Q. What is the solution to the equation y' = y(1 - y)?
-
A.
y = 1/(C - x)
-
B.
y = C/(C + x)
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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?
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A.
y = e^x + Ce^(-x)
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B.
y = e^x - Ce^(-x)
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C.
y = Ce^x - e^x
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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?
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?
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?
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?
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?
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?
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?
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?
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A.
1
-
B.
1/3
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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?
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?
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?
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A.
tan^-1(x) + C
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B.
sin^-1(x) + C
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C.
cos^-1(x) + C
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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?
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)?
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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?
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?
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?
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?
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?
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?
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A.
f(x) = 1/x
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B.
f(x) = x^2 - 4
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C.
f(x) = sin(1/x)
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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 }
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A.
Continuous
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B.
Not continuous
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C.
Depends on k
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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 }
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A.
Continuous
-
B.
Not continuous
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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
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B.
f(x) = x^2
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C.
f(x) = sin(x)
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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 }
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A.
f(1) = 1
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B.
f(1) = 0
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C.
f(1) = 2
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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?
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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?
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A.
f(x) = x^3
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B.
f(x) = e^x
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C.
f(x) = 1/x
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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?
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A.
f(x) = x^2
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B.
f(x) = 1/x
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C.
f(x) = sin(x)
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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.
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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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