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PAU Mathematics II (La Rioja) study materials
PAU Mathematics II (La Rioja) study resources

PAU Mathematics II (La Rioja) practice questions

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Question 1Matrices, determinants and linear systems

Given the matrices A = [[2, -1], [3, 4]] and B = [[1, 5], [-2, 0]], calculate the matrix expression 2A - 3B.

  1. [[1, -17], [12, 8]] Correct answer.
  2. [[1, 13], [0, 8]]
  3. [[1, -17], [0, 8]]
  4. [[7, -17], [12, 8]]

Explanation

Compute scalar multiplications first: 2A = [[4, -2], [6, 8]] and 3B = [[3, 15], [-6, 0]]. Subtract component-wise: 2A - 3B = [[4-3, -2-15], [6-(-6), 8-0]] = [[1, -17], [12, 8]].

Question 2Matrices, determinants and linear systems

What is the determinant of the upper triangular 3x3 matrix A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]]?

  1. 10
  2. 24 Correct answer.
  3. 0
  4. 15

Explanation

For any triangular matrix (upper or lower), the determinant equals the product of the diagonal elements. Therefore, det(A) = 1 * 4 * 6 = 24.

Question 3Matrices, determinants and linear systems

Calculate the determinant of the 2x2 matrix A = [[3, -2], [5, 4]].

  1. 22 Correct answer.
  2. 2
  3. -2
  4. 7

Explanation

The determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc. Here det(A) = (3)(4) - (-2)(5) = 12 - (-10) = 12 + 10 = 22.

Question 4Matrices, determinants and linear systems

Find the inverse of the matrix A = [[2, 1], [5, 3]].

  1. [[3, -1], [-5, 2]] Correct answer.
  2. [[-3, 1], [5, -2]]
  3. [[2, -5], [-1, 3]]
  4. [[3, 1], [5, 2]]

Explanation

First find det(A) = 2*3 - 1*5 = 6 - 5 = 1. The inverse of [[a, b], [c, d]] is (1/det(A)) * [[d, -b], [-c, a]]. Thus A^-1 = (1/1) * [[3, -1], [-5, 2]] = [[3, -1], [-5, 2]].

Question 5Matrices, determinants and linear systems

Solve for matrix X in the matrix equation A * X = B, where A = [[1, 2], [3, 5]] and B = [[4, 1], [2, 0]].

  1. [[-16, -5], [10, 3]] Correct answer.
  2. [[16, 5], [-10, -3]]
  3. [[-16, 5], [10, -3]]
  4. [[4, 1], [2, 0]]

Explanation

Multiply both sides by A^-1 on the left: X = A^-1 * B. First, det(A) = 1*5 - 2*3 = -1. A^-1 = (1/-1) * [[5, -2], [-3, 1]] = [[-5, 2], [3, -1]]. Multiply A^-1 * B = [[-5, 2], [3, -1]] * [[4, 1], [2, 0]] = [[-5(4)+2(2), -5(1)+2(0)], [3(4)+(-1)(2), 3(1)+(-1)(0)]] = [[-16, -5], [10, 3]].

Question 6Matrices, determinants and linear systems

If A is a 3x3 matrix with det(A) = -4, what is the value of det(2A)?

  1. -8
  2. -32 Correct answer.
  3. 32
  4. -16

Explanation

For an n x n square matrix A, det(k * A) = k^n * det(A). For n = 3 and k = 2: det(2A) = 2^3 * det(A) = 8 * (-4) = -32.

Question 7Matrices, determinants and linear systems

What is the rank of the matrix A = [[1, 2, 3], [2, 4, 6], [0, 1, 5]]?

  1. 1
  2. 2 Correct answer.
  3. 3
  4. 0

Explanation

Notice that Row 2 is 2 times Row 1, so Row 2 is linearly dependent and can be eliminated. Row 1 (1, 2, 3) and Row 3 (0, 1, 5) are linearly independent. Thus, the matrix has 2 linearly independent rows, so rank(A) = 2.

Question 8Matrices, determinants and linear systems

Calculate the determinant of the 3x3 matrix A = [[1, 0, 2], [-1, 3, 1], [2, 1, 4]].

  1. 3
  2. -3 Correct answer.
  3. 7
  4. 5

Explanation

Expand along the first row: det(A) = 1 * det([[3, 1], [1, 4]]) - 0 + 2 * det([[-1, 3], [2, 1]]) = 1 * (12 - 1) + 2 * (-1 - 6) = 11 + 2(-7) = 11 - 14 = -3.

Question 9Matrices, determinants and linear systems

Consider the system of linear equations with parameter k: x + y + z = 1, x + 2y + 3z = 2, x + 4y + k*z = 5. For what value of k is the system inconsistent (incompatible)?

  1. k = 7 Correct answer.
  2. k = 5
  3. k = 3
  4. k = 0

Explanation

Compute the determinant of the coefficient matrix: det(A) = det([[1, 1, 1], [1, 2, 3], [1, 4, k]]) = k - 7. When k = 7, det(A) = 0 so rank(A) = 2. Row reduction of the augmented matrix A* yields third row (0, 0, 0 | 1), so rank(A*) = 3. By Rouché-Capelli theorem, rank(A) = 2 != rank(A*) = 3, making the system inconsistent (incompatible) when k = 7.

Question 10Matrices, determinants and linear systems

Use Cramer's rule to find the value of y in the system 2x + y = 5, 3x - 2y = 4.

  1. 1 Correct answer.
  2. 2
  3. 3
  4. -1

Explanation

The main determinant is det(A) = det([[2, 1], [3, -2]]) = -4 - 3 = -7. The numerator determinant for y replaces column 2 with constant terms: det(A_y) = det([[2, 5], [3, 4]]) = 8 - 15 = -7. By Cramer's rule, y = det(A_y) / det(A) = -7 / -7 = 1.

Question 11Matrices, determinants and linear systems

For which value of m is the matrix A = [[1, m, 0], [2, 1, 1], [0, 3, 1]] NOT invertible?

  1. m = 1
  2. m = -1 Correct answer.
  3. m = 1/2
  4. m = 1/3

Explanation

A matrix is not invertible if and only if its determinant is zero. Calculate det(A) = 1*(1*1 - 3*1) - m*(2*1 - 0*1) + 0 = 1*(-2) - m*(2) = -2 - 2m. Set det(A) = 0: -2 - 2m = 0 => m = -1.

Question 12Matrices, determinants and linear systems

For what values of lambda does the homogeneous system x + lambda*y + z = 0, lambda*x + y + z = 0, x + y + lambda*z = 0 have non-trivial solutions?

  1. lambda = 1 and lambda = -2 Correct answer.
  2. lambda = 0 and lambda = 1
  3. lambda = -1 and lambda = 2
  4. lambda = 2 and lambda = -2

Explanation

A homogeneous system has non-trivial (infinitely many) solutions if and only if the determinant of the coefficient matrix is 0. det(A) = det([[1, lambda, 1], [lambda, 1, 1], [1, 1, lambda]]) = -(lambda - 1)^2 * (lambda + 2). Setting det(A) = 0 yields lambda = 1 or lambda = -2.

Question 13Matrices, determinants and linear systems

Given A = [[2, 1], [0, 1]], evaluate the matrix polynomial A^2 - 3A + 2I, where I is the 2x2 identity matrix.

  1. [[0, 0], [0, 0]] Correct answer.
  2. [[1, 0], [0, 1]]
  3. [[2, 1], [0, 1]]
  4. [[-1, 0], [0, -1]]

Explanation

Compute A^2 = [[2, 1], [0, 1]] * [[2, 1], [0, 1]] = [[4, 3], [0, 1]]. Next 3A = [[6, 3], [0, 3]] and 2I = [[2, 0], [0, 2]]. Now combine: A^2 - 3A + 2I = [[4-6+2, 3-3+0], [0-0+0, 1-3+2]] = [[0, 0], [0, 0]]. (This also aligns with Cayley-Hamilton theorem).

Question 14Matrices, determinants and linear systems

If A = [[1, 3], [2, 4]] and B = [[2, 0], [1, -1]], what is the entry in row 1, column 2 of the matrix product A * B?

  1. -3 Correct answer.
  2. 5
  3. 0
  4. 3

Explanation

The entry in row 1, column 2 of A * B is the dot product of row 1 of A, which is (1, 3), and column 2 of B, which is (0, -1)^T. (1)(0) + (3)(-1) = 0 - 3 = -3.

Question 15Matrices, determinants and linear systems

For a system of 3 linear equations in 3 unknowns, if rank(A) = 2 and rank(A*) = 3, what does the Rouché-Capelli theorem conclude?

  1. Compatible determinate (unique solution)
  2. Compatible indeterminate with 1 parameter
  3. Incompatible / Inconsistent (no solution) Correct answer.
  4. Compatible indeterminate with 2 parameters

Explanation

According to the Rouché-Capelli theorem, a system of linear equations has at least one solution if and only if the rank of the coefficient matrix rank(A) equals the rank of the augmented matrix rank(A*). When rank(A) != rank(A*), the system is incompatible (inconsistent) and has no solution.

Question 16Matrices, determinants and linear systems

Given A = [[1, 0], [2, 1]], find the general expression for A^n for any positive integer n.

  1. [[1, 0], [2^n, 1]]
  2. [[1, 0], [2n, 1]] Correct answer.
  3. [[n, 0], [2n, n]]
  4. [[1, 0], [n^2, 1]]

Explanation

Compute the first few powers: A^2 = [[1, 0], [4, 1]], A^3 = [[1, 0], [6, 1]]. By mathematical induction, A^n = [[1, 0], [2n, 1]].

Question 17Matrices, determinants and linear systems

Consider the system x + y + z = a, a*x + y + z = 1, x + a*y + z = 1. For a = 1, how many solutions does the system have?

  1. No solution
  2. Unique solution (1, 0, 0)
  3. Infinitely many solutions with 2 parameters (plane of solutions x + y + z = 1) Correct answer.
  4. Infinitely many solutions with 1 parameter

Explanation

Substituting a = 1 into all three equations yields the exact same equation: x + y + z = 1. Therefore, rank(A) = 1 and rank(A*) = 1. Since n = 3 unknowns, there are n - rank(A) = 3 - 1 = 2 degrees of freedom (parameters), representing infinitely many solutions corresponding to points on the plane x + y + z = 1.

Question 18Matrices, determinants and linear systems

If A is a 3x3 matrix with det(A) = 5, what is the value of det(A^-1 * A^T)?

  1. 1 Correct answer.
  2. 5
  3. 1/5
  4. 25

Explanation

Using determinant properties: det(X * Y) = det(X) * det(Y), det(A^-1) = 1 / det(A), and det(A^T) = det(A). Thus det(A^-1 * A^T) = det(A^-1) * det(A^T) = (1 / det(A)) * det(A) = 1.

Question 19Matrices, determinants and linear systems

Given the matrix equation X * A + B = C, where A, B, C are invertible square matrices of order n, solve for X.

  1. X = (C - B) * A^-1 Correct answer.
  2. X = A^-1 * (C - B)
  3. X = C * A^-1 - B
  4. X = (B - C) * A^-1

Explanation

Subtract B from both sides: X * A = C - B. Multiply both sides on the right by A^-1: (X * A) * A^-1 = (C - B) * A^-1 => X = (C - B) * A^-1. Note that matrix multiplication is non-commutative, so A^-1 must be multiplied on the right.

Question 20Matrices, determinants and linear systems

If A is a 3x4 matrix with rank(A) = 3, what is the rank of its transpose matrix A^T?

  1. 3 Correct answer.
  2. 4
  3. 12
  4. 1

Explanation

A fundamental property of matrix rank states that the row rank of a matrix equals its column rank. Therefore, rank(A^T) = rank(A) = 3.

Question 21Matrices, determinants and linear systems

Find the values of x for which det([[x, 1, 2], [0, x, 3], [0, 0, x - 2]]) = 0.

  1. x = 0 or x = 2 Correct answer.
  2. x = 1 or x = 3
  3. x = -2 or x = 0
  4. x = 2 only

Explanation

The matrix is upper triangular, so its determinant is the product of its main diagonal entries: det(A) = x * x * (x - 2) = x^2 * (x - 2). Setting x^2 * (x - 2) = 0 gives x = 0 or x = 2.

Question 22Matrices, determinants and linear systems

Solve for z in the 3x3 system: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2.

  1. z = 3 Correct answer.
  2. z = 1
  3. z = 2
  4. z = 4

Explanation

Add equation 1 and equation 3: (x+y+z) + (x+2y-z) = 6+2 => 2x + 3y = 8. Subtract eq 1 from eq 2: (2x-y+z) - (x+y+z) = 3-6 => x - 2y = -3 => x = 2y - 3. Substitute into 2x + 3y = 8: 2(2y - 3) + 3y = 8 => 7y = 14 => y = 2. Then x = 2(2) - 3 = 1. Finally, z = 6 - x - y = 6 - 1 - 2 = 3.

Question 23Matrices, determinants and linear systems

Which of the following determinant properties is ALWAYS true for any square matrices A and B of order n?

  1. (A + B)^2 = A^2 + 2AB + B^2
  2. det(A + B) = det(A) + det(B)
  3. det(A * B) = det(A) * det(B) Correct answer.
  4. A * B = B * A

Explanation

The multiplicative property det(A * B) = det(A) * det(B) holds for all square matrices of the same size. Matrix multiplication is generally non-commutative (AB != BA) and the determinant is not additive (det(A+B) != det(A) + det(B)).

Question 24Matrices, determinants and linear systems

Consider the system x + y + a*z = 1, x + a*y + z = 1, a*x + y + z = 1. For a = -2, the system is:

  1. Compatible determinate
  2. Compatible indeterminate
  3. Incompatible / Inconsistent (no solution) Correct answer.
  4. Has x = y = z = 1 as unique solution

Explanation

Compute det(A) = -(a - 1)^2 * (a + 2). For a = -2, det(A) = 0. Summing all three equations gives (a+2)x + (a+2)y + (a+2)z = 3 => 0*x + 0*y + 0*z = 3 => 0 = 3, which is a contradiction. Thus rank(A) = 2 while rank(A*) = 3, so the system is incompatible.

Question 25Matrices, determinants and linear systems

A 2x2 matrix A has trace tr(A) = 5 and determinant det(A) = 6. What are the eigenvalues of matrix A?

  1. 2 and 3 Correct answer.
  2. 1 and 4
  3. -1 and 6
  4. 0 and 5

Explanation

The characteristic polynomial of a 2x2 matrix is lambda^2 - tr(A)*lambda + det(A) = 0. Substituting: lambda^2 - 5*lambda + 6 = (lambda - 2)(lambda - 3) = 0. The eigenvalues are lambda = 2 and lambda = 3.

Question 26Differential calculus and function analysis

Calculate the limit lim_{x -> 0} (sin(3x) / (2x)).

  1. 3/2 Correct answer.
  2. 2/3
  3. 0
  4. 1

Explanation

This limit yields the indeterminate form 0/0. Applying L'Hôpital's rule: lim_{x -> 0} (3*cos(3x) / 2) = (3*1) / 2 = 3/2. Alternatively, rewrite as (3/2) * lim_{x -> 0} (sin(3x)/(3x)) = 3/2 * 1 = 3/2.

Question 27Differential calculus and function analysis

Find the derivative of the function f(x) = x^3 * e^(2x).

  1. x^2 * e^(2x) * (3 + 2x) Correct answer.
  2. 3x^2 * e^(2x)
  3. 2x^3 * e^(2x)
  4. x^2 * e^(2x) * (3 + x)

Explanation

Apply the product rule (u*v)' = u'*v + u*v' with u = x^3 and v = e^(2x): f'(x) = (3x^2)*e^(2x) + (x^3)*(2e^(2x)) = x^2 * e^(2x) * (3 + 2x).

Question 28Differential calculus and function analysis

Find the equation of the line tangent to the curve f(x) = x^2 - 4x + 5 at x = 3.

  1. y = 2x - 4 Correct answer.
  2. y = 2x - 2
  3. y = 4x - 10
  4. y = 2x + 2

Explanation

First find the point: f(3) = 3^2 - 4(3) + 5 = 9 - 12 + 5 = 2, so (3, 2). Derivative: f'(x) = 2x - 4, slope m = f'(3) = 2(3) - 4 = 2. Tangent line: y - 2 = 2(x - 3) => y = 2x - 4.

Question 29Differential calculus and function analysis

Find the value of k that makes f(x) = (x^2 - 9)/(x - 3) for x != 3, and f(3) = k, continuous at x = 3.

  1. 6 Correct answer.
  2. 3
  3. 0
  4. 9

Explanation

For f to be continuous at x = 3, we need f(3) = lim_{x -> 3} f(x). Evaluate lim_{x -> 3} (x^2 - 9)/(x - 3) = lim_{x -> 3} ((x-3)(x+3))/(x-3) = lim_{x -> 3} (x+3) = 6. Thus k = 6.

Question 30Differential calculus and function analysis

Calculate the limit lim_{x -> 0^+} x * ln(x).

  1. 0 Correct answer.
  2. -infinity
  3. -1
  4. 1

Explanation

This has indeterminate form 0 * (-infinity). Rewrite as lim_{x -> 0^+} (ln(x) / (1/x)), which is form (-infinity / infinity). Apply L'Hôpital's rule: lim_{x -> 0^+} ((1/x) / (-1/x^2)) = lim_{x -> 0^+} (-x) = 0.

Question 31Differential calculus and function analysis

Find the local (relative) minimum of the function f(x) = x^3 - 3x + 2.

  1. Local minimum at (1, 0) Correct answer.
  2. Local minimum at (-1, 4)
  3. Local minimum at (0, 2)
  4. Local minimum at (3, 20)

Explanation

Compute f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1) = 0 => critical points at x = 1 and x = -1. Second derivative f''(x) = 6x. Evaluate f''(1) = 6 > 0 (local minimum at x = 1). f(1) = 1 - 3 + 2 = 0, so local minimum is at (1, 0).

Question 32Differential calculus and function analysis

Determine the inflection point of the cubic polynomial f(x) = x^3 - 6x^2 + 9x + 1.

  1. (2, 3) Correct answer.
  2. (1, 5)
  3. (3, 1)
  4. (0, 1)

Explanation

First derivative f'(x) = 3x^2 - 12x + 9. Second derivative f''(x) = 6x - 12 = 0 => x = 2. Since f''(x) changes sign from negative to positive at x = 2, x = 2 is an inflection point. Compute y-value: f(2) = 8 - 24 + 18 + 1 = 3. Point is (2, 3).

Question 33Differential calculus and function analysis

Find the horizontal asymptote of the rational function f(x) = (3x^2 + 5) / (x^2 - 4).

  1. y = 3 Correct answer.
  2. y = 0
  3. x = 2
  4. No horizontal asymptote

Explanation

Take the limit as x -> +-infinity: lim_{x -> infinity} (3x^2 + 5)/(x^2 - 4) = 3/1 = 3. Thus, y = 3 is the horizontal asymptote.

Question 34Differential calculus and function analysis

Find the equation of the oblique (slant) asymptote of f(x) = (2x^2 + 3x - 1) / (x + 1).

  1. y = 2x + 1 Correct answer.
  2. y = 2x + 3
  3. y = 2x - 1
  4. y = x + 2

Explanation

Perform polynomial division (2x^2 + 3x - 1) / (x + 1) = 2x + 1 - 2/(x + 1). As x -> +-infinity, the remainder -2/(x + 1) -> 0. Therefore, the oblique asymptote is y = 2x + 1.

Question 35Differential calculus and function analysis

A rectangular plot of area 800 m^2 is to be fenced on three sides (one long side lies along a river wall and requires no fence). What dimensions (x parallel to river, y perpendicular) minimize fence length?

  1. x = 40 m, y = 20 m Correct answer.
  2. x = 20 m, y = 40 m
  3. x = 28.28 m, y = 28.28 m
  4. x = 50 m, y = 16 m

Explanation

Area x*y = 800 => y = 800/x. Total fence length L = x + 2y = x + 1600/x. Minimize L: L'(x) = 1 - 1600/x^2 = 0 => x^2 = 1600 => x = 40. Then y = 800/40 = 20 m. Dimensions x = 40 m, y = 20 m minimize the fencing.

Question 36Differential calculus and function analysis

The function f(x) = x^2 - 4x + 3 satisfies Rolle's theorem on [1, 3]. Find the value c in (1, 3) guaranteed by the theorem such that f'(c) = 0.

  1. c = 2 Correct answer.
  2. c = 1.5
  3. c = 2.5
  4. c = 0

Explanation

Check hypotheses: f is continuous on [1, 3], differentiable on (1, 3), and f(1) = 0 = f(3). Compute f'(x) = 2x - 4. Setting f'(c) = 2c - 4 = 0 yields c = 2, which lies in (1, 3).

Question 37Differential calculus and function analysis

For f(x) = x^3 on [0, 2], find the value c in (0, 2) guaranteed by Lagrange's Mean Value Theorem such that f'(c) = (f(2) - f(0)) / (2 - 0).

  1. c = 2 / sqrt(3) Correct answer.
  2. c = 1
  3. c = sqrt(2)
  4. c = 4/3

Explanation

Average rate of change is (f(2) - f(0))/(2 - 0) = (8 - 0)/2 = 4. Derivative is f'(x) = 3x^2. Set 3c^2 = 4 => c^2 = 4/3 => c = 2/sqrt(3) approx 1.155, which belongs to (0, 2).

Question 38Differential calculus and function analysis

For what values of a and b is the piecewise function f(x) = a*x^2 + 1 for x <= 1, and 2x + b for x > 1, differentiable everywhere?

  1. a = 1, b = 0 Correct answer.
  2. a = 2, b = -1
  3. a = 1, b = 1
  4. a = 2, b = 1

Explanation

For continuity at x = 1: a(1)^2 + 1 = 2(1) + b => a + 1 = 2 + b => a - b = 1. For differentiability at x = 1: left derivative f'(1^-) = 2a(1) = 2a, right derivative f'(1^+) = 2 => 2a = 2 => a = 1. Substitute a = 1 into a - b = 1 => 1 - b = 1 => b = 0.

Question 39Differential calculus and function analysis

Find the first derivative of f(x) = ln(cos(x)).

  1. -tan(x) Correct answer.
  2. tan(x)
  3. -cot(x)
  4. 1/cos(x)

Explanation

Using the chain rule: f'(x) = (1 / cos(x)) * d/dx(cos(x)) = (1 / cos(x)) * (-sin(x)) = -sin(x)/cos(x) = -tan(x).

Question 40Differential calculus and function analysis

Evaluate the limit lim_{x -> 0} (1 + 2x)^(1/x).

  1. e^2 Correct answer.
  2. e
  3. 1
  4. e^(1/2)

Explanation

This is of indeterminate form 1^infinity. Using the identity lim_{u -> 0} (1 + k*u)^(1/u) = e^k with k = 2 gives the limit e^2. Alternatively, L = exp(lim_{x->0} ln(1+2x)/x) = exp(lim_{x->0} (2/(1+2x))/1) = e^2.

Question 41Differential calculus and function analysis

An open-top box is made from a 12 cm by 12 cm square sheet of cardboard by cutting equal square corners of side x and folding up the flaps. What value of x maximizes volume?

  1. x = 2 cm Correct answer.
  2. x = 3 cm
  3. x = 4 cm
  4. x = 1 cm

Explanation

Volume V(x) = x * (12 - 2x)^2 = 4x^3 - 48x^2 + 144x for 0 < x < 6. Derivative V'(x) = 12x^2 - 96x + 144 = 12(x - 2)(x - 6) = 0. The critical point inside the domain is x = 2 cm. V''(2) = 24(2) - 96 = -48 < 0 (maximum). Maximum volume is V(2) = 128 cm^3.

Question 42Differential calculus and function analysis

Bolzano's Theorem guarantees that the continuous function f(x) = x^3 + 3x - 5 has at least one real root in which interval?

  1. (1, 2) Correct answer.
  2. (0, 1)
  3. (2, 3)
  4. (-1, 0)

Explanation

Evaluate f at interval endpoints: f(1) = 1 + 3 - 5 = -1 < 0, and f(2) = 8 + 6 - 5 = 9 > 0. Since f is continuous on [1, 2] and f(1) * f(2) < 0, Bolzano's theorem guarantees a root c in (1, 2).

Question 43Differential calculus and function analysis

What is the slope of the normal line to the cubic curve y = x^3 at the point (2, 8)?

  1. -1/12 Correct answer.
  2. 12
  3. -12
  4. 1/12

Explanation

Derivative dy/dx = 3x^2. Slope of the tangent at x = 2 is m_t = 3(2)^2 = 12. Slope of the normal line is m_n = -1 / m_t = -1/12.

Question 44Differential calculus and function analysis

On which set of intervals is the function f(x) = x^4 - 4x^3 concave upwards (convex)?

  1. (-infinity, 0) U (2, infinity) Correct answer.
  2. (0, 2)
  3. (2, infinity)
  4. (-infinity, 2)

Explanation

First derivative f'(x) = 4x^3 - 12x^2. Second derivative f''(x) = 12x^2 - 24x = 12x(x - 2). Function is concave upwards where f''(x) > 0, which occurs when x < 0 or x > 2.

Question 45Differential calculus and function analysis

Find the absolute maximum value of f(x) = 2x^3 - 3x^2 - 12x + 1 on the closed interval [-2, 3].

  1. 8 Correct answer.
  2. 1
  3. -19
  4. 20

Explanation

f'(x) = 6x^2 - 6x - 12 = 6(x - 2)(x + 1) = 0 => critical points at x = -1, 2. Evaluate f at endpoints and critical points: f(-2) = -3, f(-1) = 8, f(2) = -19, f(3) = -8. The absolute maximum value is 8 (attained at x = -1).

Question 46Differential calculus and function analysis

Evaluate the limit lim_{x -> infinity} (sqrt(x^2 + 4x) - x).

  1. 2 Correct answer.
  2. 0
  3. infinity
  4. 4

Explanation

Multiply and divide by conjugate: lim_{x -> infinity} ((x^2 + 4x - x^2) / (sqrt(x^2 + 4x) + x)) = lim_{x -> infinity} (4x / (x*sqrt(1 + 4/x) + x)) = 4 / (1 + 1) = 2.

Question 47Differential calculus and function analysis

For the implicit circle x^2 + y^2 = 25, calculate the second derivative y'' at the point (3, 4).

  1. -25/64 Correct answer.
  2. -25/16
  3. -9/16
  4. -16/25

Explanation

Implicit differentiation: 2x + 2y*y' = 0 => y' = -x/y. Differentiate again: y'' = -(1*y - x*y') / y^2 = -(y - x(-x/y)) / y^2 = -(y^2 + x^2) / y^3 = -25 / y^3. At (3, 4), y = 4 so y'' = -25 / 4^3 = -25/64.

Question 48Differential calculus and function analysis

Find f'(1) for the exponential function f(x) = x^x.

  1. 1 Correct answer.
  2. 0
  3. e
  4. 2

Explanation

Use logarithmic differentiation: ln(f(x)) = x * ln(x). Differentiate: f'(x)/f(x) = ln(x) + 1 => f'(x) = x^x * (ln(x) + 1). At x = 1: f'(1) = 1^1 * (ln(1) + 1) = 1 * (0 + 1) = 1.

Question 49Differential calculus and function analysis

The rational function f(x) = (x^2 - 1) / (x^2 - 3x + 2) has a vertical asymptote at:

  1. x = 2 only Correct answer.
  2. x = 1 and x = 2
  3. x = 1 only
  4. No vertical asymptotes

Explanation

Factor numerator and denominator: f(x) = ((x-1)(x+1)) / ((x-1)(x-2)). At x = 1, lim_{x->1} f(x) = 2/(-1) = -2 (hole/removable discontinuity). At x = 2, denominator goes to 0 while numerator is 3 != 0, so lim_{x->2} f(x) = +-infinity. Vertical asymptote exists at x = 2 only.

Question 50Differential calculus and function analysis

A closed cylinder has volume 16*pi cm^3. Find the base radius r that minimizes total surface area.

  1. r = 2 cm Correct answer.
  2. r = 4 cm
  3. r = sqrt(2) cm
  4. r = 1 cm

Explanation

Volume V = pi*r^2*h = 16*pi => h = 16/r^2. Total Surface Area S = 2*pi*r^2 + 2*pi*r*h = 2*pi*r^2 + 32*pi/r. S'(r) = 4*pi*r - 32*pi/r^2 = 0 => 4*pi*r^3 = 32*pi => r^3 = 8 => r = 2 cm.