Is precalculus harder than calculus? What each course is actually like, why many students find precalculus harder, which parts of precalculus calculus really uses, whether you can skip precalculus, how AP Precalculus compares with AP Calculus, and how to decide what to take

Students ask "is precalculus harder than calculus" for two reasons. Some are in precalculus, finding it harder than they expected, and frightened of what calculus must be like. Others have heard from older students that calculus was easier, which sounds impossible. Both are right in a way, and the explanation is useful for deciding what to take.

The honest answer

Calculus is harder in content. Precalculus is often harder in shape. And for many students the shape matters more.

Precalculus is a collection. Functions, then trigonometry, then logarithms, then sequences, then vectors, then conics, then maybe polar coordinates and matrices: each is a separate topic with its own rules, and the course moves from one to the next with no idea holding them together. There is a great deal to memorise and nothing to understand that makes the memorising unnecessary.

Calculus is three connected ideas, the limit, the derivative and the integral, each of which can be drawn, and everything in the course is one of those three ideas applied to something. There is much less to memorise. What there is instead is a lot of algebra inside every problem, and a new way of thinking that takes a few weeks to arrive.

So a student who learns by memorising and following procedures tends to find precalculus manageable and calculus bewildering, at least at first. A student who learns by understanding one idea and applying it everywhere tends to find precalculus tedious and hard to hold in the head, and calculus a relief. Neither student is the better mathematician; they are different, and the courses are different.

What each course is actually like

Why many students find precalculus harder

1. Trigonometry is new and strange. It fills a third of the course and is unlike anything before it: a circle, angles measured in a new unit, six functions, and dozens of identities. Students who were fine in Algebra 2 can find this the first topic in mathematics that does not come.
2. It is where weak Algebra 2 is found out. Precalculus assumes fast, exact algebra and does not reteach it. A student who passed Algebra 2 by following examples discovers in precalculus that the examples no longer cover the problems.
3. Nothing connects to anything. Each topic is learned and left. By the final, the student has to hold eight unrelated bodies of rules in mind at once.
4. It is often taught as preparation, which means the hard parts are taught without the reason they will matter, and students do not know which parts to make solid.

Why other students find calculus harder

1. The first month asks for a new kind of thinking. The limit, a value approached and never reached, is unlike anything in algebra, and the derivative is defined with it. Students who need to see why before they can do find this month slow.
2. The algebra never stops. Every calculus problem has algebra inside it, and it has to be automatic. A student with shaky algebra understands every lesson and fails every test.
3. Problems are longer. A related-rates or optimisation problem is a paragraph to be drawn, set up, differentiated, solved and checked. A student used to one-line precalculus questions has to learn to work for twenty minutes on one problem.
4. The second half is faster. Integration, and in BC the series unit, arrive when the course has built up speed. How hard each calculus course really is, with a readiness check, is in is calculus hard.

Which parts of precalculus calculus actually uses

This is the useful question, because it tells you what to make solid and what you can let go.

Roughly half of precalculus is used in calculus, and that half is used constantly. The other half belongs to later courses or to none.

Can you skip precalculus?

Sometimes. If your Algebra 2 course included trigonometry (many do, in the second half) and you finished it with an A, and you can do the check below, you can go to calculus. Students who skip precalculus and struggle in calculus are nearly always missing trigonometry, and occasionally logarithms; those two are the test.

If you skip, spend the summer on the four topics in the first half of the table above. If you cannot do the check below after that, take precalculus; it will save you a failed calculus course.

A ten-question check

No calculator, ten minutes. This is the half of precalculus that calculus uses.

1. If f(x) = 2x + 1 and g(x) = x², write f(g(x)) and g(f(x))
2. Find the inverse of f(x) = (x − 3) / 2
3. What are sin 30°, cos 45°, tan 60° and sin 90°?
4. Convert 150° to radians, and 3π/2 to degrees
5. Write sin²x + cos²x and tan x in terms of sine and cosine
6. Solve 2ˣ = 64, and solve log₃ x = 4
7. Simplify log(100) + ln(e³)
8. Write √(x³) and 1/x² as powers of x
9. Simplify (x² − 1) / (x² + 2x + 1)
10. Sketch y = sin x from 0 to 2π and mark where it crosses the axis

Answers: 1) f(g(x)) = 2x² + 1; g(f(x)) = (2x + 1)² 2) f⁻¹(x) = 2x + 3 3) ½, √2/2, √3, 1 4) 5π/6; 270° 5) 1; sin x / cos x 6) x = 6; x = 81 7) 2 + 3 = 5 8) x to the power 3/2; x⁻² 9) (x − 1) / (x + 1) 10) a wave starting at 0, peaking at π/2, crossing at π, lowest at 3π/2, back to 0 at 2π.

Nine or ten, quickly: you are ready for calculus, with or without a precalculus course. Six to eight: take precalculus, or spend a summer on the ones you missed. Five or fewer: take precalculus and take it seriously; calculus would be a year of algebra struggle.

AP Precalculus against AP Calculus AB

AP Precalculus is the newer course, built around functions and modelling in four units (polynomial and rational functions; exponential and logarithmic functions; trigonometric and polar functions; and functions involving parameters, vectors and matrices, which is not on the exam). It is more structured than a traditional precalculus course and more useful as preparation, because it spends its time on the half of precalculus that calculus uses.

AP Calculus AB is Calculus 1: harder in content, with a free-response section that asks for justification in words. A student who did well in AP Precalculus is well prepared for it. Which math course belongs in which year, and how to choose, is in what math do you take in 11th grade; the AB and BC decision is in AP Calculus AB vs BC.

How to decide what to take

• Algebra 2 secure, trigonometry included and solid, the check above easy: calculus next, AB or BC depending on your plans.
• Algebra 2 secure, no trigonometry yet: precalculus or AP Precalculus; it is a good year and calculus will be easier for it.
• Algebra 2 was a struggle: precalculus, with the first term used to make the algebra automatic. Skipping to calculus from here fails more often than it succeeds.
• Not going on to calculus at all: precalculus is still worth taking for the functions and trigonometry, which appear in physics, statistics and most college math requirements.

Where to get help

A 1-on-1 online precalculus class or calculus class with a tutor who holds a master's in mathematics: the check above in the first class, the trigonometry and logarithms made solid because calculus will use them every week, and every line of working read. For precalculus, AP Precalculus, AP Calculus AB and BC and college Calculus 1. The free 30-minute first class is a real lesson on the problem you bring.

Questions parents ask

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