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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 1

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

    Find dy/dx given that y = 4x^3 - 6x^2 + 5x - 7.

    1. A 12x^3 - 12x^2 + 5
    2. B 12x^2 - 12x + 5
    3. C 12x^2 - 12x - 7
    4. D 4x^2 - 6x + 5
  2. 21 mark

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

    1. A (3/2)x
    2. B 1 - 2/x^3
    3. C 1 - 4/x^3
    4. D 1 - 4/x
  3. 31 mark

    Given that y = x^4 - 3x^2, find the value of d2y/dx2 when x = 2.

    1. A 42
    2. B 20
    3. C 54
    4. D 36
  4. 41 mark

    Find the equation of the tangent to the curve y = x^2 - 4x + 1 at the point where x = 3.

    1. A y = -x/2 - 1/2
    2. B y = 2x - 9
    3. C y = -2x + 4
    4. D y = 2x - 8
  5. 51 mark

    Find the equation of the normal to the curve y = x^3 - x at the point where x = 1.

    1. A y = -x/2 + 1/2
    2. B y = 2x - 2
    3. C y = -2x + 2
    4. D y = x/2 - 1/2
  6. 61 mark

    The curve y = x^3 - 12x + 3 has a stationary point at x = 2. Use the second derivative to determine whether this stationary point is a maximum or a minimum.

    1. A Maximum, since d2y/dx2 = 12 which is positive.
    2. B Minimum, since d2y/dx2 = 12 which is positive.
    3. C Maximum, since dy/dx = 0 confirms a turning point.
    4. D Minimum, since d2y/dx2 = 6.
  7. 71 mark

    Find the set of values of x for which f(x) = x^3 - 6x^2 + 5 is a decreasing function.

    1. A x < 0 or x > 4
    2. B 0 < x < 12
    3. C 0 < x < 2
    4. D 0 < x < 4
  8. 81 mark

    Find the indefinite integral of f(x) = 6x^2 - 4/x^3 + 3, giving your answer in terms of x.

    1. A 2x^3 + 2/x^2 + 3x
    2. B 6x^3 + 2/x^2 + 3x + C
    3. C 2x^3 + 2/x^2 + 3x + C
    4. D 2x^3 - 4/x^2 + 3x + C
  9. 91 mark

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

    1. A 14
    2. B -14
    3. C 52
    4. D 18
  10. 101 mark

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

    1. A -32/3
    2. B 32/3
    3. C 16/3
    4. D 64/3
  11. 111 mark

    F(x) is an antiderivative of f(x), with F(2) = 5 and F(6) = 17. Find the value of the definite integral of f(x) dx from x = 2 to x = 6.

    1. A -12
    2. B 22
    3. C 12
    4. D 3
  12. 121 mark

    Given that the definite integral of f(x) dx from x = 1 to x = 5 equals 20, and the definite integral of f(x) dx from x = 5 to x = 9 equals 11, find the definite integral of f(x) dx from x = 1 to x = 9.

    1. A 9
    2. B 220
    3. C -9
    4. D 31
  13. 131 mark

    The table gives values of y = f(x) at five equally spaced values of x, using strips of width h = 1: at x = 0, 1, 2, 3, 4 the values of y are 2, 5, 8, 7, 4 respectively. Use the trapezium rule with all four strips to estimate the definite integral of f(x) dx from x = 0 to x = 4.

    1. A 23
    2. B 13
    3. C 46
    4. D 16
  14. 141 mark

    A curve y = f(x) is concave down (its second derivative is negative) for 0 <= x <= 4. 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 Overestimate. For a concave-down curve, each chord lies below the curve, so the trapezium beneath it covers more than the true area.
    2. B Underestimate. For a concave-down curve, each chord lies below the curve, so the trapezium beneath it leaves out a sliver of the true area.
    3. C Overestimate. The curve is concave down, and a concave-down curve always lies below its chords, so the trapezium rule includes extra area beyond the curve.
    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 = 3x^2 - 4x and passes through the point (2, 5). Find y in terms of x.

    1. A y = x^3 - 4x^2 + 13
    2. B y = x^3 - 2x^2 - 73
    3. C y = x^3 - 2x^2 + 5
    4. D y = x^3 - 2x^2

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: B

    1. Differentiate each term of y = 4x^3 - 6x^2 + 5x - 7 separately, using the rule d/dx(x^n) = n x^(n-1).
    2. The term 4x^3 differentiates to 3 x 4 x^(3-1) = 12x^2.
    3. The term -6x^2 differentiates to 2 x (-6) x^(2-1) = -12x.
    4. The term 5x differentiates to 5, since d/dx(x) = 1.
    5. The constant term -7 differentiates to 0, since the derivative of any constant is zero.
    6. Combining these gives dy/dx = 12x^2 - 12x + 5, so the answer is B.
    • Why not A: Multiplies each term by its exponent but forgets to reduce the power by one afterwards, so 4x^3 stays as 12x^3 instead of becoming 12x^2, and likewise -6x^2 stays as -12x^2 instead of becoming -12x.
    • Why not C: Correctly differentiates the first three terms but treats the constant term -7 as if it were unaffected by differentiation, instead of recognising that the derivative of any constant is zero.
    • Why not D: Reduces the power of each term by one but forgets to multiply by the original power at all, so 4x^3 becomes 4x^2 instead of 12x^2, and -6x^2 becomes -6x instead of -12x.

  2. Question 2Answer: C

    1. Divide each term in the numerator by x^2 before differentiating: y = x^3/x^2 + 2/x^2 = x + 2x^-2.
    2. Differentiate term by term: d/dx(x) = 1.
    3. d/dx(2x^-2) = 2 x (-2) x^(-2-1) = -4x^-3.
    4. So dy/dx = 1 - 4x^-3, which can be written as 1 - 4/x^3, so the answer is C.
    • Why not A: Attempts to differentiate the numerator and denominator separately and divide the results, treating d/dx as if it could be applied to the top and bottom of a fraction independently: 3x^2 divided by 2x gives (3/2)x. This is not a valid differentiation rule.
    • Why not B: Simplifies correctly to y = x + 2x^-2, but when differentiating 2x^-2 forgets to multiply by the existing coefficient 2, applying only the exponent rule to get -2x^-3 instead of -4x^-3.
    • Why not D: Simplifies correctly to y = x + 2x^-2, but when reducing the power for the derivative subtracts incorrectly, turning x^-2 into x^-1 instead of x^-3, so 2x^-2 differentiates to -4x^-1 instead of -4x^-3.

  3. Question 3Answer: A

    1. Differentiate y = x^4 - 3x^2 to find the first derivative: dy/dx = 4x^3 - 6x.
    2. Differentiate again to find the second derivative: d2y/dx2 = 12x^2 - 6.
    3. Substitute x=2: d2y/dx2 = 12(2)^2 - 6 = 12(4) - 6 = 48 - 6 = 42.
    4. So the answer is A.
    • Why not B: Evaluates the first derivative dy/dx = 4x^3 - 6x at x=2 (giving 32-12=20) instead of the second derivative, confusing the order of differentiation asked for.
    • Why not C: Makes a sign error substituting into the second derivative 12x^2 - 6, adding the 6 instead of subtracting it: 12(2)^2 + 6 = 54.
    • Why not D: Differentiates the -6x term of dy/dx incorrectly, leaving it unchanged instead of reducing it to the constant -6, giving a second derivative of 12x^2 - 6x and evaluating that at x=2 to get 48-12=36.

  4. Question 4Answer: D

    1. Find the y-coordinate at x=3: y = 3^2 - 4(3) + 1 = 9 - 12 + 1 = -2, giving the point (3,-2).
    2. Differentiate to find the gradient function: dy/dx = 2x - 4.
    3. Substitute x=3 into the gradient function: gradient = 2(3) - 4 = 2.
    4. Use y - y1 = m(x - x1) with m=2 and (x1,y1)=(3,-2): y - (-2) = 2(x-3), so y = 2x - 6 - 2 = 2x - 8.
    5. So the answer is D.
    • Why not A: Correctly finds the point (3,-2) and the tangent gradient 2, but then uses the gradient of the normal (the negative reciprocal, -1/2) instead of the tangent gradient itself when forming the line equation.
    • Why not B: Uses the correct gradient of 2, but miscalculates the y-coordinate at x=3 as -3 instead of -2 (an arithmetic slip evaluating 9-12+1), and carries that wrong point through the line equation.
    • Why not C: Uses the correct point (3,-2), but mistakes the y-coordinate itself, -2, for the gradient of the tangent, instead of differentiating the function to find the true gradient of 2.

  5. Question 5Answer: A

    1. Find the y-coordinate at x=1: y = 1^3 - 1 = 0, so the point is (1,0).
    2. Differentiate to find the gradient function: dy/dx = 3x^2 - 1.
    3. Substitute x=1: the gradient of the tangent is 3(1)^2 - 1 = 2.
    4. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of 2, which is -1/2.
    5. Using y - y1 = m(x - x1) with m=-1/2 and (x1,y1)=(1,0): y = -1/2(x-1) = -x/2 + 1/2.
    6. So the answer is A.
    • Why not B: Finds the tangent gradient correctly as 2 but forgets to find the normal at all, writing the equation of the tangent line, y=2x-2, instead of the normal.
    • Why not C: Takes the negative of the tangent gradient without also taking the reciprocal, using -2 as the gradient of the normal instead of the correct negative reciprocal -1/2.
    • Why not D: Takes the reciprocal of the tangent gradient without also negating it, using 1/2 as the gradient of the normal instead of the correct negative reciprocal -1/2.

  6. Question 6Answer: B

    1. Confirm the stationary point: dy/dx = 3x^2 - 12, and at x=2, dy/dx = 3(4) - 12 = 12 - 12 = 0, confirming x=2 is a stationary point.
    2. Differentiate again to get the second derivative: d2y/dx2 = 6x.
    3. Substitute x=2: d2y/dx2 = 6(2) = 12.
    4. Since d2y/dx2 is positive, the stationary point is a minimum (the second derivative test: positive means minimum, negative means maximum).
    5. So the answer is B.
    • Why not A: Correctly evaluates d2y/dx2 = 12 at x=2, but reverses the second derivative test, treating a positive value as indicating a maximum rather than a minimum.
    • Why not C: Confuses the condition that identifies a stationary point (dy/dx = 0) with the separate test that classifies it, and concludes it must be a maximum without ever checking the second derivative.
    • Why not D: Substitutes x=2 into d2y/dx2 = 6x incorrectly, writing down the coefficient 6 rather than evaluating 6 x 2 = 12.

  7. Question 7Answer: D

    1. Differentiate: f'(x) = 3x^2 - 12x.
    2. Factorise: f'(x) = 3x(x - 4).
    3. Find the critical points where f'(x)=0: x=0 and x=4.
    4. Since the coefficient of x^2 in f'(x) is positive, f'(x) is negative between its roots, so f'(x) < 0 for 0 < x < 4.
    5. A function is decreasing where its derivative is negative, so f(x) is decreasing for 0 < x < 4, and the answer is D.
    • Why not A: Correctly finds the critical points x=0 and x=4 by solving f'(x)=3x(x-4)=0, but selects the interval where f'(x) is positive rather than negative, giving the region where the function is increasing instead of decreasing.
    • Why not B: Makes an error factorising f'(x) = 3x^2 - 12x = 3x(x-4), instead dividing the -12x term by x alone rather than by 3x, wrongly obtaining a second critical point of x=12.
    • Why not C: Differentiates -6x^2 incorrectly as -6x rather than -12x, giving f'(x) = 3x^2 - 6x = 3x(x-2), which shifts the second critical point to x=2 instead of x=4.

  8. Question 8Answer: C

    1. Rewrite 4/x^3 as 4x^-3 so every term is a power of x: f(x) = 6x^2 - 4x^-3 + 3.
    2. Integrate term by term using integral of x^n dx = x^(n+1)/(n+1) for n != -1.
    3. Integral of 6x^2 = 6 x x^3/3 = 2x^3.
    4. Integral of -4x^-3 = -4 x x^-2/(-2) = 2x^-2, which is 2/x^2.
    5. Integral of 3 = 3x.
    6. Add the constant of integration, since this is an indefinite integral: 2x^3 + 2/x^2 + 3x + C, so the answer is C.
    • Why not A: Correctly finds the antiderivative but omits the constant of integration, forgetting that any indefinite integral must include a '+C'.
    • Why not B: Integrates 6x^2 by raising the power to x^3 but forgets to divide by the new power 3, so 6x^2 becomes 6x^3 instead of 2x^3.
    • Why not D: Integrates -4x^-3 by raising the power to x^-2 but forgets to divide by the new power -2, so -4x^-3 becomes -4x^-2 instead of 2x^-2.

  9. Question 9Answer: A

    1. Find the antiderivative: integral of (3x^2 - 2x) dx = x^3 - x^2 (the constant of integration cancels in a definite integral).
    2. Evaluate at the upper limit: F(3) = 3^3 - 3^2 = 27 - 9 = 18.
    3. Evaluate at the lower limit: F(2) = 2^3 - 2^2 = 8 - 4 = 4.
    4. Subtract: the definite integral is F(3) - F(2) = 18 - 4 = 14, so the answer is A.
    • Why not B: Evaluates the antiderivative correctly as F(x) = x^3 - x^2, but subtracts in the wrong order, computing F(2) - F(3) = 4 - 18 = -14 instead of F(3) - F(2).
    • 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 3x^3 - x^2, which evaluates to 72 - 20 = 52.
    • Why not D: Evaluates the antiderivative only at the upper limit, F(3) = 18, and forgets to subtract the value at the lower limit, F(2) = 4.

  10. Question 10Answer: B

    1. Find the antiderivative of x^2 - 4: F(x) = x^3/3 - 4x.
    2. Evaluate at the upper limit: F(2) = 8/3 - 8 = -16/3.
    3. Evaluate at the lower limit: F(-2) = -8/3 + 8 = 16/3.
    4. The definite integral is F(2) - F(-2) = -16/3 - 16/3 = -32/3.
    5. Since the curve lies below the x-axis on this interval, the integral is negative, but area must be positive, so the area is the absolute value: |-32/3| = 32/3.
    6. So the answer is B.
    • Why not A: Correctly evaluates the definite integral as -32/3, but reports this signed value directly as the area, forgetting that a curve lying below the x-axis gives a negative integral while the area itself must be positive.
    • Why not C: Forgets to integrate the constant term -4 at all, treating it as if it had no antiderivative, and so uses only x^3/3 as the antiderivative instead of x^3/3 - 4x.
    • Why not D: Correctly finds the area as 32/3, but then doubles it unnecessarily, mistakenly treating the interval -2 to 2 as if only half of it had been used and needed to be doubled by symmetry.

  11. Question 11Answer: C

    1. The Fundamental Theorem of Calculus states that if F is an antiderivative of f, then the definite integral of f(x) dx from a to b equals F(b) - F(a).
    2. Here a=2, b=6, F(2)=5 and F(6)=17.
    3. So the definite integral = F(6) - F(2) = 17 - 5 = 12.
    4. So the answer is C.
    • Why not A: Computes F(2) - F(6) = 5 - 17 = -12 instead of F(6) - F(2), subtracting in the wrong order.
    • Why not B: Misremembers the Fundamental Theorem of Calculus as a sum rather than a difference, adding F(6) and F(2) to get 22 instead of subtracting.
    • Why not D: Divides the difference F(6) - F(2) = 12 by the length of the interval, 6 - 2 = 4, confusing the theorem with a calculation for an average rate of change: 12/4 = 3.

  12. Question 12Answer: D

    1. The intervals [1,5] and [5,9] are contiguous, since they share the endpoint x=5, and together make up [1,9].
    2. Integrals over contiguous ranges combine by addition: integral from 1 to 9 = integral from 1 to 5 + integral from 5 to 9.
    3. = 20 + 11 = 31.
    4. So the answer is D.
    • Why not A: Subtracts the two contiguous-range integrals instead of adding them, computing 20 - 11 = 9.
    • Why not B: Multiplies the two integrals together instead of adding them, computing 20 x 11 = 220, as if combining integrals over contiguous ranges worked like a product rule.
    • Why not C: Subtracts the two integrals in the other order, computing 11 - 20 = -9.

  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=1, the end ordinates are y0=2 and y4=4, and the interior ordinates are y1=5, y2=8, y3=7.
    3. Double the sum of the interior ordinates: 2 x (5+8+7) = 2 x 20 = 40.
    4. Add the end ordinates: 40 + 2 + 4 = 46.
    5. Multiply by h/2 = 1/2: estimate = 0.5 x 46 = 23.
    6. So the answer is A.
    • Why not B: Uses the trapezium rule pattern without doubling the interior ordinates, computing 1/2 x (2+4+5+8+7) = 13 instead of doubling the values between the two ends.
    • Why not C: Correctly sums the ordinates as 2+4+2(5+8+7)=46, but forgets to multiply by the leading factor h/2, giving double the correct estimate.
    • Why not D: Doubles the wrong ordinates, applying the doubling to the two end values instead of the three interior ones: 1/2 x (2(2)+2(4)+5+8+7) = 16.

  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-down curve (f''(x) < 0), the curve bends away from a straight line joining two of its points, so the chord lies below the curve.
    4. Since each trapezium's top edge (the chord) lies below the actual curve, its area is less than the true area under the curve on that strip.
    5. So the trapezium rule underestimates the integral for a concave-down curve, and the answer is B.
    • Why not A: States the correct geometric fact, that the chord lies below a concave-down curve, but then draws the opposite conclusion: a trapezium that sits below the curve leaves out area, giving an underestimate, not an overestimate.
    • Why not C: Reverses the concavity rule: it is a concave-UP curve whose chords lie above the curve. For a concave-down curve the chord lies below it, which is the opposite of what this option claims.
    • 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: C

    1. Integrate dy/dx with respect to x to find y: y = integral of (3x^2 - 4x) dx = x^3 - 2x^2 + C.
    2. Use the given point (2,5) to find C: substitute x=2, y=5: 5 = (2)^3 - 2(2)^2 + C = 8 - 8 + C = 0 + C.
    3. So C = 5.
    4. The particular solution is y = x^3 - 2x^2 + 5, so the answer is C.
    • Why not A: Integrates -4x by raising the power to x^2 but forgets to divide by the new power 2, using -4x^2 instead of -2x^2, giving y = x^3 - 4x^2 + C, and then finding C=13 from the point (2,5) using this wrong antiderivative.
    • Why not B: Swaps the coordinates of the given point, substituting x=5 and y=2 instead of x=2 and y=5 when solving for C.
    • Why not D: Correctly integrates to y = x^3 - 2x^2 + C, but never uses the given point to find C, effectively leaving the constant of integration as 0.

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