Q. Find the value of k such that the function f(x) = { kx + 1, x < 1; 2x - 1, x >= 1 } is continuous at x = 1.
Solution
Setting k(1) + 1 = 2(1) - 1 gives k + 1 = 1, so k = 0.
Correct Answer: B — 1
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Q. Find the value of k such that the function f(x) = { kx + 1, x < 1; 3, x = 1; x^2 + 1, x > 1 is continuous at x = 1.
Solution
Setting k(1) + 1 = 3 gives k = 2 for continuity.
Correct Answer: C — 3
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Q. Find the value of k such that the function f(x) = { kx + 1, x < 2; x^2 - 3, x >= 2 } is continuous at x = 2.
Solution
Setting k(2) + 1 = 2^2 - 3 gives 2k + 1 = 1, leading to k = 0.
Correct Answer: B — 2
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Q. Find the value of k such that the function f(x) = { kx + 2, x < 1; 3, x = 1; 2x + 1, x > 1 } is continuous at x = 1.
Solution
Setting k(1) + 2 = 3 gives k = 1.
Correct Answer: A — 1
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Q. Find the value of k such that the function f(x) = { kx, x < 0; 0, x = 0; x^2 + k, x > 0 is continuous at x = 0.
Solution
Setting k = 0 for continuity at x = 0 gives f(0) = 0.
Correct Answer: B — 0
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Q. Find the value of k such that the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0.
Solution
Setting k(0) = 0^2 + 1 gives k = 1.
Correct Answer: A — 0
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Q. Find the value of m for which the function f(x) = { 2x + m, x < 1; mx + 3, x >= 1 is continuous at x = 1.
Solution
Setting 2(1) + m = m(1) + 3 gives m = 1.
Correct Answer: B — 2
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Q. Find the value of m for which the function f(x) = { 2x + m, x < 1; x^2 + 1, x >= 1 is continuous at x = 1.
Solution
Setting 2(1) + m = 1^2 + 1 gives m = 0.
Correct Answer: A — 0
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Q. Find the value of m for which the function f(x) = { 2x + m, x < 3; x^2 - 3, x >= 3 } is continuous at x = 3.
Solution
Setting the two pieces equal at x = 3 gives us 6 + m = 6. Thus, m = 0.
Correct Answer: C — 2
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Q. Find the value of m for which the function f(x) = { 3x + m, x < 1; 2x^2, x >= 1 is continuous at x = 1.
Solution
Setting 3(1) + m = 2(1)^2 gives m = -1.
Correct Answer: B — 0
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Q. Find the value of m such that the function f(x) = { x^2 + m, x < 1; 4 - x, x >= 1 } is continuous at x = 1.
Solution
Setting the two pieces equal at x = 1: 1^2 + m = 4 - 1. Solving gives m = 2.
Correct Answer: B — 1
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Q. Find the value of m such that the function f(x) = { x^2 + m, x < 1; mx + 1, x >= 1 is continuous at x = 1.
Solution
Setting x^2 + m = mx + 1 at x = 1 gives m = 1 for continuity.
Correct Answer: A — 0
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Q. Find the value of the derivative of f(x) = x^4 - 4x^3 + 6x^2 at x = 1.
Solution
f'(x) = 4x^3 - 12x^2 + 12x. Evaluating at x = 1 gives f'(1) = 4 - 12 + 12 = 4.
Correct Answer: A — 0
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Q. Find the value of the integral ∫(0 to 1) (1 - x^2)dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
The integral evaluates to [x - (1/3)x^3] from 0 to 1 = 1 - 1/3 = 2/3.
Correct Answer: C — 2/3
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Q. Find the value of the integral ∫(0 to 1) (3x^2)dx.
Solution
The integral ∫(3x^2)dx from 0 to 1 = [x^3] from 0 to 1 = 1.
Correct Answer: A — 1
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Q. Find the value of the integral ∫(0 to 1) (x^2 + 2x)dx.
Solution
The integral evaluates to [(1/3)x^3 + x^2] from 0 to 1 = (1/3 + 1) - 0 = 4/3.
Correct Answer: B — 2
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Q. Find the value of ∫ from 0 to 1 of (1 - x^2) dx.
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer: C — 2/3
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Q. Find the value of ∫ 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. Find the value of ∫ from 0 to 1 of (x^2 * e^x) dx.
Solution
Using integration by parts, the integral evaluates to (e - 1)/2.
Correct Answer: B — e - 1
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Q. Find the value of ∫ from 0 to 1 of (x^2 + 1/x^2) dx.
Solution
The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.
Correct Answer: C — 3
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Q. Find the value of ∫ from 0 to 1 of (x^2 + 3x + 2) dx.
Solution
The integral evaluates to [x^3/3 + (3/2)x^2 + 2x] from 0 to 1 = (1/3 + 3/2 + 2) - 0 = 4/3 + 3/2 = 2.5.
Correct Answer: D — 4
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Q. Find the value of ∫ from 0 to 1 of (x^2 - 2x + 1) dx.
Solution
The integral evaluates to [x^3/3 - x^2 + x] from 0 to 1 = (1/3 - 1 + 1) = 1/3.
Correct Answer: A — 0
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Q. Find the value of ∫ from 0 to 1 of (x^3 - 3x^2 + 2) dx.
Solution
The integral evaluates to [x^4/4 - x^3 + 2x] from 0 to 1 = (1/4 - 1 + 2) - (0) = 1/4.
Correct Answer: B — 1
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Q. Find the value of ∫ from 0 to 1 of (x^3 - 3x^2 + 3x - 1) dx.
Solution
The integral evaluates to [x^4/4 - x^3 + (3/2)x^2 - x] from 0 to 1 = 0.
Correct Answer: A — 0
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Q. Find the value of ∫ from 0 to 1 of (x^3 - 3x^2 + 3x) dx.
Solution
The integral evaluates to [x^4/4 - x^3 + (3/2)x^2] from 0 to 1 = (1/4 - 1 + 3/2) = 1/4.
Correct Answer: B — 1
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Q. Find the value of ∫ from 0 to 1 of (x^3 - 4x + 4) dx.
Solution
The integral evaluates to [x^4/4 - 2x^2 + 4x] from 0 to 1 = (1/4 - 2 + 4) = (1/4 + 2) = 9/4.
Correct Answer: B — 1
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Q. Find the value of ∫ from 0 to 1 of (x^4 + 2x^2) dx.
-
A.
1/5
-
B.
1/3
-
C.
1/2
-
D.
1
Solution
The integral evaluates to [x^5/5 + (2/3)x^3] from 0 to 1 = (1/5 + 2/3) = 11/15.
Correct Answer: B — 1/3
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Q. Find the value of ∫ from 0 to 1 of (x^4 - 4x^3 + 6x^2 - 4x + 1) dx.
Solution
The integral evaluates to [x^5/5 - x^4 + 2x^3 - 2x^2 + x] from 0 to 1 = (1/5 - 1 + 2 - 2 + 1) = 0.
Correct Answer: B — 1
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Q. Find the value of ∫ from 0 to 1 of (x^4) dx.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The integral evaluates to [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Find the value of ∫ from 0 to 2 of (x^2 - 2x + 1) dx.
Solution
The integral evaluates to [x^3/3 - x^2 + x] from 0 to 2 = (8/3 - 4 + 2) = 2/3.
Correct Answer: C — 2
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