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University of Toronto · St. George · Independent study guide

MAT148H1: Calculus I with Proofs

Practise turning a calculation into a justified argument. Start with a short exercise on continuity and derivatives, then choose the step you want to work on.

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What to work on

The official scope includes logic, quantifiers, sets, induction, limits, continuity, differentiation and the mean value theorem. Your section’s outline sets the teaching order and assessment scope.

Looking for MAT137Y1? Check the course-code FAQ.

Check continuity and differentiability at a join

Try this after you have met continuity and the derivative definition. It is a focused sample of reasoning, not a test of readiness for the whole course. The exercise is original practice, not a U of T assessment.

Let x be a real input and let a and b be real constants. Define f on all real numbers by:

When x<1x<1: f(x)=ax+bf(x)=ax+b.

f(1)=3f(1)=3.

When x>1x>1: f(x)=x2+2f(x)=x^2+2.

  1. Find all pairs (a,b)(a,b) for which f is continuous at 1.
  2. Among those pairs, find every pair for which f is differentiable at 1. Use the difference quotient to justify your answer.
Show a hint

compare the two one-sided function limits with f(1)f(1), then compare the two one-sided difference quotients.

Show the method

  1. Check continuity

    As x approaches 1 from below, ax+bax+b approaches a+ba+b. From above, x2+2x^2+2 approaches 3. Continuity requires both limits to equal the assigned value f(1)=3f(1)=3. Thus continuity holds exactly when a+b=3a+b=3.

  2. Use the difference quotient

    Let h be a nonzero real displacement from 1, so the nearby input is 1+h1+h. As h approaches zero through negative values, 1+h1+h approaches 1 from the left; through positive values, it approaches from the right. The derivative is the limit, as h approaches zero, of:

    f(1+h)−f(1)h.\frac{f(1+h)-f(1)}{h}.

    Using a+b=3a+b=3, for h<0h<0 this quotient is:

    a(1+h)+b−3h=ahh=a.\begin{gathered}\frac{a(1+h)+b-3}{h}\\=\frac{ah}{h}=a.\end{gathered}

    For h>0h>0 it is:

    (1+h)2+2−3h=2h+h2h=2+h.\begin{gathered}\frac{(1+h)^2+2-3}{h}\\=\frac{2h+h^2}{h}\\=2+h.\end{gathered}

  3. Compare the two finite limits

    The two quotient limits are a and 2. They must be the same finite number, so a=2a=2; continuity then gives b=1b=1. Conversely, these values make the left quotient 2 and the right quotient 2+h2+h, both tending to 2. Therefore the unique differentiable pair is (2,1)(2,1), and f′(1)=2f^\prime(1)=2.

What this checks: continuity compares nearby values with f(1)f(1). Differentiability also compares rates of change. Matching the values alone does not settle the derivative.

Try the next step

1. Keep continuity; test the slope

Set a=0a=0 and b=3b=3. Is f continuous at 1? Is it differentiable there? Explain both answers before opening the solution.

Show the solution

Continuity holds because a+b=3a+b=3. The one-sided difference-quotient limits are 0 and 2, so no derivative exists at 1. The graph has a corner there. This gives a counterexample to the claim that every continuous function is differentiable.

2. Find the missing condition

A student differentiates the two branch formulas, matches their slopes at 1, and chooses a=2a=2, b=0b=0. Why is the argument incomplete?

Show the solution

The left function limit is 2, while f(1)f(1) and the right limit are 3. The join is not continuous. More directly, the left difference quotient is:2(1+h)−3h=2−1h.\begin{gathered}\frac{2(1+h)-3}{h}\\=2-\frac{1}{h}.\end{gathered}This quotient has no finite limit as h approaches zero from below. The derivative of a branch away from the join does not establish the derivative at the join.

3. Optional proof: why differentiability forces continuity

Let g be a real-valued function defined on an open interval containing 1. Suppose g′(1)g^\prime(1) exists and is finite. Use the identity below to show that g is continuous at 1.

g(1+h)−g(1)=h⋅g(1+h)−g(1)h,h≠0.\begin{gathered}g(1+h)-g(1)\\=h\cdot\frac{g(1+h)-g(1)}{h},\\h\ne0.\end{gathered}

Show the solution

The quotient tends to g′(1)g^\prime(1), while h tends to zero. The product limit is 0 · g′(1)g^\prime(1) = 0. Hence g(1+h)g(1+h) tends to g(1)g(1), which is continuity at 1. The first practice question shows why the converse fails.

Independent practice

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These are shared topic resources. Their other course connections do not make those courses equivalent to MAT148H1.

What to work on with a tutor

Bring a question or a concept you want explained, even if you have not started a solution. If you have tried a definition or theorem, you can include that work. Your tutor can help you:

  • Separate the assumptions from the conclusion before you begin a proof.
  • Check which side of a piecewise function applies and where a quotient is defined.
  • Locate the first unjustified step, then let you repair the argument.
  • Change one assumption and ask you to explain whether the conclusion still follows.

You can choose the first step together or review the point where your reasoning became unclear.

Course and tutoring questions

Which code should I use: MAT137Y1, MAT148H1 or MAT149H1?

The MAT148H1 and MAT149H1 calendar entries both list MAT137Y1 as their previous course number. This guide focuses on MAT148H1; MAT149H1 is a separate course covering integral calculus and series. Check your current outline or enrolment record, especially when using older MAT137 materials.

Is MAT148H1 the same as MAT135H1 or MAT136H1?

These are different courses. MAT135H1 develops applied differential calculus, including linear approximation and Euler’s method. MAT136H1 continues the applied sequence with integration, series and differential equations. MAT148H1 includes foundations based on mathematical proofs. Use the guide matching your course, and ask U of T about eligibility, exclusions or substitutions.

What is the prerequisite?

The MAT148H1 calendar lists high-school-level calculus. For study preparation, review function notation, algebra and how to state assumptions clearly. Those study suggestions are not extra enrolment requirements. Consult the official entry for exclusions.

Does this page confirm a Fall 2026 section?

U of T lists MAT148H1 among new courses approved for Fall 2026. That approval does not by itself verify a particular section or an available seat. Check the official timetable and your course site for current arrangements.

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Is this official U of T material?

No. Nablio is independent of the university. This page provides original practice and links to official course information. Follow your instructor’s rules for assessed work; a tutor can help you understand and revise your own reasoning.