Chapter Exam

Chapter 6: Applications of Derivatives — Online MCQ Test

Q: If both x and y vary with respect to time t, then the relationship between dy/dx, dy/dt, and dx/dt is:

Answer: B. dy/dx = (dy/dt) ÷ (dx/dt), where dx/dt ≠ 0 — By the chain rule, when two variables vary with respect to a third variable, dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Q: A water tank is being filled such that the radius increases at 2 cm/s. The rate of change of volume when r = 5 cm is:

Answer: C. 200π cm³/s — Using dV/dt = 4πr²(dr/dt) = 4π(5)²(2) = 4π(25)(2) = 200π cm³/s.

Q: A cylindrical can has volume V cm³. The material cost for the lateral surface area is Rs. 10/cm² and for the base is Rs. 20/cm². The total cost function is minimized when:

Answer: C. h = r/2 — Cost = 20(2πr²) + 10(2πrh) = 40πr² + 20πrh. With constraint πr²h = V, substitute h = V/(πr²) and minimize with respect to r to find h = r/2.

MATHS · CLASS 12 INTER II YEAR · Andhra State Board

Practice Chapter 6: Applications of Derivatives with a free chapter-wise online MCQ test for Andhra State Board CLASS 12 INTER II YEAR MATHS. This chapter covers: 1. Rate of Change of QuantitiesCore Concept: The derivative \(\frac{dy}{dx}\) represents the rate of change of \(y\) with respect to \(x\).Chain Rule Application: If two variables.... AI-generated questions from basic to board-exam level, with instant results and explanations.

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Chapter 6: Applications of Derivatives — Important Questions & Answers (FAQ)

Frequently asked questions from Andhra State Board CLASS 12 INTER II YEAR MATHS — Chapter 6: Applications of Derivatives, with answers and explanations. These are sample questions; the exam has its own separate question set.

The derivative dy/dx represents which of the following?

A. The instantaneous rate of change of y with respect to x ✓
B. The average value of y over an interval
C. The total change in y
D. The slope of the function at infinity

Answer: A. The instantaneous rate of change of y with respect to x
The derivative dy/dx is defined as the instantaneous rate of change of y with respect to x at any given point.

If both x and y vary with respect to time t, then the relationship between dy/dx, dy/dt, and dx/dt is:

A. dy/dx = (dy/dt) × (dx/dt)
B. dy/dx = (dy/dt) ÷ (dx/dt), where dx/dt ≠ 0 ✓
C. dy/dx = (dx/dt) ÷ (dy/dt)
D. dy/dx = (dy/dt) + (dx/dt)

Answer: B. dy/dx = (dy/dt) ÷ (dx/dt), where dx/dt ≠ 0
By the chain rule, when two variables vary with respect to a third variable, dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

A water tank is being filled such that the radius increases at 2 cm/s. The rate of change of volume when r = 5 cm is:

A. 20π cm³/s
B. 100π cm³/s
C. 200π cm³/s ✓
D. 50π cm³/s

Answer: C. 200π cm³/s
Using dV/dt = 4πr²(dr/dt) = 4π(5)²(2) = 4π(25)(2) = 200π cm³/s.

A cylindrical can has volume V cm³. The material cost for the lateral surface area is Rs. 10/cm² and for the base is Rs. 20/cm². The total cost function is minimized when:

A. h = 2r
B. h = r
C. h = r/2 ✓
D. h = 4r

Answer: C. h = r/2
Cost = 20(2πr²) + 10(2πrh) = 40πr² + 20πrh. With constraint πr²h = V, substitute h = V/(πr²) and minimize with respect to r to find h = r/2.

The function f(x) = |x - 2| is NOT differentiable at x = 2 because:

A. The function is not continuous at x = 2
B. The left and right derivatives at x = 2 are not equal ✓
C. The function value at x = 2 is undefined
D. The function is decreasing on both sides of x = 2

Answer: B. The left and right derivatives at x = 2 are not equal
At x = 2, the left derivative is -1 and the right derivative is +1. Since these are not equal, the function is not differentiable at x = 2, making it a critical point.