UCSD MATH 20A Calculus: Topics, Study Guide and Practice

MATH 20A is the first course in UC San Diego's calculus sequence for science and engineering. As the title says, it is the version of Calculus I built for science and engineering students, who then continue to MATH 20B and MATH 20C. This guide maps every MATH 20A topic to free practice, so you can learn each idea by solving problems and getting instant feedback.

What UCSD MATH 20A covers

The UC San Diego course catalog describes MATH 20A as the foundations of one-variable differential and integral calculus. The listed topics are functions and their graphs, continuity, limits, the derivative and the tangent line, and optimization problems, worked with algebraic, exponential, logarithmic and trigonometric functions. The course closes with an introduction to the integral.

That lines up closely with a standard Calculus 1 course. The main difference is pace: UCSD uses ten-week quarters, so most of the integral material (Riemann sums, the Fundamental Theorem and substitution) is finished in MATH 20B rather than here.

MATH 20A topic map

Each row links to a lesson you can practice and to a reference page or guide that explains the idea.

MATH 20A in the UCSD calculus sequence

MATH 20A is the entry point for the science and engineering track. MATH 10A also opens the door to 20B, but if you take 20A after 10A you only earn two units for it, so it pays to pick the right course the first time. Continue with the MATH 20B guide when you finish.

Example problems at the MATH 20A level

1. A limit that needs factoring. Evaluate \lim_{x \to 3} \frac{x^2 - 9}{x - 3}.

Direct substitution gives \frac{0}{0}, so factor first:

2. A tangent line. Find the tangent line to f(x) = e^x + x^2 at x = 0.

Here f(0) = 1 and f'(x) = e^x + 2x, so f'(0) = 1. The tangent line is

3. An optimization problem. A rectangle has perimeter 40. What dimensions give the largest area?

With sides x and 20 - x, the area is A(x) = x(20 - x) = 20x - x^2. Then A'(x) = 20 - 2x = 0 gives x = 10, and A''(x) = -2 < 0 confirms a maximum. The best rectangle is a 10 \times 10 square with area 100.

How to do well in UCSD MATH 20A

• Keep pace with the quarter. Ten weeks goes fast. Short daily sessions on practice problems beat a long weekend catch-up.
• Master the derivative rules early. The chain rule shows up in almost every problem after week four, so drill it until it is automatic.
• Watch for algebra slips. Factoring, signs and exponent rules cost more points than calculus ideas. Read our list of common calculus mistakes.
• Keep a formula sheet handy while you study. The calculus formula sheet puts every derivative on one page.
• Prepare for midterms deliberately. Our guide on how to prepare for a calculus exam walks through a one-week plan.

Tip: optimization problems feel hard because of the setup, not the calculus. Always write the quantity to maximize as a function of one variable before you differentiate.