For f(x) = x^3 - 3x^2 + 5, what is f'(2)?
- 0 Correct answer.
- 2
- 4
- 8
Explanation
Differentiate first: f'(x) = 3x^2 - 6x. Substituting x = 2 gives 12 - 12 = 0.
Kaoyan Mathematics I practice questions
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Evaluate lim as x -> 0 of sin(3x)/x.
- 0
- 1
- 3 Correct answer.
- 1/3
Explanation
Write sin(3x)/x = 3[sin(3x)/(3x)]. Since sin u/u -> 1 as u -> 0, the limit is 3.
What is integral_0^1 2x dx?
- 1/2
- 1 Correct answer.
- 2
- 4
Explanation
An antiderivative of 2x is x^2. Evaluating from 0 to 1 gives 1^2 - 0^2 = 1.
Define f(x) = (e^x - 1)/x for x != 0. What value of f(0) makes f continuous at 0?
- 0
- 1 Correct answer.
- e
- Does not exist
Explanation
Continuity requires f(0) to equal lim as x -> 0 (e^x - 1)/x. This standard derivative limit equals 1.
For f(x) = ln(1 + x^2), what is f'(1)?
- 1/2
- 1 Correct answer.
- 2
- ln 2
Explanation
By the chain rule, f'(x) = 2x/(1 + x^2). At x = 1, this is 2/(2) = 1.
Find the sum of the geometric series 1 + 1/2 + 1/4 + 1/8 + ... .
- 1
- 3/2
- 2 Correct answer.
- The series diverges
Explanation
The first term is 1 and the common ratio is 1/2. Since |r| < 1, the sum is a/(1 - r) = 1/(1/2) = 2.
If y' = 2x and y(0) = 1, what is y(2)?
- 3
- 4
- 5 Correct answer.
- 6
Explanation
Integrating y' = 2x gives y = x^2 + C. Since y(0) = 1, C = 1, so y(2) = 4 + 1 = 5.
For f(x, y) = x^2 y + y^3, what is the partial derivative f_x at (1, 2)?
- 2
- 4 Correct answer.
- 8
- 12
Explanation
Treat y as constant. f_x = 2xy, and at (1, 2) this equals 2*1*2 = 4.
Evaluate the double integral of x + y over the unit square 0 <= x <= 1, 0 <= y <= 1.
- 1/2
- 1 Correct answer.
- 3/2
- 2
Explanation
The integral splits as integral of x over the square plus integral of y over the square. Each part is 1/2, so the total is 1.
What is the area under y = x from x = 0 to x = 2?
- 1
- 2 Correct answer.
- 3
- 4
Explanation
The area is integral_0^2 x dx = x^2/2 evaluated from 0 to 2, which is 2.
Find the tangent plane to z = x^2 + y^2 at (1, 1, 2).
- z = x + y
- z = 2x + 2y - 2 Correct answer.
- z = 2x - 2y + 2
- z = x^2 + y^2
Explanation
For z = f(x,y), f_x = 2x and f_y = 2y. At (1,1), the plane is z - 2 = 2(x - 1) + 2(y - 1), so z = 2x + 2y - 2.
Solve y' + y = 0 with y(0) = 3.
- y = 3e^x
- y = 3e^(-x) Correct answer.
- y = e^(-3x)
- y = 3 - x
Explanation
The equation gives y' = -y, so y = Ce^(-x). Using y(0) = 3 gives C = 3.
Evaluate lim as n -> infinity of (2n + 1)/(n - 3).
- 0
- 1
- 2 Correct answer.
- Does not exist
Explanation
Divide numerator and denominator by n. The limit becomes (2 + 1/n)/(1 - 3/n), which approaches 2.
For the circle x^2 + y^2 = 25, what is dy/dx at the point (3, 4)?
- -3/4 Correct answer.
- 3/4
- -4/3
- 4/3
Explanation
Implicit differentiation gives 2x + 2y y' = 0, so y' = -x/y. At (3,4), y' = -3/4.
A line passes through the origin with direction vector (1, 2, 2). Which point is on the line?
- (1, 1, 2)
- (2, 4, 4) Correct answer.
- (2, 2, 4)
- (4, 2, 2)
Explanation
Points on the line have form t(1, 2, 2). Taking t = 2 gives (2, 4, 4).
Compute the dot product of (1, 2, 3) and (4, 0, -1).
- 1 Correct answer.
- 4
- 7
- 11
Explanation
The dot product is 1*4 + 2*0 + 3*(-1) = 4 - 3 = 1.
Evaluate lim as x -> 0 of (1 - cos x)/x^2.
- 0
- 1/2 Correct answer.
- 1
- 2
Explanation
Using cos x = 1 - x^2/2 + o(x^2), the numerator is x^2/2 + o(x^2), so the limit is 1/2.
For x > 0, let f(x) = x^x. What is f'(1)?
- 0
- 1 Correct answer.
- e
- 2
Explanation
Use logarithmic differentiation: ln f = x ln x, so f'/f = ln x + 1. At x = 1, f = 1 and ln 1 + 1 = 1, hence f'(1) = 1.
Evaluate integral_0^(pi/2) sin^2 x dx.
- 1/2
- pi/8
- pi/4 Correct answer.
- pi/2
Explanation
Use sin^2 x = (1 - cos 2x)/2. The integral is [x/2 - sin 2x/4]_0^(pi/2) = pi/4.
Which statement about the series sum from n = 1 to infinity of 1/n^2 is correct?
- It diverges by comparison with 1/n
- It converges because it is a p-series with p = 2 Correct answer.
- It converges only conditionally
- Its terms do not tend to 0
Explanation
The series is a p-series with p = 2 > 1, so it converges absolutely.
In the Taylor expansion of e^x at x = 0, what is the coefficient of x^2?
- 1/2 Correct answer.
- 1
- 2
- 1/6
Explanation
e^x = 1 + x + x^2/2! + x^3/3! + ... . The coefficient of x^2 is 1/2.
For which real a does integral_0^1 x^a dx converge?
- a > -1 Correct answer.
- a >= 0 only
- a < -1
- All real a
Explanation
The improper integral near 0 converges exactly when the power exponent a is greater than -1.
What is the maximum of f(x, y) = x^2 + y^2 on the disk x^2 + y^2 <= 4?
- 0
- 2
- 4 Correct answer.
- 8
Explanation
On the disk, f(x,y) equals the squared distance from the origin. Its maximum occurs on the boundary x^2 + y^2 = 4, so the maximum is 4.
For f(x, y) = x^2 y, find the directional derivative at (1, 2) in the unit direction (3/5, 4/5).
- 4/5
- 12/5
- 16/5 Correct answer.
- 5
Explanation
The gradient is (2xy, x^2), so at (1,2) it is (4,1). Dotting with (3/5,4/5) gives 12/5 + 4/5 = 16/5.
What is the double integral of 1 over the triangle x >= 0, y >= 0, x + y <= 1?
- 1/4
- 1/2 Correct answer.
- 1
- 2
Explanation
The integral of 1 over a region equals its area. The triangle has legs 1 and 1, so its area is 1/2.
Evaluate the double integral of x^2 + y^2 over the unit disk x^2 + y^2 <= 1.
- pi/4
- pi/2 Correct answer.
- pi
- 2pi
Explanation
In polar coordinates, x^2 + y^2 = r^2 and dA = r dr dtheta. The integral is integral_0^(2pi) integral_0^1 r^3 dr dtheta = 2pi*(1/4) = pi/2.
For F = (y, x), compute the line integral of F dot dr along the straight line from (0,0) to (1,1).
- 0
- 1/2
- 1 Correct answer.
- 2
Explanation
Parameterize r(t) = (t,t), 0 <= t <= 1. Then F(r(t)) = (t,t), dr = (1,1)dt, so F dot dr = 2t dt and the integral is 1.
For F(x,y,z) = (x^2, y^2, z^2), what is div F at (1,1,1)?
- 3
- 6 Correct answer.
- 9
- 12
Explanation
div F = partial_x x^2 + partial_y y^2 + partial_z z^2 = 2x + 2y + 2z. At (1,1,1), this is 6.
For F(x,y,z) = (-y, x, 0), what is curl F?
- (0, 0, 0)
- (0, 0, 1)
- (0, 0, 2) Correct answer.
- (2, 0, 0)
Explanation
curl F has k-component partial_x Q - partial_y P = partial_x x - partial_y(-y) = 1 - (-1) = 2; the other components are 0.
Solve y'' + y = 0 with y(0) = 0 and y'(0) = 1.
- y = cos x
- y = sin x Correct answer.
- y = e^x
- y = x^2
Explanation
The general solution is y = C1 cos x + C2 sin x. y(0)=0 gives C1=0, and y'(0)=C2=1, so y = sin x.
Solve y' - y = e^x with y(0) = 0.
- y = x e^x Correct answer.
- y = e^x - 1
- y = e^x
- y = x
Explanation
Multiplying by e^(-x) gives (y e^(-x))' = 1. Thus y e^(-x) = x + C, and y(0)=0 gives C=0, so y = x e^x.
A process satisfies y' + 2y = 4 with y(0) = 1. Which formula gives y(x)?
- y = 2 - e^(-2x) Correct answer.
- y = 2 + e^(-2x)
- y = 4 - 3e^(-2x)
- y = e^(-2x)
Explanation
The steady value is 2. The solution is y = 2 + Ce^(-2x). From y(0)=1, C = -1, so y = 2 - e^(-2x).
Find the sum of sum from n = 1 to infinity of (1/3)^n.
- 1/3
- 1/2 Correct answer.
- 2/3
- 1
Explanation
This is a geometric series with first term 1/3 and ratio 1/3. The sum is (1/3)/(1 - 1/3) = 1/2.
What is the radius of convergence of the power series sum from n = 1 to infinity of n x^n?
- 0
- 1 Correct answer.
- 2
- Infinity
Explanation
The coefficients grow like n, and the root test gives lim n^(1/n) = 1. Thus the radius of convergence is 1.
What is the volume under z = 1 over the unit disk x^2 + y^2 <= 1?
- 1
- pi/2
- pi Correct answer.
- 2pi
Explanation
The volume is the integral of height 1 over the unit disk, which equals the disk area pi.
Find the plane through (1,0,0) with normal vector (2,-1,3).
- 2x - y + 3z = 0
- 2x - y + 3z = 2 Correct answer.
- x + 2y - z = 1
- 2x + y + 3z = 2
Explanation
Use n dot ((x,y,z) - (1,0,0)) = 0. This gives 2(x - 1) - y + 3z = 0, or 2x - y + 3z = 2.
What is the distance from point (1,2,3) to the plane x + 2y + 2z = 9?
- 1/3
- 2/3 Correct answer.
- 1
- 2
Explanation
The distance is |1 + 2*2 + 2*3 - 9|/sqrt(1^2 + 2^2 + 2^2) = |2|/3 = 2/3.
What is the angle between vectors (1,1,0) and (1,0,1)?
- 30 degrees
- 45 degrees
- 60 degrees Correct answer.
- 90 degrees
Explanation
The dot product is 1. Each vector has length sqrt(2), so cos theta = 1/2. Hence theta = 60 degrees.
Evaluate lim as n -> infinity of n * integral_0^(1/n) e^(-x^2) dx.
- 0
- 1 Correct answer.
- e^(-1)
- Infinity
Explanation
On the small interval [0,1/n], e^(-x^2) tends uniformly to 1. The integral is asymptotic to 1/n, so multiplying by n gives 1.
Evaluate the double integral of xy over the region 0 <= y <= x <= 1.
- 1/12
- 1/8 Correct answer.
- 1/6
- 1/4
Explanation
Compute integral_0^1 integral_0^x xy dy dx. The inner integral is x*y^2/2 from 0 to x, which is x^3/2. Integrating from 0 to 1 gives 1/8.
Let S be the sphere x^2 + y^2 + z^2 = 4 with outward orientation. For F = (x,y,z), what is the flux of F through S?
- 8pi
- 16pi
- 32pi Correct answer.
- 64pi
Explanation
By the divergence theorem, div F = 3. The ball has radius 2 and volume (4/3)pi*8 = 32pi/3. The flux is 3*(32pi/3) = 32pi.
For positive x, y, z with x + y + z = 3, what is the maximum possible value of xyz?
- 1/3
- 1 Correct answer.
- 3
- 9
Explanation
By AM-GM, (x + y + z)/3 >= (xyz)^(1/3). Since the average is 1, xyz <= 1, with equality at x = y = z = 1.
Solve y'' - 3y' + 2y = 0 with y(0) = 0 and y'(0) = 1.
- e^x - e^(2x)
- e^(2x) - e^x Correct answer.
- e^x + e^(2x)
- x e^x
Explanation
The characteristic equation r^2 - 3r + 2 = 0 has roots 1 and 2. Thus y = C1e^x + C2e^(2x). The conditions give C1 = -1 and C2 = 1, so y = e^(2x) - e^x.
Which statement best describes the series sum from n = 1 to infinity of (-1)^(n-1)/n?
- It diverges because 1/n diverges
- It converges absolutely
- It converges conditionally Correct answer.
- Its terms do not approach 0
Explanation
The alternating harmonic series converges by the alternating series test, but the series of absolute values is the harmonic series, which diverges. Therefore it converges conditionally.
For F = (2xy, x^2 + 2y), compute the line integral of F dot dr from (0,0) to (1,1).
- 1
- 2 Correct answer.
- 3
- Path-dependent; cannot be determined
Explanation
The field is conservative because partial_y(2xy) = 2x and partial_x(x^2 + 2y) = 2x. A potential is phi = x^2 y + y^2. The value is phi(1,1) - phi(0,0) = 2.
Evaluate the exact differential integral of M dx + N dy from (0,0) to (1,2), where M = 2xy + y^2 and N = x^2 + 2xy.
- 3
- 4
- 6 Correct answer.
- 8
Explanation
A potential function is phi = x^2 y + xy^2, since phi_x = 2xy + y^2 and phi_y = x^2 + 2xy. The integral is phi(1,2) - phi(0,0) = 2 + 4 = 6.
What is the volume of the tetrahedron x >= 0, y >= 0, z >= 0, x + y + z <= 1?
- 1/12
- 1/6 Correct answer.
- 1/3
- 1/2
Explanation
The intercept tetrahedron has mutually perpendicular edge lengths 1, 1, and 1 from the origin, so its volume is 1*1*1/6 = 1/6.
Let C be the unit circle oriented counterclockwise. What is integral_C (-y dx + x dy)?
- 0
- pi
- 2pi Correct answer.
- 4pi
Explanation
By Green's theorem, the integral is double integral over the disk of (partial_x x - partial_y(-y)) dA = double integral of 2 dA = 2pi.
Find the determinant of the matrix [[1, 2], [3, 4]].
- -2 Correct answer.
- 2
- 10
- 14
Explanation
For a 2 by 2 matrix, det = ad - bc. Here det = 1*4 - 2*3 = 4 - 6 = -2.
What is the sum of the eigenvalues of [[2, 1], [0, 3]]?
- 1
- 3
- 5 Correct answer.
- 6
Explanation
The sum of eigenvalues equals the trace. The trace is 2 + 3 = 5.