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Calculus

Each subtopic includes About section, revision page link, 10 preview questions, and practice CTAs.

Limits, Continuity and Differentiation from First Principles

Subtopic

Limits, Continuity and Differentiation from First Principles under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Determine the limit: lim⁡x→0x3x\lim_{x \to 0} \frac{x^3}{x}.

    A.

    0

    B.

    1

    C.

    ∞\infty

    D.

    Undefined

  2. 2.

    Evaluate lim⁡x→∞(1+1x)\lim_{x \to \infty} (1 + \frac{1}{x}).

    A.

    0

    B.

    1

    C.

    ee

    D.

    ∞\infty

  3. 3.

    Find lim⁡x→1x10−1x−1\lim_{x \to 1} \frac{x^{10} - 1}{x - 1} using the derivative of x10x^{10} at x=1x=1.

    A.

    1

    B.

    0

    C.

    10

    D.

    9

Download the worksheet for Calculus - Limits, Continuity and Differentiation from First Principles to practice offline. It includes additional chapter-level practice questions.

Rules of Differentiation (Product, Quotient, Chain)

Subtopic

Rules of Differentiation (Product, Quotient, Chain) under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the derivative of y=e4xsin⁡xy = e^{4x} \sin x.

    A.

    e4x(4sin⁡x+cos⁡x)e^{4x}(4\sin x + \cos x)

    B.

    e4x(4cos⁡x+sin⁡x)e^{4x}(4\cos x + \sin x)

    C.

    4e4xcos⁡x4e^{4x} \cos x

    D.

    e4x(sin⁡x+cos⁡x)e^{4x}(\sin x + \cos x)

  2. 2.

    Differentiate f(x)=4xx2+1f(x) = \frac{4x}{x^2 + 1}.

    A.

    4−4x2(x2+1)2\frac{4-4x^2}{(x^2+1)^2}

    B.

    4x2−4(x2+1)2\frac{4x^2-4}{(x^2+1)^2}

    C.

    4(x2+1)2\frac{4}{(x^2+1)^2}

    D.

    8x2(x2+1)2\frac{8x^2}{(x^2+1)^2}

  3. 3.

    Find dydx\frac{dy}{dx} if y=x2tan⁡xy = x^2 \tan x.

    A.

    2xtan⁡x+x2sec⁡2x2x \tan x + x^2 \sec^2 x

    B.

    2xsec⁡2x2x \sec^2 x

    C.

    2xtan⁡x−x2sec⁡2x2x \tan x - x^2 \sec^2 x

    D.

    x2sec⁡2xx^2 \sec^2 x

Download the worksheet for Calculus - Rules of Differentiation (Product, Quotient, Chain) to practice offline. It includes additional chapter-level practice questions.

Implicit Differentiation and Higher Derivatives

Subtopic

Implicit Differentiation and Higher Derivatives under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Calculate y′′y'' for y=13x3+5xy = \frac{1}{3}x^3 + 5x.

    A.

    x2+5x^2 + 5

    B.

    2x+52x + 5

    C.

    2x2x

    D.

    xx

  2. 2.

    Find dydx\frac{dy}{dx} for the curve y2=x3y^2 = x^3.

    A.

    dydx=3x22y\frac{dy}{dx} = \frac{3x^2}{2y}

    B.

    dydx=2y3x2\frac{dy}{dx} = \frac{2y}{3x^2}

    C.

    dydx=3x2y\frac{dy}{dx} = \frac{3x}{2y}

    D.

    dydx=3x2\frac{dy}{dx} = 3x^2

  3. 3.

    Differentiate 3x2+2y2=63x^2 + 2y^2 = 6 implicitly to find dydx\frac{dy}{dx}.

    A.

    dydx=−3x2y\frac{dy}{dx} = -\frac{3x}{2y}

    B.

    dydx=−2x3y\frac{dy}{dx} = -\frac{2x}{3y}

    C.

    dydx=3x2y\frac{dy}{dx} = \frac{3x}{2y}

    D.

    dydx=−3xy\frac{dy}{dx} = -\frac{3x}{y}

Download the worksheet for Calculus - Implicit Differentiation and Higher Derivatives to practice offline. It includes additional chapter-level practice questions.

Applications of Differentiation (Tangents, Normals, Optimization, Kinematics)

Subtopic

Applications of Differentiation (Tangents, Normals, Optimization, Kinematics) under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    What is the gradient of the tangent to y=x4y = x^4 at the point (1,1)(1, 1)?

    A.

    1

    B.

    2

    C.

    3

    D.

    4

  2. 2.

    Find the gradient of the curve y=cos⁡(2x)y = \cos(2x) at x=0x = 0.

    A.

    0

    B.

    1

    C.

    2

    D.

    -2

  3. 3.

    The displacement of a particle is s(t)=1ts(t) = \frac{1}{t} for t>0t > 0. Find the velocity at t=1t = 1.

    A.

    1

    B.

    -1

    C.

    0

    D.

    -0.5

Download the worksheet for Calculus - Applications of Differentiation (Tangents, Normals, Optimization, Kinematics) to practice offline. It includes additional chapter-level practice questions.

Indefinite and Definite Integration

Subtopic

Indefinite and Definite Integration under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the constant of integration CC for ∫2xdx\int 2x dx if the integral passes through (1,5)(1, 5).

    A.

    4

    B.

    5

    C.

    1

    D.

    0

  2. 2.

    Evaluate ∫(2x3)dx\int (2x^3) dx.

    A.

    12x4+C\frac{1}{2}x^4 + C

    B.

    6x2+C6x^2 + C

    C.

    x4+Cx^4 + C

    D.

    23x4+C\frac{2}{3}x^4 + C

  3. 3.

    Calculate ∫5e0.5xdx\int 5e^{0.5x} dx.

    A.

    2.5e0.5x+C2.5e^{0.5x} + C

    B.

    10e0.5x+C10e^{0.5x} + C

    C.

    5e0.5x+C5e^{0.5x} + C

    D.

    10ex+C10e^x + C

Download the worksheet for Calculus - Indefinite and Definite Integration to practice offline. It includes additional chapter-level practice questions.

Techniques of Integration (Substitution, By Parts)

Subtopic

Techniques of Integration (Substitution, By Parts) under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find ∫xsin⁡(x2)dx\int x \sin(x^2) dx using the substitution u=x2u = x^2.

    A.

    12cos⁡(x2)+C\frac{1}{2} \cos(x^2) + C

    B.

    −12cos⁡(x2)+C-\frac{1}{2} \cos(x^2) + C

    C.

    −2cos⁡(x2)+C-2 \cos(x^2) + C

    D.

    cos⁡(x2)+C\cos(x^2) + C

  2. 2.

    Evaluate ∫12x+5dx\int \frac{1}{\sqrt{2x+5}} dx.

    A.

    2x+5+C\sqrt{2x+5} + C

    B.

    22x+5+C2\sqrt{2x+5} + C

    C.

    122x+5+C\frac{1}{2}\sqrt{2x+5} + C

    D.

    (2x+5)3/2+C(2x+5)^{3/2} + C

  3. 3.

    Find ∫x2ex3dx\int x^2 e^{x^3} dx.

    A.

    3ex3+C3 e^{x^3} + C

    B.

    ex3+Ce^{x^3} + C

    C.

    x3ex3+Cx^3 e^{x^3} + C

    D.

    13ex3+C\frac{1}{3} e^{x^3} + C

Download the worksheet for Calculus - Techniques of Integration (Substitution, By Parts) to practice offline. It includes additional chapter-level practice questions.

Applications of Integration (Area between curves, Volumes of Revolution)

Subtopic

Applications of Integration (Area between curves, Volumes of Revolution) under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the area of the region bounded by y=x2−2xy = x^2 - 2x and the xx-axis from x=2x = 2 to x=3x = 3.

    A.

    43\frac{4}{3}

    B.

    23\frac{2}{3}

    C.

    11

    D.

    53\frac{5}{3}

  2. 2.

    Calculate the volume of the solid formed by revolving y=xxy = x\sqrt{x} about the xx-axis from x=0x = 0 to x=1x = 1.

    A.

    π4\frac{\pi}{4}

    B.

    π3\frac{\pi}{3}

    C.

    π2\frac{\pi}{2}

    D.

    π\pi

  3. 3.

    Find the area of the region bounded by y=2xy = 2x, the xx-axis, x=1x = 1, and x=4x = 4.

    A.

    1515

    B.

    1616

    C.

    1717

    D.

    1414

Download the worksheet for Calculus - Applications of Integration (Area between curves, Volumes of Revolution) to practice offline. It includes additional chapter-level practice questions.

Differential Equations (Separation of Variables)

Subtopic

Differential Equations (Separation of Variables) under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the general solution of dydx=4x\frac{dy}{dx} = \frac{4}{x} for x>0x > 0.

    A.

    y=4ln⁡x+Cy = 4\ln x + C

    B.

    y=ln⁡(x4)+Cy = \ln(x^4) + C

    C.

    y=−4x2+Cy = -\frac{4}{x^2} + C

    D.

    Both A and B are correct

  2. 2.

    What is the general solution of dydx=9x2\frac{dy}{dx} = 9x^2?

    A.

    y=3x3+Cy = 3x^3 + C

    B.

    y=18x+Cy = 18x + C

    C.

    y=x3+Cy = x^3 + C

    D.

    y=9x3+Cy = 9x^3 + C

  3. 3.

    Find the general solution of dydx=csc⁡2x\frac{dy}{dx} = \csc^2 x.

    A.

    y=−cot⁡x+Cy = -\cot x + C

    B.

    y=cot⁡x+Cy = \cot x + C

    C.

    y=tan⁡x+Cy = \tan x + C

    D.

    y=−sin⁡x+Cy = -\sin x + C

Download the worksheet for Calculus - Differential Equations (Separation of Variables) to practice offline. It includes additional chapter-level practice questions.

Maclaurin and Taylor Series

Subtopic

Maclaurin and Taylor Series under Calculus for Grade 12 IB.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    A Maclaurin series is simply a Taylor series centered at which value of xx?

    A.

    x=1x = 1

    B.

    x=ex = e

    C.

    x=0x = 0

    D.

    x=πx = \pi

  2. 2.

    The linear approximation of f(x)=sin⁡(x)f(x) = \sin(x) near x=0x=0 is:

    A.

    f(x)≈xf(x) \approx x

    B.

    f(x)≈1f(x) \approx 1

    C.

    f(x)≈1−xf(x) \approx 1 - x

    D.

    f(x)≈0f(x) \approx 0

  3. 3.

    If f(x)=x3+x2+x+1f(x) = x^3 + x^2 + x + 1, what is the value of f′′′(0)f'''(0)?

    A.

    11

    B.

    33

    C.

    66

    D.

    00

Download the worksheet for Calculus - Maclaurin and Taylor Series to practice offline. It includes additional chapter-level practice questions.

Calculus - IB Grade 12 Maths Notes & Revision | Krit.club