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. Determine the value of n for which the function f(x) = { n^2 - 1, x < 0; 2x + 3, x >= 0 } is continuous at x = 0.

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

Solution

Setting the two pieces equal at x = 0: n^2 - 1 = 3. Solving gives n = ±2.

Correct Answer: A — 1

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Q. Determine the value of p for which the function f(x) = { 2x + 3, x < 2; px + 1, x = 2; x^2 - 1, x > 2 is continuous at x = 2.

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

Solution

Setting 2(2) + 3 = p(2) + 1 gives p = 3 for continuity.

Correct Answer: C — 3

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Q. Determine the value of p for which the function f(x) = { 3x - 1, x < 2; px + 4, x = 2; x^2 - 2, x > 2 is continuous at x = 2.

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

Solution

Setting 3(2) - 1 = p(2) + 4 gives p = 2.

Correct Answer: C — 3

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

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

Solution

Setting p = 1 for continuity at x = 0 gives f(0) = 1.

Correct Answer: B — 0

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Q. Determine the value of p for which the function f(x) = { x^2 - 1, x < 1; p, x = 1; 2x + 1, x > 1 is continuous at x = 1.

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

Solution

Setting the left limit (1 - 1 = 0) equal to the right limit (2(1) + 1 = 3), we find p = 2.

Correct Answer: C — 2

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Q. Determine the value of p for which the function f(x) = { x^3 - 3x + p, x < 1; 2x + 1, x >= 1 is continuous at x = 1.

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

Solution

Setting -3 + p = 3 gives p = 0.

Correct Answer: A — -1

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Q. Determine the value of p for which the function f(x) = { x^3 - 3x + p, x < 1; 2x^2 + 1, x >= 1 is continuous at x = 1.

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

Solution

Setting 1 - 3 + p = 2 + 1 gives p = 4.

Correct Answer: A — -1

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Q. Evaluate the definite integral ∫(0 to 1) (3x^2)dx.

A. 1
B. 0.5
C. 0.33
D. 0.25

Solution

The integral evaluates to 1.

Correct Answer: A — 1

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Q. Evaluate the definite integral ∫(1 to 2) (3x^2)dx.

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

Solution

The integral evaluates to 6.

Correct Answer: B — 6

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Q. Evaluate the derivative of f(x) = e^x + ln(x) at x = 1.

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

Solution

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

Correct Answer: A — 1

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Q. Evaluate the integral ∫ (1/x) dx.

A. ln
B. x
C. + C
D. ln(x) + C
. 1/x + C
. x + C

Solution

The integral of 1/x is ln|x|. Therefore, ∫ (1/x) dx = ln|x| + C.

Correct Answer: A — ln

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Q. Evaluate the integral ∫ (2x + 1)/(x^2 + x) dx.

A. ln
B. x^2 + x
C. + C
D. ln
. x
. + C
. ln
. x^2 + x
. + 1 + C
. ln
. x^2 + x
. + 1

Solution

Using partial fraction decomposition, we can integrate to find that ∫ (2x + 1)/(x^2 + x) dx = ln|x^2 + x| + C.

Correct Answer: A — ln

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Q. Evaluate the integral ∫ (3x^2 + 2x + 1) dx.

A. x^3 + x^2 + x + C
B. x^3 + x^2 + C
C. x^3 + x^2 + x
D. 3x^3 + 2x^2 + x + C

Solution

The integral of 3x^2 is x^3, the integral of 2x is x^2, and the integral of 1 is x. Therefore, ∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.

Correct Answer: A — x^3 + x^2 + x + C

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Q. Evaluate the integral ∫ (sec^2(x)) dx.

A. tan(x) + C
B. sec(x) + C
C. sin(x) + C
D. cos(x) + C

Solution

The integral of sec^2(x) is tan(x) + C.

Correct Answer: A — tan(x) + C

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Q. Evaluate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.

A. (1/3)x^3 + x^2 + C
B. x^2 + x + C
C. (1/3)x^3 + (1/2)x^2 + C
D. x^2 + 2x + C

Solution

By simplifying the integrand, we can integrate to find that ∫ (x^2 + 2x + 1)/(x + 1) dx = (1/3)x^3 + x^2 + C.

Correct Answer: A — (1/3)x^3 + x^2 + C

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Q. Evaluate the integral ∫ cos(3x) dx.

A. (1/3)sin(3x) + C
B. sin(3x) + C
C. (1/3)cos(3x) + C
D. -(1/3)sin(3x) + C

Solution

The integral of cos(kx) is (1/k)sin(kx). Here, k = 3, so ∫ cos(3x) dx = (1/3)sin(3x) + C.

Correct Answer: A — (1/3)sin(3x) + C

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Q. Evaluate the integral ∫ cos(5x) dx.

A. 1/5 sin(5x) + C
B. -1/5 sin(5x) + C
C. 5 sin(5x) + C
D. sin(5x) + C

Solution

The integral of cos(kx) is (1/k)sin(kx). Here, k = 5, so ∫ cos(5x) dx = (1/5)sin(5x) + C.

Correct Answer: A — 1/5 sin(5x) + C

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Q. Evaluate the integral ∫ e^(3x) dx.

A. (1/3)e^(3x) + C
B. (1/3)e^(3x)
C. 3e^(3x) + C
D. e^(3x) + C

Solution

The integral of e^(kx) is (1/k)e^(kx). Here, k = 3, so ∫ e^(3x) dx = (1/3)e^(3x) + C.

Correct Answer: A — (1/3)e^(3x) + C

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Q. Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.

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

Solution

The integral evaluates to [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.

Correct Answer: B — 2

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Q. Evaluate the integral ∫ from 0 to 1 of e^x dx.

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

Solution

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

Correct Answer: A — e - 1

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Q. Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.

A. 6
B. 8
C. 10
D. 12

Solution

The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.

Correct Answer: B — 8

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Q. Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.

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

Solution

The integral evaluates to [x^4/4 + 2x^3/3] from 0 to 1 = 1/4 + 2/3 = 11/12.

Correct Answer: B — 1/3

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Q. Evaluate the integral ∫(1 to 2) (2x + 3)dx.

A. 7
B. 8
C. 9
D. 10

Solution

∫(2x + 3)dx = [x^2 + 3x] from 1 to 2 = (4 + 6) - (1 + 3) = 10 - 4 = 6.

Correct Answer: B — 8

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Q. Evaluate 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) = 5.

Correct Answer: A — 3

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Q. Evaluate the integral ∫(1 to 2) (x^2 + 2x)dx.

A. 7
B. 8
C. 9
D. 10

Solution

The integral ∫(x^2 + 2x)dx = [(1/3)x^3 + x^2] from 1 to 2 = 8.

Correct Answer: B — 8

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Q. Evaluate the integral ∫(2x^3 - 4x)dx.

A. (1/2)x^4 - 2x^2 + C
B. (1/4)x^4 - 2x^2 + C
C. (1/2)x^4 - 4x^2 + C
D. (1/3)x^4 - 2x^2 + C

Solution

The integral ∫(2x^3 - 4x)dx = (1/2)x^4 - 2x^2 + C.

Correct Answer: A — (1/2)x^4 - 2x^2 + C

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Q. Evaluate the integral ∫_0^1 (x^2 + 2x) dx.

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

Solution

∫_0^1 (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.

Correct Answer: B — 2

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Q. Evaluate the integral ∫_0^1 (x^3 - 3x^2 + 3x - 1) dx.

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

Solution

∫_0^1 (x^3 - 3x^2 + 3x - 1) dx = [x^4/4 - x^3 + (3/2)x^2 - x] from 0 to 1 = 0.

Correct Answer: A — 0

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Q. Evaluate the integral ∫_0^π/2 cos^2(x) dx.

A. π/4
B. π/2
C. 1
D. 0

Solution

∫_0^π/2 cos^2(x) dx = π/4.

Correct Answer: A — π/4

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Q. Evaluate the integral ∫_1^2 (3x^2 - 2) dx.

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

Solution

∫_1^2 (3x^2 - 2) dx = [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.

Correct Answer: A — 1

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Calculus

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