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Topic-based questions, mock exams and step-by-step solutions for UBC MATH 100, first-year differential calculus. Free, no account needed.

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LimitsIntroductory2 min

Evaluate the limit.

lim⁡x→1x3−1x−1\lim_{x \to 1} \frac{x^{3} - 1}{x - 1}
Start practisingTry a mock midterm

473 questions in 23 topics, and 56 mock papers.

What do you want to practise?

In course order, by the test they come before. 473 questions across them, each with hints and a worked solution.

Before the midterm

Practise a mixed set for the midterm
  1. Power functions and first sketches
  2. Functions, graphs and inverses
  3. Limits
  4. Continuity
  5. The definition of the derivative
  6. Differentiation rules
  7. Exponential and trigonometric functions
  8. The chain rule and logarithms
  9. Implicit differentiation and inverse trigonometric functions
  10. Related rates

Before Quiz 2

Practise a mixed set for Quiz 2
  1. L’Hôpital’s rule
  2. Curve sketching

Before the final

  1. Optimization
  2. Linear approximation
  3. Taylor polynomials
  4. Newton’s methodflavours A and B
  5. Exponential growth and decayflavours A and B
  6. Slope fields and Euler’s methodflavours A and B
  7. Logistic growth and phase linesflavour B
  8. Functions of two variables and partial derivativesflavour C
  9. Optimization in two variables and Lagrange multipliersflavour C

Also in the library, though the course does not examine them: the Mean Value Theorem and antiderivatives.

  1. Every step has a reason

    A worked solution starts from the plan, gives the reason for each line beside it, and ends on the mistake the question is built to catch.

    Practise the chain rule

    Let y=(3x2+1)5y = \bigl(3x^{2} + 1\bigr)^{5}. Find dydx\dfrac{dy}{dx} at x=1x = 1.

    Strategy

    Differentiate the outside function with the inside left alone, then multiply by the derivative of the inside. Substitute only at the end.

    Working

    1. dydx=5(3x2+1)4⋅ddx(3x2+1)\frac{dy}{dx} = 5\bigl(3x^{2} + 1\bigr)^{4} \cdot \frac{d}{dx}\bigl(3x^{2} + 1\bigr)

      Outer derivative first. The bracket is carried through unchanged.

    2. =5(3x2+1)4⋅6x=30x(3x2+1)4= 5\bigl(3x^{2} + 1\bigr)^{4} \cdot 6x = 30x\bigl(3x^{2} + 1\bigr)^{4}

      The inner derivative is 6x. Multiply the constants.

    3. 3(1)2+1=43(1)^{2} + 1 = 4

      Evaluate the inside at x = 1 before raising to the fourth power.

    4. dydx∣x=1=30(1)(4)4=30⋅256=7680\left.\frac{dy}{dx}\right|_{x=1} = 30(1)(4)^{4} = 30 \cdot 256 = 7680

      Substitute and compute. 44 = 256.

    Answer

    dydx∣x=1=7680\left.\frac{dy}{dx}\right|_{x=1} = 7680

    The answer is 7680.

    Where this goes wrong

    Stopping at 5(3x² + 1)4 and forgetting the inner derivative, which gives 1280. The chain rule is a product of two derivatives, and the second one is easy to leave out.

  2. See it again

    A question you miss comes back the next day, then 3 days later, then 7 days after that. A slip on any of those days starts it over.

    See what is due

    1. Day 1: missed, so it goes into review.
    2. Day 2: back, and answered cleanly.
    3. Day 5: back, and answered cleanly.
    4. Day 12: back, and answered cleanly. It leaves review.

  3. Sit a paper first

    Mock papers for the midterm, Quiz 2 and the final, from questions kept out of the practice library. After one, see your marks by topic and every solution.

    All 56 papers

    Midterm

    Everything up to the midterm, set out like a written test: five short answers, then three longer problems in parts, on limits and continuity, on differentiation, and on related rates. 10 papers, each 14 questions, 28 marks and 60 minutes, timed or untimed.

    Also publishedMidterm skills checkQuiz 2Final, flavour AFinal, flavour BFinal, flavour CPrecalculus check

  4. Pick up a pen

    Work each question on paper and type only the answer.

    Start practising