### Application of Derivatives Exercise 04

## Question:

If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating in surface area

## Answer:

Step 1: Find the surface area of the sphere using the given radius. Surface area = 4πr2

Step 2: Substitute the given radius in the formula and calculate the surface area. Surface area = 4π(9)2 = 4π(81) = 304π

Step 3: Find the approximate error in calculating the surface area. Approximate error = 0.03 x 4π = 0.12π m2

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (0.999)^1/10

## Answer:

Step 1: Calculate the value of (0.999)^1/10. Answer: 0.9512

Step 2: Use differentials to approximate the sum of digits in 0.9512. Answer: 9 + 5 + 1 + 2 = 17

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (0.0037)^1/2

## Answer:

Answer: Step 1: Calculate the square root of 0.0037.

Step 2: Use differentials to approximate the value of the square root up to 3 places of decimal.

Step 3: Add the digits of the approximate value to get the sum.

Answer: The sum of digits of the approximate value of (0.0037)^1/2 is 6.

## Question:

If the radius of a sphere is measured as 7m with an error of 0.02m, then find the approximate error in calculating its volume.

## Answer:

Step 1: Calculate the volume of the sphere.

Volume = (4/3)πr3

Substituting the radius as 7m,

Volume = (4/3) × 3.14 × (7)3

Volume = 1436.76 m3

Step 2: Calculate the approximate error in calculating the volume.

Error = (Error in radius) × (Derivative of volume with respect to radius)

Error = (0.02) × (4πr2)

Error = (0.02) × (4 × 3.14 × (7)2)

Error = 101.11 m3

Therefore, the approximate error in calculating the volume of the sphere is 101.11 m3.

## Question:

The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is : A 0.06x3m^3 B 0.6x3m^3 C 0.09x3m^3 D 0.9x3m^3

## Answer:

Step 1: Find the original volume of the cube. The volume of a cube is given by V = x3, where x is the side length of the cube.

Step 2: Find the new side length of the cube. The new side length of the cube will be increased by 3%. Therefore, the new side length will be 1.03x.

Step 3: Find the new volume of the cube. Using the formula for the volume of a cube, the new volume of the cube will be (1.03x)3.

Step 4: Find the approximate change in the volume of the cube. The approximate change in the volume of the cube is given by the difference between the new volume and the original volume. Therefore, the approximate change in the volume of the cube is (1.03x)3 - x3 = 0.09x3m^3.

Answer: C 0.09x3m^3

## Question:

Using differentials, find the approximate value of each of the following up to 3 places of decimal. (i) √25.3 (ii) √49.5 (iii) √0.6 (iv) (0.009)^1/3 (v) (0.999)^1/10 (vi) (15)^1/4 (vii) (26)^1/3 (viii) (255)^1/4 (ix) (82)^1/4 (x) (401)^1/2 (xi) (0.0037)^1/2

## Answer:

(i) √25.3 ≈ 5.04 (ii) √49.5 ≈ 7.02 (iii) √0.6 ≈ 0.78 (iv) (0.009)^1/3 ≈ 0.10 (v) (0.999)^1/10 ≈ 0.95 (vi) (15)^1/4 ≈ 2.25 (vii) (26)^1/3 ≈ 2.96 (viii) (255)^1/4 ≈ 8.05 (ix) (82)^1/4 ≈ 2.86 (x) (401)^1/2 ≈ 20.05 (xi) (0.0037)^1/2 ≈ 0.06

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (401)^1/2

## Answer:

Answer: Step 1: Calculate the exact value of (401)^1/2

Exact Value = 20.04987562112089

Step 2: Calculate the differentials of (401)^1/2

Differentials = 0.04987562112089

Step 3: Add the differentials to the exact value to get the approximate value

Approximate Value = 20.09975124223178

Step 4: Round the approximate value to 3 decimal places

Rounded Approximate Value = 20.100

## Question:

If f(x)=3x^2+15x+5, then the approximate value of f(3.02) is. A 47.66 B 57.66 C 67.66 D 77.66

## Answer:

Step 1: Substitute x = 3.02 into the function f(x)

f(3.02) = 3(3.02)^2 + 15(3.02) + 5

Step 2: Simplify the equation

f(3.02) = 28.9608 + 45.3 + 5

Step 3: Calculate the value of f(3.02)

f(3.02) = 79.2608

Step 4: Choose the correct answer

Answer: D 77.66

## Question:

Find the approximate change in the volume V of a cube of side x metres caused by increasing side by 1% percent.

## Answer:

Step 1: Calculate the initial volume of the cube: V = x3

Step 2: Calculate the new side length of the cube after the 1% increase: x’ = x + 0.01x

Step 3: Calculate the new volume of the cube: V’ = (x + 0.01x)3

Step 4: Calculate the approximate change in volume: ΔV = V’ - V = (x + 0.01x)3 - x3

Step 5: Simplify the expression to get the final answer: ΔV = 3x2(0.01x)

## Question:

Using differentials, find the approximate value of the following up to 3 decimal places. √49.5

## Answer:

Answer: Step 1: Calculate the square root of 49.5 using a calculator: √49.5 = 7.04

Step 2: Use differentials to approximate the value of the square root: 7.04 + 0.01 = 7.05

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (82)^1/4

## Answer:

Answer:

Step 1: Calculate the exact value of (82)^1/4

(82)^1/4 = 2.82843

Step 2: Use differentials to approximate the value of (82)^1/4

Differentials = (82 + Δ)^1/4 - (82)^1/4

Differentials = 2.83 - 2.82843 = 0.00157

Step 3: Round the value of the differentials to 3 places of decimal

Differentials = 0.002

Step 4: Add the digits of the differentials

0 + 0 + 2 = 2

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (0.999)^1/3

## Answer:

Answer: Step 1: Calculate the derivative of (0.999)^1/3 Derivative of (0.999)^1/3 = 0.333(0.999)^-2/3

Step 2: Calculate the differential of (0.999)^1/3 Differential of (0.999)^1/3 = 0.333(0.999)^-2/3 * 0.001

Step 3: Calculate the sum of digits approximate value of (0.999)^1/3 Sum of digits approximate value = (0.999)^1/3 + 0.333(0.999)^-2/3 * 0.001 = 0.999 + 0.000333 = 0.999333 (rounded to 3 decimal places)

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (255)^1/4

## Answer:

Answer:

Step 1: Calculate the exact value of (255)^1/4

Exact value = 5

Step 2: Calculate the differentials of (255)^1/4

Differentials = (1/4) * (255)^(-3/4)

Step 3: Calculate the approximate value of (255)^1/4

Approximate value = 5.000

Step 4: Calculate the sum of digits of the approximate value

Sum of digits = 5 + 0 + 0 + 0 = 5

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (26)^1/3

## Answer:

Answer:

Step 1: Calculate the value of (26)^1/3

(26)^1/3 = 3.717

Step 2: Use differentials to approximate the sum of the digits of 3.717

Differential = 0.001

3.717 + 0.001 = 3.718

Step 3: Calculate the sum of the digits of 3.718

Sum of the digits = 3 + 7 + 1 + 8 = 19

Step 4: Round the result to 3 places of decimal

19.000

## Question:

Find the approximate value of f(5.001), where f(x)=x^3−7x^2+15.

## Answer:

f(5.001) = (5.001)^3 - 7(5.001)^2 + 15

f(5.001) ≈ 141.926

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (15)^1/4

## Answer:

Answer:

Step 1: Calculate the value of (15)^1/4

(15)^1/4 = 2.902

Step 2: Use differentials to approximate the sum of the digits of the value

The sum of the digits of the value is 11 (2 + 9 + 0 + 2).

Step 3: Round the sum of the digits to 3 decimal places

The sum of the digits rounded to 3 decimal places is 11.000.

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. √0.6

## Answer:

Step 1: Find the square root of 0.6.

Answer: 0.7745966692414834

Step 2: Find the sum of the digits of the square root.

Answer: 22 (0 + 7 + 7 + 4 + 5 + 9 + 6 + 6 + 6 + 9 + 2 + 4 + 1 + 4 + 8 + 3 + 4)

## Question:

Find the approximate value of f(2.01), where f(x)=4x^2+5x+2.

## Answer:

f(2.01) = 4(2.01)^2 + 5(2.01) + 2

f(2.01) = 16.0824 + 10.05 + 2

f(2.01) = 28.1324

## Question:

Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. (401)^1/2

## Answer:

Answer:

Step 1: Calculate the square root of 401 using a calculator.

The square root of 401 is 20.049875621.

Step 2: Use differentials to approximate the sum of digits up to 3 decimal places.

The approximate sum of digits is 20.049.

## Question:

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1 percent.

## Answer:

Step 1: Calculate the original surface area of the cube.

Surface area = 6x2 = 6x2xx = 6x2x2 = 24x2 = 48x

Step 2: Calculate the change in side length caused by decreasing the side by 1 percent.

Change in side length = 0.01x

Step 3: Calculate the new surface area of the cube.

Surface area = 6(x-0.01x)2 = 6(x2 - 0.02x2) = 6x2 - 0.12x2 = 24x2 - 0.12x2 = 48x2 - 0.12x2

Step 4: Calculate the approximate change in surface area.

Approximate change in surface area = 48x2 - (48x2 - 0.12x2) = 0.12x2