Application of Derivatives – important questions for Boards 2027
CBSE Class 12 Maths
For y = −2x + 5, calculate dy/dx and determine whether y increases or decreases as x increases.
Answer
dy/dx = −2 < 0; therefore, y decreases as x increases.
Which condition on f′(x) guarantees that a differentiable function f is strictly increasing on an interval (a, b)? (A) f′(x) < 0 for every x ∈ (a, b) (B) f′(x) > 0 for every x ∈ (a, b) (C) f′(x) = 0 for every x ∈ (a, b) (D) f(x) > 0 for every x ∈ (a, b)
Answer
(B) f′(x) > 0 for every x ∈ (a, b) — a positive derivative guarantees that f is strictly increasing on the interval.
For y = x² + 3x, calculate the rate of change of y with respect to x at x = 2.
Answer
y = x² + 3x dy/dx = 2x + 3 At x = 2, dy/dx = 2(2) + 3 ∴ dy/dx = 7
If y = x³ − 2x, find the rate of change of y with respect to x at x = 2.
Answer
y = x³ − 2x dy/dx = 3x² − 2 At x = 2, dy/dx = 3(2)² − 2 = 12 − 2 ∴ dy/dx = 10
The radius of a circle is increasing at 3 cm/s. Find the rate at which its area is increasing when the radius is 4 cm.
Answer
A = πr² dA/dt = 2πr(dr/dt) At r = 4 cm and dr/dt = 3 cm/s, dA/dt = 2π(4)(3) ∴ dA/dt = 24π cm²/s
Find the intervals in which f(x) = 4x³ − 6x² − 72x + 30 is increasing and the intervals in which it is decreasing.
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Suppose x = t² + 1 and y = t³ + 2t. Find dx/dt and dy/dt, and hence calculate dy/dx when t = 1. Verify that the result agrees with dy/dx = (dy/dt)/(dx/dt).
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Show that f(x) = cos x is neither increasing nor decreasing on (0, 2π).
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On a given interval, the absolute maximum and absolute minimum values of a function are, respectively: (A) its least and greatest values (B) its greatest and least values (C) its values at the endpoints only (D) its local maximum and local minimum values
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The total cost of producing x items is C(x) = 2x² + 5x + 100. Find the marginal cost when 3 items are produced.
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63 more probable questions in this chapter, every one with a full answer.
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