Differential Calculus MCQ Quiz - Objective Question with Answer for Differential Calculus - Download Free PDF

Concept:

•   Problems where two or more quantities change with respect to time.
• The formula is , where is volume, is radius, and is height.
• Given , the radius and height are proportional, so express in terms of .
• Differentiate with respect to time to find how fast the height changes as the volume increases.

Calculation:

Given,

Height and radius relation: so

Volume of cone:

⇒ Substitute :

⇒

⇒

Differentiate both sides w.r.t :

Substitute and :

⇒

⇒

⇒

⇒

The height increases at a rate of . 

⇒ 96πh = 2

Hence 2 is the correct answer. 

Calculation:

Given,

and .

Differentiate both sides with respect to implicitly:

Left side:

Right side (product rule):

Rearrange to collect terms and use

.

After cancellation of the common factor , you obtain:

∴ 

Hence, the correct answer is Option 1. 

Calculation:

Given,

Differentiate implicitly w.r.t. :

Rearrange to collect :

Use to simplify:

∴ , independent of and .

Hence, the correct answer is Option 4.

Given:  f(x) = (x - 4) (x - 5)

Expanding the given equation 

⇒  f(x) = x² - 9x + 20

Differentiating both sides, we have:

f'(x) = 2x -9

Substituting the value of x = 5, we have

f'(5) = 2 x 5 - 9 = 1

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Product Rule: 
• Division rule: 

Formulas:

Calculation:

= 

= 

= 

⇒ 

⇒

= 

Concept:

The equation of the tangent to a curve y = f(x) at a point (a, b) is given by (y - b) = m(x - a), where m = y'(b) = f'(a) [value of the derivative at point (a, b)].

Calculation:

⇒ y' = f'(x) = 3x²

m = f'(1) = 3 × 1² = 3.

(y - b) = m(x - a)

⇒ (y - 1) = 3(x - 1)

⇒ y - 1 = 3x - 3

Calculations:

Given, 

Differentiating with respect to x, we get

⇒ f'(x) = 1 - 

⇒ 1 + 

Put x = -1

⇒ f'(-1) = 1 +  = 1 + 1 = 2

∴ f'(-1) = 2

Concept:

Following steps to finding minima using derivatives.

• Find the derivative of the function.
• Set the derivative equal to 0 and solve. This gives the values of the maximum and minimum points.
• Now we have to find the second derivative: If f"(x) Is greater than 0 then the function is said to be minima

Calculation:

f(x) = x2 - x + 2

f'(x) = 2x - 1

Set the derivative equal to 0, we get

f'(x) = 2x - 1 = 0

⇒ x = 

Now, f''(x) = 2 > 0

So, we get minimum value at x =

f() = ()² - + 2 =

Hence, option (3) is correct. 

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Chain Rule: 
• Product Rule: 

Calculation:

y = x^x

Taking log both sides, we get

⇒ log y = log x^x                          (∵ log mⁿ = n log m)

⇒ log y = x log x

Differentiating with respect to x, we get

put x = 1

                (∵ log 1 = 0)

Concept:

Rate of change of 'x' is given by

Calculation:

Given that, y = 2x – x² and = 3 units/sec

Then, the slope of the curve, = 2 - 2x = m

⇒  = 0 - 2 × 

= -2(3)

= -6 units per second

Hence, the slope of the curve is decreasing at the rate of 6 units per second when x is increasing at the rate of 3 units per second.

Hence, option (2) is correct.

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Chain Rule: 

Calculation:

Given:

y = sin x°

We know that,

180° = π radian

∴ 1° =  radian

Now, x° =  radian

⇒ y = 

Differenatiate with respect to x, we get

Calculation:

Given: x = t² , y = t³.

⇒  and 

Again differentiating with respect to x:

⇒ 

⇒   (∵ )

∴  

The correct answer is .

Concept:

Chain Rule of Derivatives: 

.

 = e^x.

Calculation:

It is given that y = .

∴ y = 

Differentiating both sides with respect to x and using the chain rule, we get:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ .

Concept:

Calculation:

Given:  y = e^{log (log x)}

To Find: 

As we know that, e^{log x} = x

∴ e^{log (log x)} = log x

Now, y = log x

Differentiating with respect to x, we get