What UBC Calculus Students Should Review Before Their First Midterm

Most students who struggle with their first university calculus midterm do not struggle with calculus. They struggle with the algebra underneath it, at a speed that never allowed for hesitation.

That is an uncomfortable thing to hear a few weeks into first year, and it is also good news, because it points at something fixable. The chain rule is not the problem. Simplifying the expression the chain rule produced, quickly and without error, usually is.

Here is what is worth reviewing before the first midterm, why each item matters, and worked examples with the reasoning shown. If you would rather not face that jump alone, our university math tutoring is built for exactly this school-to-university step.

What first-year calculus actually assumes

The course teaches limits, derivatives and their applications. It assumes — without ever saying so — that you can already:

• Manipulate exponents and radicals without stopping to think
• Factor quickly, including differences of squares and cubes
• Handle trigonometric identities and exact values from memory
• Rearrange an equation with several symbols and no numbers in it
• Work fluently with logarithms and exponentials

None of that is calculus. All of it is Pre-Calculus 12, and it is the layer where midterm marks are actually lost. The calculus step is often a single line; the four lines of algebra afterwards are where the answer dies.

Limits: know why, not just how

Early limit questions look trivial and are not, because they are testing whether you know what a limit means.

The cancellation is legal precisely because $x\to 2$ means $x$ approaches 2 without ever equalling it — so $x-2\ne 0$ and dividing is allowed.

Note that the function is undefined at $x=2$ and the limit still exists and equals 4. A limit captures the value a function is approaching, which need not equal the value it takes at that point — approaching and arriving are separate questions. Students who never internalise that distinction find continuity, and later the definition of the derivative, permanently mysterious.

The other limit technique: the conjugate

Factoring gets you most of the way through a limits section. Then a square root turns up and factoring does nothing, and this is reliably where students stall on a midterm.

Substitute and you get 0/0 — the top is √4 − 2 = 0 and so is the bottom. Nothing factors. The move is to multiply top and bottom by the conjugate, which turns a difference of square roots into a plain difference:

The numerator collapses to x, and now it cancels:

The cancellation is legal for the same reason it was legal in the difference quotient: x is approaching zero, not sitting on it, so it is never actually zero and you may divide by it. The whole technique is one idea — multiplying by a disguised 1 to make the awkward part cancel.

Limits at infinity are a different question

These look like the same topic and are not. "What happens as x approaches 2" asks about a point. "What happens as x approaches infinity" asks about long-run behaviour, and infinity is not a place you can substitute.

The technique: divide every term by the highest power of x in the denominator. Here that is x²:

Everything with an x underneath dies, and what survives is the ratio of the leading coefficients. The shortcut is worth having: if the degrees match, the limit is that ratio. If the bottom wins, the limit is 0. If the top wins, it diverges. But learn the division first, because the shortcut is only safe once you know why it is true.

This is also what a horizontal asymptote is — the same calculation, wearing a graph's clothing. Questions that ask for asymptotes are asking for this limit, and students who learned them as separate topics do twice the work.

And the one you must simply know

Also 0/0, and neither factoring nor conjugates touch it. It is proved geometrically and then used constantly, so learn the result. One warning that costs marks every year: it is true in radians and false in degrees. In degrees that limit is π/180, not 1 — which is a large part of why calculus abandons degrees entirely.

The chain rule: the one that decides your grade

More marks turn on the chain rule than on anything else in the first half of the course, because it appears inside almost every other rule.

Narrate it as you go: differentiate the outer function first with the inner part untouched, then multiply by the inner function's own derivative. The common failure is stopping halfway — writing cos(x²) and forgetting the 2x.

Combined with the product rule

Two things happened there. The product rule ran, the chain rule ran inside its second term, and then — the step students skip — the result was factored. On a midterm, an unsimplified answer costs marks and makes the next part of the question harder than it needs to be.

Related rates: a modelling problem in disguise

Related rates questions are where students who have been coping by pattern-matching stop coping, because no two look alike. The calculus is easy; setting it up is not.

Worked example: the sliding ladder

A 5 m ladder leans against a wall. The bottom slides away at $0.5m/s$. How fast is the top descending when the bottom is 3 m from the wall?

Start with the relationship that is always true — not at one instant, but at every instant. That is the whole trick:

Differentiate both sides with respect to time. Every variable gets a chain-rule factor, because everything is moving:

Only now substitute the instant you care about. When $x=3$, the 3–4–5 triangle gives $y=4$:

The top descends at 0.375 m/s. The negative sign is not decoration — it is the physics, saying y decreases as x grows.

The mistake that ruins these: substituting x = 3 before differentiating. Do that and x becomes a constant, its derivative is zero, and the equation collapses into nonsense. Differentiate the general relationship first; substitute the instant last.

Continuity: the definition that is actually examined

Continuity gets described as "you can draw it without lifting your pen", which is a picture, not a definition, and it is useless on the question they actually ask. The real definition has three parts, and a function is continuous at a point only if all three hold:

• f(a) exists — the function is defined there at all.
• The limit as x approaches a exists — and that means the left and right limits agree.
• They are equal to each other.

Three separate conditions, and the exam question is built to break exactly one of them while the other two look fine.

The standard question

Find k so that this is continuous at x = 1:

For every x other than 1, the top factors and cancels:

So the limit as x approaches 1 is 2. Condition one says f(1) must exist — it does, it is k. Condition three says they must match. Therefore $k=2$, and any other value leaves a hole with a dot floating above or below it.

Notice why the function had to be defined piecewise in the first place: at x = 1 the original formula is 0/0, genuinely undefined. The piecewise line is not decoration — it is patching a hole that the algebra cannot patch by itself. Understanding that is the difference between answering this question and pattern-matching it.

What to review, in priority order

1. Algebra speed — factoring, exponents, radicals. Not because it is hard, but because slow algebra is what runs you out of time.
2. The chain rule until it is automatic, including nested inside the product and quotient rules.
3. Exact trig values and the identities. Looking them up mid-question breaks your train of thought.
4. Logarithm and exponential rules, especially for implicit and logarithmic differentiation.
5. Related rates setup — practise writing the relationship before touching the numbers.

A note on how to practise

Reading worked solutions feels productive and mostly is not. It builds recognition, which vanishes under exam conditions, rather than recall, which does not.

The more useful test: do a problem, then close the book and do it again from scratch. If the second attempt stalls, you had recognised the solution rather than learned it. That distinction is worth more than another twenty problems skimmed.

Why first year feels so much faster

It is not an illusion. A high-school course spreads a topic over a week with practice built in; a university course may spend one lecture on it and expect fluency by the problem set. Nothing is retaught, and the pace does not pause for anyone.

That is why gaps compound so quietly here. A shaky week in September does not announce itself — it surfaces in October on a question that assumed you had closed it. The same pattern shows up in first-year physics, for the same structural reason.

Getting help before the midterm

If the calculus makes sense in lecture but falls apart on the problem set, the gap is almost always in the algebra layer underneath — and that is quick to diagnose and quick to fix, which is exactly what one-on-one university math support is good for.

We work with UBC and SFU students online and in person in Burnaby. Book a free 30-minute consultation and we will find out whether it is the calculus or the algebra — it is usually the algebra.