How to Catch Up in Precalculus Before Calculus Starts

If you are finishing precalculus with gaps and calculus starts in a few months, the useful thing to know is this: calculus does not use all of precalculus equally.

Some precalculus topics you will use in almost every problem, every week, for a year. Others you will barely touch. Students who try to "review all of precalculus" usually run out of time and energy on the topics that mattered least, which is the worst possible outcome.

So this article is a ranking. What calculus actually leans on, in order, with the specific skill and a worked example for each. Start at the top and work down as far as you get.

Tier 1: You Will Use This Constantly

Function notation, especially $f(x+h)$

This is the single most under-practised skill among students entering calculus, and it appears in week one.

You are comfortable with $f(2)$. But calculus immediately asks for $f(x+h)$ — substituting an expression into a function rather than a number. The rule is identical: replace every $x$ with whatever is in the brackets.

If $f(x) = x^2 - 3x$, then:

$$f(x+h) = (x+h)^2 - 3(x+h) = x^2 + 2xh + h^2 - 3x - 3h$$

Notice you must expand $(x+h)^2$ properly. It is $x^2 + 2xh + h^2$, never $x^2 + h^2$.

Here is why this matters. The definition of the derivative is:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Put our function into it. The numerator is:

$$\left(x^2 + 2xh + h^2 - 3x - 3h\right) - \left(x^2 - 3x\right) = 2xh + h^2 - 3h$$

Every term without an $h$ cancels — that always happens, and it is a useful check. Now divide by $h$:

$$\frac{2xh + h^2 - 3h}{h} = 2x + h - 3$$

And let $h \to 0$:

$$f'(x) = 2x - 3$$

That is the whole derivative, done from the definition. The only calculus in it was the final limit. Everything else was precalculus algebra. If $f(x+h)$ is shaky, week one of calculus will feel impossible for reasons that have nothing to do with calculus.

Practise: take five functions and expand $f(x+h)$ for each. Include one with a square root and one with a fraction.

Algebraic fluency

Factoring, simplifying rational expressions, handling negative and fractional exponents. Calculus problems end in algebra far more often than they end in calculus.

This deserves its own treatment, so it has one: Common Algebra Gaps That Make Calculus Hard covers the specific errors and how to test yourself on them. If you only have time for two things on this page, make it that article and the section above.

The graph library

You need to be able to sketch these from memory, quickly, without a calculator and without plotting points:

$$y = x^2 \qquad y = x^3 \qquad y = \sqrt{x} \qquad y = \frac{1}{x} \qquad y = |x|$$ $$y = e^x \qquad y = \ln x \qquad y = \sin x \qquad y = \cos x$$

For each, know: the general shape, where it crosses the axes, its domain, and what it does at the extremes.

This is not busywork. Calculus asks constant questions of the form "where is this function increasing?" or "does this limit exist?" and the fastest route to an answer is usually a rough sketch. A student who has to plot points to remember what $\frac{1}{x}$ looks like is spending their thinking budget in the wrong place.

You also want the transformations: $f(x) + 3$ shifts up, $f(x+3)$ shifts left, $-f(x)$ flips vertically, $2f(x)$ stretches vertically.

Tier 2: You Will Use This Weekly

Exponentials and logarithms

Two things: the rules, and the fact that they undo each other.

$$\ln(e^x) = x \qquad e^{\ln x} = x \ (x > 0)$$ $$\ln(ab) = \ln a + \ln b \qquad \ln\!\left(\frac{a}{b}\right) = \ln a - \ln b \qquad \ln(a^n) = n \ln a$$

You need to solve equations with these without stopping to think:

$$e^{2x} = 7 \ \Longrightarrow \ 2x = \ln 7 \ \Longrightarrow \ x = \frac{\ln 7}{2}$$ $$\ln(x) = 3 \ \Longrightarrow \ x = e^3$$

Exponential growth and decay problems, logistic models, and half of all optimisation problems finish with a step exactly like this.

Trigonometry — in radians

Two separate things students under-prepare here.

First, radians are not optional. Calculus uses radians and only radians. The reason is concrete rather than stylistic: the rule

$$\frac{d}{dx}\sin x = \cos x$$

is only true in radians. In degrees the derivative picks up an ugly constant factor of $\frac{\pi}{180}$. So every trig derivative and integral you will meet assumes radian measure. Get comfortable thinking of $\frac{\pi}{6}$ rather than $30°$ now, because the conversion habit will slow you down all year.

Second, exact values. You need $\sin$ and $\cos$ at $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$ and their reflections, without a calculator — a large part of the AP exam is calculator-free.

$$\sin\frac{\pi}{6} = \frac{1}{2} \qquad \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} \qquad \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$$ $$\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} \qquad \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2} \qquad \cos\frac{\pi}{3} = \frac{1}{2}$$

Notice sine climbs while cosine falls across those three — that pattern is easier to hold than six separate facts.

The identities that earn their keep in calculus:

$$\sin^2\theta + \cos^2\theta = 1 \qquad \sin(2\theta) = 2\sin\theta\cos\theta$$

Composition — and decomposition

The chain rule is where most students first get properly stuck, and the reason is usually not the rule itself. It is that they cannot see the composition.

Given $y = \sqrt{x^2 + 1}$, you need to see it instantly as two functions stacked:

• outside: $\sqrt{u}$
• inside: $u = x^2 + 1$

The chain rule then says: differentiate the outside, leave the inside alone, multiply by the derivative of the inside.

$$\frac{dy}{dx} = \frac{1}{2\sqrt{x^2+1}} \cdot 2x = \frac{x}{\sqrt{x^2+1}}$$

The calculus is mechanical. The hard part was the decomposition, and that is a precalculus skill you can drill right now, before you know any calculus at all.

Practise: for each of these, name the outside and inside function. $(3x+1)^5$; $\sin(x^2)$; $e^{\cos x}$; $\ln(5x - 2)$; $\sqrt[3]{x^2+x}$. Do not differentiate anything — just decompose.

Domain and piecewise functions

Limits and continuity are largely questions about where a function is and is not defined. Know how to find the domain of a rational function (denominator $\neq 0$), a square root (radicand $\geq 0$), and a logarithm (argument $> 0$), and be comfortable reading a piecewise definition.

Tier 3: Lower Priority Than Students Assume

Worth saying plainly, because it saves time: conic sections, matrices, complex numbers, and synthetic division get very little use in AP Calculus AB or BC. They are legitimate precalculus topics and your teacher was right to teach them. They are just not what to revise in a limited window before calculus.

Two caveats. If you are heading into BC, sequences and series are worth a look — they are the largest topic in that course. And if you are heading into a university engineering programme, matrices come back quickly in linear algebra, just not in calculus.

A Four-Week Plan

This assumes about forty minutes a day. Adjust the pace, keep the order — later weeks depend on earlier ones.

Week 1 — Algebra mechanics. Factoring, negative and fractional exponents, simplifying rational expressions, complex fractions. Work through the self-check in the algebra gaps article and fix whatever it exposes.

Week 2 — Functions. $f(x+h)$ expansions until they are automatic. Domain. Composition and decomposition. Sketch the nine graphs above from memory each day until it is boring.

Week 3 — Exponentials, logs, and trig. Log and exponent rules, solving equations with them, the unit circle in radians, exact values, the Pythagorean identity.

Week 4 — Mixed practice. No topic labels. Take problems from all three weeks in random order. This is the week that matters most, because in calculus nobody tells you which precalculus skill a problem needs.

The reason for the ordering: week 2 needs week 1's algebra, and week 4 tests whether you can retrieve a skill without a prompt — which is the thing that actually breaks down under exam conditions.

Diagnostic: Where Are You Now?

Do these on paper, no calculator. Then check against the answers.

1. If $f(x) = x^2 + 2x$, expand and simplify $f(x+h)$.
2. For that same $f$, simplify $\dfrac{f(x+h) - f(x)}{h}$.
3. Factor $x^3 - 4x$ completely.
4. Write $\dfrac{5}{\sqrt[3]{x}}$ as a constant times a power of $x$.
5. Solve $\ln x = 3$.
6. Solve $3^x = 81$.
7. Evaluate $\cos\dfrac{\pi}{3}$ and $\sin\dfrac{\pi}{2}$.
8. Decompose $y = (2x - 7)^4$ into an outside and an inside function.
9. State the domain of $f(x) = \dfrac{1}{x^2 - 9}$.
10. Simplify $\dfrac{x^2 - x - 6}{x - 3}$.

Answers

1. $x^2 + 2xh + h^2 + 2x + 2h$
2. $2x + h + 2$ — subtract $f(x)$, get $2xh + h^2 + 2h$, divide by $h$. (As $h \to 0$ this becomes $2x+2$, the derivative.)
3. $x(x-2)(x+2)$ — pull out $x$ first, then difference of squares.
4. $5x^{-1/3}$
5. $x = e^3$
6. $x = 4$
7. $\cos\frac{\pi}{3} = \frac{1}{2}$ and $\sin\frac{\pi}{2} = 1$
8. Outside $u^4$, inside $u = 2x - 7$
9. All real numbers except $x = 3$ and $x = -3$
10. $x + 2$, for $x \neq 3$ — factor the numerator as $(x-3)(x+2)$

How to read your score. Eight or more, quickly and correctly: you are in good shape, spend your remaining time on mixed practice. Five to seven: you have specific gaps rather than a general problem — find which questions failed and drill exactly those. Four or fewer: start at Week 1 and go in order; this is very fixable, but it needs consistent daily work rather than one long session.

One Last Thing

Being behind in precalculus does not predict how you will do in calculus. What predicts it is whether you close the gaps before the new material starts stacking on top of them.

Calculus is relentlessly cumulative. Week 6 assumes week 3. If you enter with $f(x+h)$ unsteady, you do not get a fresh start in November — you get a bigger gap. Which is exactly why four focused weeks now is worth far more than the same four weeks spent later, in a panic, while also trying to learn integration.

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Related

• Common Algebra Gaps That Make Calculus Hard
• AP Calculus AB vs. BC: Which Course Should You Take?
• Why Limits Feel Impossible (And Why They're Not)