dt = π*(2r)*(dr/dt)
= π*(24)*2
= 48 πcm² /s
Rate of increase of area is 48 πcm² /s (increasing as it is positive).
Question 8
1 / -0.25

Using approximation find the value of

A
2.025

B
2.001

C
2.01

D
2.0025

Solution

Let x=4, Δx=0.01
y=x^½= 2
y+Δy = (x+ Δx)^½= (4.01)^½
Δy = (dy/dx) * Δx
Δy = (x^(-1/2))/2 * Δx
Δy = (½)*(½) * 0.01
Δy = 0.25 * 0.01
Δy = 0.0025
So, (4.01)^½= 2 + 0.0025 = 2.0025
Question 9
1 / -0.25

Find the approximate value of f(10.01) where f(x) = 5x² +6x + 3

A
564.06

B
564.01

C
563.00

D
563.01

Solution

f(x) = 5x² +6x + 3
f(10.01) = 5*(10.01)² + 6*(10.01) + 3
To find (10.01)²
Let p=10, Δp=0.01
y=p2 = 100
y+Δy = (p+ Δp)² = (10.01)²
Δy = (dy/dp) * Δp
Δy = 2*p* Δx
Δy = 2*10* 0.01
Δy = 20 * 0.01
Δy = 0.2
So, (10.01)² = y + Δy
= 100.2
So,
f(10.01) = 5*(100.2) + 6*(10.01) + 3
= 501 + 60.06 + 3
= 564.06
Question 10
1 / -0.25

Given a function y = f(x) . Let Δx be the very small change in the value of x , then the corresponding change in the value of y that is Δy is approximately given by

A

B

C

D
F`(x)

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Re-Attempt Weekly Quiz Competition

Applications of Derivatives Test - 4

Result

Applications of Derivatives Test - 4

Score

-

Rank

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SHARING IS CARING

Question 1

4 πcm3 /s

22 πcm3 /s

2 πcm3 /s

πcm3 /s

Question 2

Rs 7

Rs 127

Rs 92

Rs 48

Solution

Question 3

12 m2

0.12x2 m2

1.2x m2

12x m2

Question 4

25 cm3 /sec

25 cm/s

1/25 cm/s

1/25 cm3 /s

Solution

Question 5

Rs.64

Rs.70

Rs.50

Rs.(10x + 14)

Solution

Question 6

0.342

0.284

0.154

0.986

Solution

Question 7

Decreasing at the rate of 48 πcm2 / sec

Increasing at the rate of 24 πcm2 / sec

Increasing at the rate of 48 πcm2 / sec

Decreasing at the rate of 24 πcm2 / sec

Solution

Question 8

2.025

2.001

2.01

2.0025

Solution

Question 9

564.06

564.01

563.00

563.01

Solution

Question 10

F`(x)

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4 πcm³ /s

22 πcm³ /s

2 πcm³ /s

πcm³ /s

12 m²

0.12x² m²

1.2x m²

12x m²

25 cm³ /sec

1/25 cm³ /s

Decreasing at the rate of 48 πcm² / sec

Increasing at the rate of 24 πcm² / sec

Increasing at the rate of 48 πcm² / sec

Decreasing at the rate of 24 πcm² / sec