Key Concepts to Review
You don't need to re-learn everything. You need to know what's actually on the exam and where you're weakest. Here's a targeted breakdown of the three big topic areas.
Limits
Limits are the foundation of everything in calculus. If your exam is cumulative, expect at least a couple of limit problems, usually designed to test whether you actually understand the concept or just memorized procedures.
The formal definition
lim (x → a) f(x) = L means that as x gets arbitrarily close to a (from both sides), f(x) gets arbitrarily close to L. You probably won't need the epsilon-delta definition on a Calc I final, but you absolutely need to understand what "the limit exists" means: the left-hand limit and right-hand limit must be equal.
L'Hopital's Rule is your best friend for indeterminate forms. If you plug in and get 0/0 or infinity/infinity, take the derivative of the top and bottom separately, then re-evaluate. Two things students forget: (1) you can apply it more than once, and (2) it only works on indeterminate forms. If you get 5/0, that's not indeterminate, that's just undefined.
The Squeeze Theorem comes up less often, but when it does, students freeze because they never practiced it. The idea is simple: if g(x) is always between h(x) and f(x), and h(x) and f(x) have the same limit at a point, then g(x) must also have that limit. The classic example is lim (x → 0) of x sin(1/x). You know -1 is less than or equal to sin(1/x) which is less than or equal to 1, so -|x| is less than or equal to x sin(1/x) which is less than or equal to |x|. Both sides go to 0, so the limit is 0.
Derivatives
Derivatives are the core of Calc I and show up constantly in Calc II. You need to be fast and accurate. If you're spending more than 60 seconds on a straightforward derivative, you haven't practiced enough.
Power Rule: d/dx [x^n] = n x^(n-1). This is your bread and butter. Make sure you can handle negative and fractional exponents too. The derivative of 1/x is -1/x², and the derivative of the square root of x is 1/(2 times the square root of x).
Chain Rule: The derivative of f(g(x)) is f'(g(x)) times g'(x). This is the single most tested rule and the one students mess up most often. Whenever you see a function inside another function, you need the chain rule. Practice it until it's automatic.
Product Rule: d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x). Remember it as "derivative of the first times the second, plus the first times the derivative of the second."
Quotient Rule: d/dx [f(x)/g(x)] = [f'(x) g(x) - f(x) g'(x)] / [g(x)]². Low-d-high minus high-d-low, over the square of what's below. Alternatively, rewrite the fraction as a product using negative exponents and use the product rule. Both work.
Integrals
Integration is where Calc I ends and Calc II begins. At minimum, you need basic antiderivatives, u-substitution, and the Fundamental Theorem of Calculus. If you're in Calc II, add integration by parts, partial fractions, and trig substitution to your list.
Basic rules: The integral of x^n dx = x^(n+1)/(n+1) + C (where n is not equal to -1). The integral of 1/x dx = ln|x| + C. The integral of e^x dx = e^x + C. The integral of sin(x) dx = -cos(x) + C. Know these cold.
U-substitution: Look for a function and its derivative sitting in the same integral. Set u equal to the inner function, find du, substitute, integrate, then substitute back. The hardest part is choosing the right u. If your first choice doesn't simplify things, try a different one.
Integration by parts: Integral of u dv = uv - integral of v du. Use LIATE to choose u (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). The first function in that list becomes u. For example, for the integral of x e^x dx, let u = x (algebraic) and dv = e^x dx.
Common Exam Mistakes
These aren't obscure gotchas. These are the mistakes that cost students 10-20 points on every calculus exam. Read them, memorize them, and check for them on every single problem.
Forgetting the chain rule. The derivative of sin(3x) is not cos(3x). It's 3cos(3x). The derivative of (x² + 1)^5 is not 5(x² + 1)^4. It's 10x(x² + 1)^4. Every time you differentiate a composite function, multiply by the derivative of the inner function. Every. Time.
Sign errors. These are the most frustrating way to lose points because you understood the concept perfectly. Go slowly on algebra steps. When you factor out a negative sign, double-check it. When you subtract fractions, make sure you distributed the negative to every term in the numerator.
Not checking limits from both sides. If a problem asks whether lim (x → 2) f(x) exists, you need to verify that lim (x → 2-) f(x) equals lim (x → 2+) f(x). This is especially important for piecewise functions and absolute value functions. One-sided limits must agree for the two-sided limit to exist.
Forgetting +C on indefinite integrals. It sounds trivial, but professors take off points for this every single time. If there are no bounds on the integral, you need the constant of integration. Write "+C" at the end of every indefinite integral. Make it a reflex.
Study Strategy
Having the right study plan matters more than the number of hours you put in. Here's how to use your last week before the exam effectively.
Days 7-5: Identify your gaps
Don't start by reviewing everything from lecture 1. Grab a practice exam or the table of contents from your textbook and go topic by topic. For each one, try a problem. If you solve it confidently, move on. If you struggle, mark it. You now have your priority list.
Days 4-3: Drill your weak spots
Spend 80% of your time on the topics you marked. Don't just read examples. Work problems with your notes closed. When you get stuck, check your notes, then close them and try a similar problem. This is active recall, and it's the most effective study technique that exists.
Days 2-1: Simulate the exam
Take a full practice exam under real conditions. Set a timer. No notes. No calculator (unless your exam allows one). Grade yourself honestly. Any topics you still got wrong? Those get your final review session.
Day of: Don't cram
Review your formula sheet for 20-30 minutes, then stop. Eat something. Show up early. If you've been doing problems all week, you're ready. Last-minute cramming creates anxiety without adding knowledge.
What to focus on
Calculus exams are predictable. Your professor is going to test: one or two limit problems, several derivatives (at least one requiring chain rule), an optimization or related rates problem, and two or three integrals. If you can do all of those quickly and accurately, you'll do well.
Office hours are underrated. Go with specific questions. "I don't get integrals" is not a good question. "I can set up u-substitution but I keep getting stuck choosing u for trig functions" is a great one. Professors and TAs respect students who've done the work and have targeted questions.
Formula Sheet
Print this out or copy it onto an index card. These are the formulas you need on exam day.
tan' = sec², sec' = sec tan
d/dx [ln x] = 1/x
∫ cos x dx = sin x + C
1 + tan²x = sec²x
