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Class 12 Application of Derivatives worksheet
CBSE Mathematics. Rates of change, increasing and decreasing, tangents and normals, maxima and minima. Every sheet has 20 questions in three sections and an answer key, and the numbers change every time.
Sample questions
Real questions from this chapter, as the app sets them. Tap Answer to see the answer and working.
Section A · multiple choice
- 1.
The radius of a circle grows at 3 cm/s. When the radius is 3 cm, its area grows at:
- 18π cm²/s
- 36π cm²/s
- 27π cm²/s
- 9π cm²/s
Answer
(a) 18π cm²/s
differentiate the formula with respect to t
- 2.
f(x) = 2x³ − 27x² + 120x + 6 is strictly increasing on:
- (4, 5)
- (−∞, −5) ∪ (−4, ∞)
- (−∞, 4) ∪ (5, ∞)
- (−∞, 5)
Answer
(c) (−∞, 4) ∪ (5, ∞)
f′(x) = 6(x − 4)(x − 5)
- 3.
The slope of the tangent to y = −x³ − 5x² + x − 9 at x = 3 is:
- −56
- −78
- 56
- −55
Answer
(a) −56
dy/dx = −3x² − 10x + 1
Section B · short answer
- 4.
Air is pumped into a spherical balloon at 25 cm³/s. How fast is the radius increasing when the radius is 15 cm?
Answer
1/(36π) cm/s
dV/dt = 4πr² dr/dt ⇒ dr/dt = 25/(4π × 225)
- 5.
Find the intervals on which f(x) = 2x³ − 24x² + 90x + 7 is strictly increasing and strictly decreasing.
Answer
Increasing on (−∞, 3) ∪ (5, ∞); decreasing on (3, 5)
f′(x) = 6(x − 3)(x − 5)
Section C · long answer
- 6.
An open box is made from a square sheet of cardboard of side 78 cm by cutting equal squares from each corner and folding up the sides. Find the side of the square to be cut off for the box to have the greatest volume, and that volume.
Answer
13 cm; 35,152 cm³
V = x(78 − 2x)²; V′ = 0 at x = 13 (x = 39 gives no box); V″ < 0 there
Fresh sheets for your child
In the app, every Application of Derivatives sheet is new: the app remembers the questions your child has already had and leaves them out. Print the sheet as a PDF, or use notebook mode and write the answers in a notebook.
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