Differential Calculus — Limits, Derivatives & Applications — Practice

Question 1

The value of lim(x→0) [sin 4x / 2x] is:
(a) 1  (b) 2  (c) 4  (d) 1/2

Accepted Answer

sin 4x / 2x = (4/2) · [sin 4x / 4x] → 2 × 1 = 2, using lim(x→0) sin kx/kx = 1.

Answer: (b) 2. Trap (c) 4 comes from applying sin kx/x → k to the numerator's coefficient while ignoring the denominator's own coefficient of 2.

Question 2

lim(x→3) [(x² − 9) / (x − 3)] equals:
(a) 0  (b) 3  (c) 6  (d) 9

Accepted Answer

Factor: (x−3)(x+3)/(x−3) = x+3 for x ≠ 3. As x → 3, this → 6.

Answer: (c) 6. Trap (b) 3 comes from invalidly cancelling x² with x directly instead of factoring the difference of squares.

Question 3

f(x) = (x² − 1)/(x − 1) for x ≠ 1, and f(1) = k. For f to be continuous at x = 1, k equals:
(a) 0  (b) 1  (c) 2  (d) No such k exists

Accepted Answer

lim(x→1) (x−1)(x+1)/(x−1) = lim(x→1) (x+1) = 2.

For continuity, f(1) must equal this limit, so k = 2.

Answer: (c) 2.

Question 4

Using the first-principles definition, the derivative of f(x) = 1/x is:
(a) 1/x²  (b) −1/x²  (c) −1/x  (d) 1/x

Accepted Answer

f'(x) = lim(h→0) [1/(x+h) − 1/x] / h = lim(h→0) [x − (x+h)] / [h·x(x+h)] = lim(h→0) −h / [h·x(x+h)] = lim(h→0) −1/[x(x+h)] = −1/x².

Answer: (b) −1/x². Trap (a) drops the negative sign that comes from x − (x+h) = −h.

Question 5

If y = x³ cos x, then dy/dx equals:
(a) 3x² cos x + x³ sin x  (b) 3x² cos x − x³ sin x  (c) x³ sin x − 3x² cos x  (d) 3x² sin x − x³ cos x

Accepted Answer

u = x³, u' = 3x²; v = cos x, v' = −sin x.

dy/dx = u'v + uv' = 3x² cos x + x³(−sin x) = 3x² cos x − x³ sin x.

Answer: (b) 3x² cos x − x³ sin x. Trap (a) uses v' = +sin x, forgetting the negative sign in d/dx(cos x).

Question 6

If y = x² / (x + 1), the value of dy/dx at x = 1 is:
(a) 1/4  (b) 3/4  (c) 1  (d) 3/2

Accepted Answer

dy/dx = [2x(x+1) − x²(1)] / (x+1)² = (x² + 2x) / (x+1)².

At x = 1: (1 + 2)/4 = 3/4.

Answer: (b) 3/4. Trap (c) 1 comes from forgetting the −x²(1) term in the numerator; trap (d) 3/2 comes from forgetting to square the denominator.

Question 7

If y = sin(2x² + 1), then dy/dx equals:
(a) cos(2x² + 1)  (b) 4x cos(2x² + 1)  (c) 2x cos(2x² + 1)  (d) 4x sin(2x² + 1)

Accepted Answer

Outer derivative: cos(2x²+1). Inner derivative: d/dx(2x²+1) = 4x.

dy/dx = 4x cos(2x²+1).

Answer: (b) 4x cos(2x² + 1). Trap (a) omits the chain-rule multiplier entirely; trap (c) uses inner derivative 2x, dropping the coefficient 2 on x².

Question 8

For y = x⁴ − 6x², the value of d²y/dx² at x = 1 is:
(a) −12  (b) 0  (c) 12  (d) 24

Accepted Answer

y' = 4x³ − 12x. y'' = 12x² − 12.

At x = 1: 12(1) − 12 = 0.

Answer: (b) 0. Trap (c) 12 comes from evaluating only the 12x² term and forgetting to subtract 12.

Question 9

The equation of the tangent to the curve y = x² − 4x + 3 at the point where x = 3 is:
(a) y = 2x − 6  (b) y = 2x + 6  (c) y = −2x + 6  (d) y = 2x − 3

Accepted Answer

y(3) = 9 − 12 + 3 = 0, so the point is (3, 0).

y' = 2x − 4, so at x = 3: slope m = 2.

Tangent: y − 0 = 2(x − 3) ⇒ y = 2x − 6.

Answer: (a) y = 2x − 6. Trap (c) comes from a sign error giving slope −2 instead of 2.

Question 10

The radius of a circular ripple on a pond is increasing at 4 cm/s. The rate at which the area is increasing when the radius is 6 cm is:
(a) 24π cm²/s  (b) 48π cm²/s  (c) 96π cm²/s  (d) 12π cm²/s

Accepted Answer

A = πr² ⇒ dA/dt = 2πr · dr/dt = 2π(6)(4) = 48π cm²/s.

Answer: (b) 48π cm²/s. Trap (a) drops the factor of 2 from differentiating r²; trap (c) comes from using the diameter (12 cm) in place of the radius.

Question 11

The local maximum value of f(x) = −x² + 4x + 1 is:
(a) 3  (b) 4  (c) 5  (d) 6

Accepted Answer

f'(x) = −2x + 4 = 0 ⇒ x = 2. f''(x) = −2 < 0 ⇒ local maximum at x = 2.

f(2) = −4 + 8 + 1 = 5.

Answer: (c) 5. Trap (b) 4 comes from dropping the constant term (+1) when evaluating f(2).

Practice — Differential Calculus — Limits, Derivatives & Applications

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities