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Admissions tests / ESAT / Maths 2 / Differentiation and integration

Test standard. 15 questions, 15 marks, about 22 minutes.

ESAT Mathematics 2: Differentiation and integration, set 2

The derivative as a gradient, differentiating powers of x, tangents, normals, stationary points, increasing and decreasing functions, indefinite and definite integration, and areas.

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  • Answer all questions. No calculator.
  • Each question has exactly one correct answer.
  1. 11 mark

    Given that y = x^3 - 5x + 7, find the rate of change of y with respect to x at the instant when x = 3.

    1. A 12
    2. B 19
    3. C 22
    4. D 29
  2. 21 mark

    Given that y = (3x^4 + 2x)/x^2 for x != 0, find dy/dx, simplifying your answer fully.

    1. A 6x - 2/x^2
    2. B 6x + 2/x^2
    3. C 3x^2 - 2/x^2
    4. D 6x - 2/x
  3. 31 mark

    Given that dy/dx = 6x^2 - 4x + 1 for a curve y = f(x), find the value of d2y/dx2 when x = -1.

    1. A -10
    2. B -8
    3. C 11
    4. D -16
  4. 41 mark

    Find the x-coordinate of the point on the curve y = x^2 - 6x + 8 where the tangent is parallel to the line y = 2x - 5.

    1. A 3
    2. B 4
    3. C 2
    4. D 0.5
  5. 51 mark

    Find the gradient of the normal to the curve y = x^3/3 - 2x at the point where x = 3.

    1. A -1/7
    2. B 7
    3. C -7
    4. D 1/7
  6. 61 mark

    The curve y = x^3 - 3x^2 - 9x + 2 has a stationary point at x = 3. By considering the sign of dy/dx just before and just after x = 3, determine whether this is a maximum or a minimum.

    1. A Maximum, since dy/dx changes from negative to positive as x increases through 3.
    2. B Maximum, since dy/dx changes from positive to negative as x increases through 3.
    3. C Minimum, since d2y/dx2 = 0 at x = 3.
    4. D Minimum, since dy/dx changes from negative to positive as x increases through 3.
  7. 71 mark

    Find the set of values of x for which f(x) = 2x^3 - 3x^2 - 12x + 5 is an increasing function.

    1. A -1 < x < 2
    2. B x < -1 or x > 2
    3. C x < -2 or x > 1
    4. D x > 2
  8. 81 mark

    The curve y = x^3 - 4x crosses the x-axis at x = -2, x = 0, and x = 2, and lies above the x-axis for -2 <= x <= 0 and below the x-axis for 0 <= x <= 2. Find the total area enclosed between the curve and the x-axis over -2 <= x <= 2.

    1. A 0
    2. B 4
    3. C 8
    4. D -8
  9. 91 mark

    Find the indefinite integral of f(x) = (x + 3)(x - 2), giving your answer in terms of x.

    1. A x^3/3 + x^2/2 - 6x
    2. B x^3/3 + x^2/2 - 6x + C
    3. C x^3/3 - x^2/2 - 6x + C
    4. D x^3 + x^2/2 - 6x + C
  10. 101 mark

    Evaluate the definite integral of (4x - 3x^2) dx from x = 1 to x = 2.

    1. A -1
    2. B 1
    3. C 0
    4. D -15
  11. 111 mark

    F(x) is an antiderivative of f(x), with F(1) = 3 and the definite integral of f(x) dx from x = 1 to x = 4 equals 15. Find F(4).

    1. A -12
    2. B 45
    3. C 12
    4. D 18
  12. 121 mark

    Given that the definite integral of f(x) dx from x = 3 to x = 8 equals 14, find the definite integral of f(x) dx from x = 8 to x = 3.

    1. A 14
    2. B 0
    3. C -14
    4. D 28
  13. 131 mark

    The table gives values of y = f(x) at four equally spaced values of x, using strips of width h = 4: at x = 0, 4, 8, 12 the values of y are 3, 7, 10, 4 respectively. Use the trapezium rule with all three strips to estimate the definite integral of f(x) dx from x = 0 to x = 12.

    1. A 82
    2. B 48
    3. C 41
    4. D 62
  14. 141 mark

    A curve y = f(x) is concave up (its second derivative is positive) for 1 <= x <= 5. When the trapezium rule is used to estimate the definite integral of f(x) dx over this interval, does it give an overestimate or an underestimate of the true value?

    1. A Underestimate. For a concave-up curve, each chord lies above the curve, so the trapezium bounded by it leaves out a sliver of the true area.
    2. B Overestimate. For a concave-up curve, each chord lies above the curve, so the trapezium bounded by it covers more than the true area.
    3. C Underestimate. The curve is concave up, and a concave-up curve always lies above its chords, so the trapezium rule misses part of the true area.
    4. D Overestimate. The trapezium rule always overestimates a definite integral, regardless of the shape of the curve.
  15. 151 mark

    A curve satisfies dy/dx = 4x - 3x^2 and passes through the point (2, 3). Find the value of y when x = 1.

    1. A 1
    2. B 12
    3. C 18
    4. D 4

Worked solutions

Every question below carries the reasoning, not just the answer. The official material for this test publishes a correct option letter and nothing else.

  1. Question 1Answer: C

    1. The rate of change of y with respect to x is the derivative dy/dx.
    2. Differentiate y = x^3 - 5x + 7 term by term: d/dx(x^3) = 3x^2, d/dx(-5x) = -5, and d/dx(7) = 0, since the derivative of any constant is zero.
    3. So dy/dx = 3x^2 - 5.
    4. Substitute x=3: dy/dx = 3(9) - 5 = 27 - 5 = 22.
    5. So the answer is C.
    • Why not A: Forgets to differentiate the term -5x properly, treating its derivative as if it were still -5x rather than the constant -5, and evaluates 3x^2 - 5x at x=3 to get 27-15=12.
    • Why not B: Confuses the rate of change with the value of y itself, and evaluates y(3) = 27-15+7 = 19 instead of differentiating first.
    • Why not D: Forgets that a constant differentiates to zero, and carries the original constant term +7 through into the derivative unchanged, computing 3x^2-5+7=29.

  2. Question 2Answer: A

    1. Divide each term in the numerator by x^2 before differentiating: y = 3x^4/x^2 + 2x/x^2 = 3x^2 + 2x^-1.
    2. Differentiate term by term: d/dx(3x^2) = 6x.
    3. d/dx(2x^-1) = 2 x (-1) x^(-1-1) = -2x^-2.
    4. So dy/dx = 6x - 2x^-2, which can be written as 6x - 2/x^2, so the answer is A.
    • Why not B: Makes a sign error differentiating the term 2x^-1, forgetting that the negative exponent brings out a negative sign, so computes d/dx(2x^-1) = 2x^-2 instead of -2x^-2, giving 6x + 2/x^2.
    • Why not C: Forgets to differentiate the first term 3x^2 at all, carrying it through unchanged into the derivative, and applies the power rule only to the second term.
    • Why not D: Forgets to reduce the power by one when differentiating 2x^-1, multiplying by the exponent -1 but leaving the power at -1 rather than reducing it to -2, giving -2x^-1 instead of -2x^-2.

  3. Question 3Answer: D

    1. d2y/dx2 is found by differentiating the given first derivative dy/dx = 6x^2 - 4x + 1 a second time.
    2. Differentiate term by term: d/dx(6x^2) = 12x, d/dx(-4x) = -4, and d/dx(1) = 0.
    3. So d2y/dx2 = 12x - 4.
    4. Substitute x=-1: d2y/dx2 = 12(-1) - 4 = -12 - 4 = -16.
    5. So the answer is D.
    • Why not A: Differentiates the term 6x^2 by reducing the power by one but forgetting to multiply by the original power 2, giving 6x instead of 12x, so uses a second derivative of 6x - 4 and evaluates 6(-1)-4=-10.
    • Why not B: Makes a sign error differentiating the term -4x, treating its derivative as +4 instead of -4, so uses a second derivative of 12x + 4 and evaluates 12(-1)+4=-8.
    • Why not C: Confuses which derivative is being asked for, and evaluates the GIVEN first derivative dy/dx = 6x^2-4x+1 directly at x=-1 (getting 6-(-4)+1=11) instead of differentiating a second time to find d2y/dx2.

  4. Question 4Answer: B

    1. Two lines are parallel exactly when they have the same gradient, so the tangent to the curve must have gradient 2, the gradient of y = 2x - 5.
    2. Differentiate the curve: dy/dx = 2x - 6.
    3. Set this equal to the required gradient: 2x - 6 = 2.
    4. Solve: 2x = 8, so x = 4.
    5. So the answer is B.
    • Why not A: Sets dy/dx = 0 to find a stationary point, instead of setting dy/dx equal to the target gradient 2, confusing 'parallel to a given line' with 'gradient zero'.
    • Why not C: Makes a sign error, setting the derivative equal to -2 instead of +2, and solves 2x - 6 = -2 to get x = 2.
    • Why not D: Confuses the line's gradient (2) with its y-intercept (-5), setting the derivative 2x - 6 equal to -5 instead of to 2, and solves 2x - 6 = -5 to get x = 0.5.

  5. Question 5Answer: A

    1. Differentiate to find the gradient function: dy/dx = x^2 - 2.
    2. Substitute x=3: the gradient of the tangent is 3^2 - 2 = 9 - 2 = 7.
    3. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of 7, which is -1/7.
    4. So the answer is A.
    • Why not B: Correctly finds the tangent's gradient as 7, but stops there, giving the tangent's gradient instead of finding the normal at all.
    • Why not C: Takes the negative of the tangent gradient without also taking the reciprocal, using -7 as the gradient of the normal instead of the correct negative reciprocal -1/7.
    • Why not D: Takes the reciprocal of the tangent gradient without also negating it, using 1/7 as the gradient of the normal instead of the correct negative reciprocal -1/7.

  6. Question 6Answer: D

    1. Differentiate: dy/dx = 3x^2 - 6x - 9 = 3(x-3)(x+1).
    2. Just before x=3 (say x=2.9): (x-3) is negative and (x+1) is positive, so dy/dx is negative.
    3. Just after x=3 (say x=3.1): (x-3) is positive and (x+1) is positive, so dy/dx is positive.
    4. Since dy/dx changes from negative to positive as x increases through 3, the stationary point is a minimum.
    5. So the answer is D.
    • Why not A: Correctly identifies that dy/dx changes from negative to positive as x increases through 3, but reverses the classification rule, concluding this indicates a maximum rather than a minimum.
    • Why not B: States the classification rule backwards (treating negative-to-positive as a maximum) and also gets the sign change itself backwards, claiming dy/dx changes from positive to negative through x=3.
    • Why not C: Assumes the second derivative must be zero at every stationary point, rather than actually computing d2y/dx2 = 6x - 6, which equals 6(3)-6=12, not 0, at x=3.

  7. Question 7Answer: B

    1. Differentiate: f'(x) = 6x^2 - 6x - 12.
    2. Factorise: f'(x) = 6(x^2-x-2) = 6(x-2)(x+1).
    3. Find the critical points where f'(x)=0: x=2 and x=-1.
    4. Since the coefficient of x^2 in f'(x) is positive, f'(x) is positive outside its roots, so f'(x) > 0 for x < -1 or x > 2.
    5. A function is increasing where its derivative is positive, so f(x) is increasing for x < -1 or x > 2, and the answer is B.
    • Why not A: Correctly finds the critical points x=-1 and x=2, but selects the interval BETWEEN them, where f'(x) is negative, giving the region where f is decreasing rather than increasing.
    • Why not C: Misreads the signs when factorising f'(x) = 6x^2 - 6x - 12 = 6(x^2-x-2), factorising the quadratic as (x-1)(x+2) instead of the correct (x-2)(x+1), which shifts the critical points to x=1 and x=-2.
    • Why not D: Correctly identifies x=2 as the point beyond which f is increasing, but forgets that a cubic with a positive leading coefficient is also increasing below its smaller critical point, and omits the 'x<-1' branch entirely.

  8. Question 8Answer: C

    1. Find the antiderivative of x^3 - 4x: F(x) = x^4/4 - 2x^2.
    2. For the piece above the axis, -2 <= x <= 0: F(0) - F(-2) = 0 - (4 - 8) = 0 - (-4) = 4.
    3. For the piece below the axis, 0 <= x <= 2: F(2) - F(0) = (4 - 8) - 0 = -4, so the area of this piece is the absolute value, |-4| = 4.
    4. The total area is the sum of the two pieces' areas: 4 + 4 = 8.
    5. So the answer is C.
    • Why not A: Computes the plain definite integral of (x^3-4x) dx from x=-2 to x=2 directly, getting 0 (since the curve is symmetric about the origin over this interval), and reports this signed total as the area, without splitting the interval where the curve changes sign.
    • Why not B: Correctly finds the area of one piece of the region (either the part above the axis or the part below it) but stops there, forgetting that the total area must include both pieces.
    • Why not D: Attempts to use the symmetry of the interval by doubling the signed integral from x=0 to x=2 (2 x (-4) = -8), rather than adding the ABSOLUTE VALUE of each piece's contribution separately.

  9. Question 9Answer: B

    1. Expand the product first: (x+3)(x-2) = x^2 + x - 6.
    2. Integrate term by term using integral of x^n dx = x^(n+1)/(n+1) for n != -1.
    3. Integral of x^2 = x^3/3. Integral of x = x^2/2. Integral of -6 = -6x.
    4. Add the constant of integration, since this is an indefinite integral: x^3/3 + x^2/2 - 6x + C, so the answer is B.
    • Why not A: Correctly expands and integrates but forgets the constant of integration, omitting the required '+C'.
    • Why not C: Misreads the signs of the two factors, expanding (x-3)(x+2) instead of the correct (x+3)(x-2), which gives x^2 - x - 6 instead of x^2 + x - 6, and integrates that instead.
    • Why not D: Integrates the x^2 term by raising the power to x^3 but forgets to divide by the new power 3, so x^2 becomes x^3 instead of x^3/3.

  10. Question 10Answer: A

    1. Find the antiderivative: integral of (4x - 3x^2) dx = 2x^2 - x^3 (the constant of integration cancels in a definite integral).
    2. Evaluate at the upper limit: F(2) = 2(4) - 8 = 8 - 8 = 0.
    3. Evaluate at the lower limit: F(1) = 2(1) - 1 = 2 - 1 = 1.
    4. Subtract: the definite integral is F(2) - F(1) = 0 - 1 = -1, so the answer is A.
    • Why not B: Correctly finds the antiderivative F(x) = 2x^2 - x^3, but subtracts in the wrong order, computing F(1) - F(2) = 1 - 0 = 1 instead of F(2) - F(1).
    • Why not C: Evaluates the antiderivative only at the upper limit, F(2) = 0, and forgets to subtract the value at the lower limit, F(1) = 1.
    • Why not D: Integrates -3x^2 by raising the power to x^3 but forgets to divide by the new power 3, using an antiderivative of 2x^2 - 3x^3 instead of 2x^2 - x^3.

  11. Question 11Answer: D

    1. The Fundamental Theorem of Calculus states that the definite integral of f(x) dx from a to b equals F(b) - F(a).
    2. Here a=1, b=4, F(1)=3, and the integral equals 15.
    3. So 15 = F(4) - 3, which rearranges to F(4) = 15 + 3 = 18.
    4. So the answer is D.
    • Why not A: Rearranges the Fundamental Theorem of Calculus incorrectly as F(4) = F(1) - integral, instead of F(4) = F(1) + integral, giving 3 - 15 = -12.
    • Why not B: Misremembers the Fundamental Theorem of Calculus as a product rather than a sum, multiplying the integral by F(1): 15 x 3 = 45, instead of adding them.
    • Why not C: Subtracts F(1) from the integral rather than adding it, computing 15 - 3 = 12 instead of 15 + 3 = 18.

  12. Question 12Answer: C

    1. Reversing the limits of a definite integral negates its value: the integral from b to a equals the negative of the integral from a to b.
    2. Here the integral from 3 to 8 is 14, so the integral from 8 to 3 is -14.
    3. So the answer is C.
    • Why not A: Assumes that reversing the limits of a definite integral leaves its value unchanged, forgetting that swapping the limits negates the integral.
    • Why not B: Assumes that integrating forward and then backward over the same interval must cancel to zero, rather than recognising that reversing the limits simply negates the single value already given.
    • Why not D: Treats the reversed-limit integral as an ADDITIONAL forward traversal rather than its negative, adding 14 + 14 = 28 instead of negating to get -14.

  13. Question 13Answer: A

    1. The trapezium rule for n strips of width h is: integral ~ (h/2) x [y0 + yn + 2(y1+...+y_(n-1))].
    2. Here h=4, the end ordinates are y0=3 and y3=4, and the interior ordinates are y1=7, y2=10.
    3. Double the sum of the interior ordinates: 2 x (7+10) = 34.
    4. Add the end ordinates: 34 + 3 + 4 = 41.
    5. Multiply by h/2 = 2: estimate = 2 x 41 = 82.
    6. So the answer is A.
    • Why not B: Applies the trapezium rule pattern without doubling the interior ordinates, computing (h/2) x (y0+y1+y2+y3) = 2 x (3+7+10+4) = 48.
    • Why not C: Correctly forms the bracket total y0+y3+2(y1+y2) = 3+4+2(7+10) = 41, but forgets to multiply by the leading factor h/2 = 2, leaving the estimate at half its correct value.
    • Why not D: Doubles the wrong ordinates, applying the doubling to the two end values instead of the two interior ones: (h/2) x (2y0+2y3+y1+y2) = 2 x (6+8+7+10) = 62.

  14. Question 14Answer: B

    1. The trapezium rule approximates the area under a curve by joining points on the curve with straight-line chords and summing the areas of the resulting trapezia.
    2. Whether this over- or under-estimates the true area depends on the concavity of the curve.
    3. For a concave-up curve (f''(x) > 0), the curve bends away from a straight line joining two of its points, so the chord lies above the curve.
    4. Since each trapezium's top edge (the chord) lies above the actual curve, its area is more than the true area under the curve on that strip.
    5. So the trapezium rule overestimates the integral for a concave-up curve, and the answer is B.
    • Why not A: States the correct geometric fact, that a chord on a concave-up curve lies above the curve, but then draws the opposite conclusion: a trapezium whose top edge lies above the curve encloses MORE area than the true region, not less, so this is an overestimate, not an underestimate.
    • Why not C: Reverses the concavity rule: a concave-up curve lies BELOW its chords (it bows away from them), not above. This option's claim that the curve 'lies above its chords' actually describes a concave-down curve.
    • Why not D: Assumes the trapezium rule has a fixed bias in every case, ignoring that whether it over- or under-estimates the true integral depends entirely on the concavity of the curve.

  15. Question 15Answer: D

    1. Integrate dy/dx with respect to x to find y: y = integral of (4x - 3x^2) dx = 2x^2 - x^3 + C.
    2. Use the given point (2,3) to find C: substitute x=2, y=3: 3 = 2(4) - 8 + C = 8 - 8 + C = C.
    3. So C = 3, and y = 2x^2 - x^3 + 3.
    4. Substitute x=1: y = 2(1) - 1 + 3 = 2 - 1 + 3 = 4.
    5. So the answer is D.
    • Why not A: Correctly integrates dy/dx = 4x - 3x^2 to y = 2x^2 - x^3 + C, but never uses the given point to find C, effectively leaving the constant of integration as 0.
    • Why not B: Swaps the coordinates of the given point, substituting x=3, y=2 instead of x=2, y=3 when solving for C, which gives C=11 instead of the correct C=3.
    • Why not C: Integrates -3x^2 by raising the power to x^3 but forgets to divide by the new power 3, using an antiderivative of 2x^2 - 3x^3 + C instead of 2x^2 - x^3 + C, and finds C accordingly.

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