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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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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