Q. What is the equation of the tangent line to the curve y = x^2 + 2x at the point where x = 1?
A.
y = 3x - 2
B.
y = 2x + 1
C.
y = 2x + 2
D.
y = x + 3
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Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 4. The point is (1, 3). The tangent line is y - 3 = 4(x - 1) => y = 4x - 1.
Correct Answer: A — y = 3x - 2
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Q. What is the first derivative of f(x) = e^(2x)?
A.
2e^(2x)
B.
e^(2x)
C.
e^(x)
D.
2x*e^(2x)
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Solution
Using the chain rule, f'(x) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. What is the general solution of the differential equation dy/dx = 3y?
A.
y = Ce^(3x)
B.
y = Ce^(-3x)
C.
y = 3x + C
D.
y = Cx^3
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Solution
The differential equation is separable. Integrating both sides gives ln|y| = 3x + C, hence y = Ce^(3x).
Correct Answer: A — y = Ce^(3x)
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Q. What is the integral of cos(x)dx?
A.
sin(x) + C
B.
cos(x) + C
C.
-sin(x) + C
D.
-cos(x) + C
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Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. What is the integral of f(x) = cos(x)?
A.
sin(x) + C
B.
-sin(x) + C
C.
tan(x) + C
D.
sec(x) + C
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Solution
The integral is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. What is the integrating factor for the equation dy/dx + 2y = 3x?
A.
e^(2x)
B.
e^(-2x)
C.
e^(3x)
D.
e^(-3x)
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Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
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Q. What is the integrating factor for the equation dy/dx + 3y = 6x?
A.
e^(3x)
B.
e^(-3x)
C.
e^(6x)
D.
e^(-6x)
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Solution
The integrating factor is e^(∫3dx) = e^(3x).
Correct Answer: A — e^(3x)
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Q. What is the limit of (1/x) as x approaches infinity?
A.
0
B.
1
C.
Infinity
D.
-Infinity
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Solution
As x approaches infinity, (1/x) approaches 0.
Correct Answer: A — 0
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Q. What is the limit of (x^2 - 1)/(x - 1) as x approaches 1?
A.
0
B.
1
C.
2
D.
Infinity
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Solution
Using L'Hôpital's Rule, lim (x -> 1) (x^2 - 1)/(x - 1) = lim (x -> 1) (2x)/(1) = 2.
Correct Answer: C — 2
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Q. What is the limit of f(x) = 1/x as x approaches 0 from the right?
A.
0
B.
Infinity
C.
1
D.
Does not exist
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Solution
As x approaches 0 from the right, f(x) approaches infinity, indicating a discontinuity at x = 0.
Correct Answer: B — Infinity
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Q. What is the maximum slope of the function f(x) = -x^2 + 4x?
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Solution
f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. The maximum slope occurs at x = 2, f'(2) = 2.
Correct Answer: A — 4
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Q. What is the maximum value of f(x) = -x^2 + 4x?
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Solution
The vertex form gives the maximum at x = -b/(2a) = 4/2 = 2, f(2) = -2^2 + 4*2 = 4.
Correct Answer: A — 4
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Q. What is the maximum value of the function f(x) = -x^2 + 4x + 1?
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Solution
The vertex occurs at x = -b/(2a) = 4/(-2) = 2. f(2) = -2^2 + 4*2 + 1 = 6.
Correct Answer: B — 6
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Q. What is the minimum value of the function f(x) = 2x^2 - 8x + 10?
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Solution
The vertex form gives the minimum at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 6.
Correct Answer: B — 4
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Q. What is the minimum value of the function f(x) = x^2 - 4x + 5?
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Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 5 = 1.
Correct Answer: A — 1
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Q. What is the minimum value of the function f(x) = x^4 - 8x^2 + 16?
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Solution
f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x(x^2 - 4) = 0, so x = 0, 2, -2. f(0) = 16, f(2) = 0, f(-2) = 0. Minimum value is 0.
Correct Answer: A — 0
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Q. What is the particular solution of dy/dx = 4x with the initial condition y(0) = 1?
A.
y = 2x^2 + 1
B.
y = 4x^2 + 1
C.
y = 2x^2
D.
y = 4x^2 + C
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Solution
Integrating gives y = 2x^2 + C. Using the initial condition, C = 1.
Correct Answer: A — y = 2x^2 + 1
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Q. What is the particular solution of the equation dy/dx = 2y with the initial condition y(0) = 1?
A.
y = e^(2x)
B.
y = e^(2x) + 1
C.
y = 1 + e^(2x)
D.
y = 1 + 2x
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Solution
The general solution is y = Ce^(2x). Using the initial condition y(0) = 1, we find C = 1.
Correct Answer: A — y = e^(2x)
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Q. What is the particular solution of the equation dy/dx = 2y with y(0) = 5?
A.
y = 5e^(2x)
B.
y = 2e^(2x)
C.
y = 5e^(-2x)
D.
y = 5 + 2x
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Solution
The general solution is y = Ce^(2x). Using y(0) = 5 gives C = 5.
Correct Answer: A — y = 5e^(2x)
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Q. What is the second derivative of f(x) = e^x sin(x)?
A.
e^x (sin(x) + cos(x))
B.
e^x (2sin(x) + cos(x))
C.
e^x (sin(x) - cos(x))
D.
e^x (sin(x) + 2cos(x))
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Solution
Using the product rule and chain rule, the second derivative is f''(x) = e^x (sin(x) + cos(x)).
Correct Answer: A — e^x (sin(x) + cos(x))
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Q. What is the second derivative of f(x) = e^x?
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Solution
The second derivative f''(x) = e^x.
Correct Answer: A — e^x
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Q. What is the second derivative of f(x) = x^4 - 4x^3 + 6x^2?
A.
12x - 12
B.
12x - 8
C.
12x - 6
D.
12x - 4
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Solution
f''(x) = 12x - 12.
Correct Answer: A — 12x - 12
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Q. What is the slope of the tangent line to the curve y = x^2 + 2x at the point (1, 3)?
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Solution
The derivative y' = 2x + 2. At x = 1, y' = 2(1) + 2 = 4, which is the slope of the tangent line.
Correct Answer: B — 2
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Q. What is the solution of the differential equation dy/dx = (x^2 + 1)y?
A.
y = Ce^(x^3/3 + x)
B.
y = Ce^(x^2 + 1)
C.
y = Ce^(x^2/2)
D.
y = Ce^(x^3)
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Solution
This is a separable equation. Integrating gives y = Ce^(x^3/3 + x).
Correct Answer: A — y = Ce^(x^3/3 + x)
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Q. What is the solution of the equation y'' - 4y' + 4y = 0?
A.
y = (C1 + C2x)e^(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 e^(4x) + C2 e^(-4x)
D.
y = C1 cos(2x) + C2 sin(2x)
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Solution
The characteristic equation has a repeated root, leading to the solution form (C1 + C2x)e^(2x).
Correct Answer: A — y = (C1 + C2x)e^(2x)
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Q. What is the solution to the differential equation dy/dx = (x^2 + 1)y?
A.
y = Ce^(x^3/3 + x)
B.
y = Ce^(x^2 + 1)
C.
y = Ce^(x^2/2)
D.
y = Ce^(x^3)
Show solution
Solution
This is a separable equation. Integrating gives y = Ce^(x^3/3 + x).
Correct Answer: A — y = Ce^(x^3/3 + x)
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Q. What is the solution to the differential equation dy/dx = 2y + 3?
A.
y = Ce^(2x) - 3/2
B.
y = Ce^(2x) + 3/2
C.
y = 3e^(2x)
D.
y = 3/2e^(2x)
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Solution
Using the integrating factor method, we find y = Ce^(2x) + 3/2.
Correct Answer: B — y = Ce^(2x) + 3/2
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Q. What is the solution to the differential equation dy/dx = y^2?
A.
y = 1/(C - x)
B.
y = C/(x - 1)
C.
y = Cx
D.
y = e^(x + C)
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Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. What is the solution to the differential equation y' = y^2 - 1?
A.
y = tan(x + C)
B.
y = 1/(C - x)
C.
y = C/(1 - Cx)
D.
y = C/(x - 1)
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Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: B — y = 1/(C - x)
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Q. What is the solution to the differential equation y'' + 4y = 0?
A.
y = C1 cos(2x) + C2 sin(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 e^(x) + C2 e^(-x)
D.
y = C1 sin(2x) + C2 cos(2x)
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Solution
This is a second-order linear homogeneous differential equation. The characteristic equation has roots ±2i, leading to the solution y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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