Calculus Study Materials for Competitive Exam Prep

Calculus

Application of Derivatives (AOD) Area Under Curves Continuity Definite Integration Derivatives (Basics) Differentiability Differential Equations Indefinite Integration Limits

Q. What is the solution to the equation y' = y(1 - y)?

A. y = 1/(C - x)
B. y = C/(C + x)
C. y = C/(1 + Cx)
D. y = C/(1 - Cx)

Solution

This is a separable equation. Integrating gives y = C/(C + x).

Correct Answer: B — y = C/(C + x)

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Q. What is the solution to the first-order linear differential equation dy/dx + y = e^x?

A. y = e^x + Ce^(-x)
B. y = e^x - Ce^(-x)
C. y = Ce^x - e^x
D. y = Ce^(-x) + e^x

Solution

Using the integrating factor e^x, we solve to get y = e^x + Ce^(-x).

Correct Answer: A — y = e^x + Ce^(-x)

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Q. What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4, x >= 2 is continuous at x = 2?

A. 0
B. 1
C. 2
D. 3

Solution

Setting 2k + 2 = 0 gives k = 2.

Correct Answer: C — 2

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Q. What is the value of k for which the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0?

A. -1
B. 0
C. 1
D. 2

Solution

Setting k(0) = 0^2 + 1 gives k = 1.

Correct Answer: B — 0

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Q. What is the value of k for which the function f(x) = { kx, x < 2; x^2, x >= 2 } is continuous at x = 2?

A. 0
B. 2
C. 4
D. 1

Solution

Setting k(2) = 2^2 gives 2k = 4, thus k = 2.

Correct Answer: C — 4

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Q. What is the value of p for which the function f(x) = { 3x + p, x < 2; x^2 - 4, x >= 2 } is continuous at x = 2?

A. -1
B. 0
C. 1
D. 2

Solution

Setting the two pieces equal at x = 2: 3(2) + p = 2^2 - 4. Solving gives p = -2.

Correct Answer: A — -1

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Q. What is the value of q for which the function f(x) = { 5 - q, x < 1; 3x + 2, x >= 1 } is continuous at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting the two pieces equal at x = 1: 5 - q = 3(1) + 2. Solving gives q = 0.

Correct Answer: C — 2

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Q. What is the value of the derivative of f(x) = ln(x^2 + 1) at x = 1?

A. 0
B. 1/2
C. 1
D. 1/3

Solution

f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = 2/(1 + 1) = 1.

Correct Answer: B — 1/2

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Q. What is the value of the integral ∫(0 to 1) (3x^2 + 2)dx?

A. 5/3
B. 2
C. 1
D. 4/3

Solution

Integral = [x^3 + 2x] from 0 to 1 = (1 + 2) - (0) = 3.

Correct Answer: A — 5/3

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Q. What is the value of the integral ∫(0 to 1) (3x^2)dx?

A. 1
B. 1/3
C. 1/2
D. 3/4

Solution

The integral evaluates to [x^3] from 0 to 1 = 1^3 - 0^3 = 1.

Correct Answer: A — 1

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Q. What is the value of the integral ∫(1 to 2) (3x^2 - 2)dx?

A. 3
B. 4
C. 5
D. 6

Solution

The integral evaluates to [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.

Correct Answer: B — 4

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Q. What is the value of the integral ∫(1 to 2) (x^2 + 2x)dx?

A. 7
B. 6
C. 5
D. 8

Solution

The integral evaluates to 7.

Correct Answer: A — 7

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Q. What is the value of the integral ∫(1/(x^2 + 1))dx?

A. tan^-1(x) + C
B. sin^-1(x) + C
C. cos^-1(x) + C
D. ln(x) + C

Solution

The integral evaluates to tan^-1(x) + C.

Correct Answer: A — tan^-1(x) + C

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Q. What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?

A. 1
B. 2
C. 3
D. 4

Solution

Evaluating the integral gives [x^3 + x^2]_0^1 = (1 + 1) - (0 + 0) = 2.

Correct Answer: B — 2

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Q. What is the value of the limit lim (x -> 1) (x^2 - 1)/(x - 1)?

A. 0
B. 1
C. 2
D. Infinity

Solution

Using L'Hôpital's Rule, the limit evaluates to 2.

Correct Answer: C — 2

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Q. What is the value of the limit: lim (x -> ∞) (1/x)?

A. 0
B. 1
C. Infinity
D. Undefined

Solution

As x approaches infinity, 1/x approaches 0.

Correct Answer: A — 0

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Q. What value of a makes the function f(x) = { 2x + 1, x < 1; a, x = 1; x^2 + 1, x > 1 continuous at x = 1?

A. 1
B. 2
C. 3
D. 4

Solution

Setting 2(1) + 1 = a and a = 2 for continuity.

Correct Answer: B — 2

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Q. What value of a makes the function f(x) = { 2x + a, x < 3; 5, x = 3; x^2 - 1, x > 3 continuous at x = 3?

A. 0
B. 1
C. 2
D. 3

Solution

Setting 2(3) + a = 5 gives a = -1.

Correct Answer: C — 2

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Q. What value of a makes the function f(x) = { 4 - x^2, x < 0; ax + 2, x = 0; x + 1, x > 0 continuous at x = 0?

A. -1
B. 0
C. 1
D. 2

Solution

Setting 4 = 2 gives a = 1 for continuity.

Correct Answer: B — 0

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Q. What value of k makes the function f(x) = { kx, x < 1; 2, x = 1; x + 1, x > 1 continuous at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting the left limit (k(1) = k) equal to the right limit (1 + 1 = 2), we find k = 2.

Correct Answer: B — 1

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Q. What value of m makes the function f(x) = { 3x + 1, x < 2; mx + 4, x = 2; x^2 - 1, x > 2 continuous at x = 2?

A. 1
B. 2
C. 3
D. 4

Solution

Setting the left limit (3(2) + 1 = 7) equal to the right limit (2^2 - 1 = 3), we find m = 3.

Correct Answer: D — 4

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Q. Which of the following functions is continuous at x = 2?

A. f(x) = 1/x
B. f(x) = x^2 - 4
C. f(x) = sin(1/x)
D. f(x) =
. x
.

Solution

f(x) = x^2 - 4 is a polynomial function and is continuous everywhere, including at x = 2.

Correct Answer: B — f(x) = x^2 - 4

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Q. Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }

A. Continuous
B. Not continuous
C. Depends on k
D. None of the above

Solution

At x = 2, f(2) = 0 and limit from left is 0, limit from right is also 0. Hence, it is continuous.

Correct Answer: A — Continuous

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Q. Which of the following functions is continuous at x = 2? f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }

A. Continuous
B. Not continuous
C. Depends on k
D. None of the above

Solution

To check continuity at x = 2, we find the left limit (4), right limit (4), and f(2) (4). All are equal, so f(x) is continuous.

Correct Answer: A — Continuous

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Q. Which of the following functions is continuous everywhere?

A. f(x) = 1/x
B. f(x) = x^2
C. f(x) = sin(x)
D. f(x) =
. x
.

Solution

f(x) = x^2 is a polynomial function and is continuous everywhere.

Correct Answer: B — f(x) = x^2

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Q. Which of the following functions is differentiable at x = 1? f(x) = { x^2, x < 1; 2x - 1, x >= 1 }

A. f(1) = 1
B. f(1) = 0
C. f(1) = 2
D. f(1) = 3

Solution

Check continuity and differentiability at x = 1 by equating left and right derivatives.

Correct Answer: A — f(1) = 1

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Q. Which of the following functions is differentiable everywhere?

A. f(x) =
B. x
C.
D. f(x) = x^2
. f(x) = sqrt(x)
. f(x) = 1/x

Solution

f(x) = x^2 is a polynomial and differentiable everywhere.

Correct Answer: B — x

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Q. Which of the following functions is not continuous at x = 0?

A. f(x) = x^3
B. f(x) = e^x
C. f(x) = 1/x
D. f(x) = ln(x)

Solution

The function f(x) = 1/x is not defined at x = 0, hence it is not continuous there.

Correct Answer: C — f(x) = 1/x

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Q. Which of the following functions is not continuous at x = 1?

A. f(x) = x^2
B. f(x) = 1/x
C. f(x) = sin(x)
D. f(x) = { x, x < 1; 2, x >= 1 }

Solution

The function has a jump discontinuity at x = 1, hence it is not continuous.

Correct Answer: D — f(x) = { x, x < 1; 2, x >= 1 }

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Q. Which of the following functions is not differentiable at x = 0? f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0.

A. f(x)
B. g(x) =
C. x
D.
. h(x) = x^3
. k(x) = x^2

Solution

The function f(x) is not differentiable at x = 0 due to the oscillatory nature of sin(1/x) as x approaches 0.

Correct Answer: A — f(x)

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Calculus

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