UMN MATH 1271 Calculus I: Topics, Study Guide and Practice

MATH 1271 is Calculus I at the University of Minnesota Twin Cities, the first course in the calculus sequence for most science, math and pre-health students. It is worth 4 credits and runs every fall, spring and summer. This guide maps every MATH 1271 topic to free practice, so you can work through each idea by solving problems with instant feedback.

What MATH 1271 covers

The University of Minnesota catalog describes MATH 1271 in two halves. The first half is differential calculus of one variable: derivatives of polynomial, rational, exponential and trigonometric functions, with applications to optimization and related rates. The second half starts integral calculus using antiderivatives and simple substitution, with applications that may include area, volume and work.

That lines up closely with a standard Calculus 1 course. The main difference is that MATH 1271 can reach volume and work, which some universities save for Calculus 2. Limits and continuity are not named in the catalog line, but every derivative course is built on them, so they come first in the topic map below.

MATH 1271 topic map

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

MATH 1271 in the Minnesota calculus sequence

All four courses carry 4 credits. MATH 1272 requires MATH 1271 or an equivalent with at least a C-. See the MATH 1272 guide and the MATH 2263 guide for what comes next.

Example problems at the MATH 1271 level

1. A related rates problem. A circle's radius grows at 3 cm/s. How fast is the area growing when r = 5 cm?

With A = \pi r^2, differentiate with respect to time:

2. An optimization problem. Find two positive numbers that add to 20 with the largest product. Let P(x) = x(20 - x) = 20x - x^2. Then P'(x) = 20 - 2x = 0 gives x = 10, and P''(x) = -2 < 0 confirms a maximum. The numbers are 10 and 10.

3. Simple substitution. Evaluate \int 2x\cos(x^2)\,dx. Let u = x^2, so du = 2x\,dx:

How to do well in MATH 1271

• Make the derivative rules automatic. Optimization and related rates are hard only when the differentiation is slow. Drill the power rule, product rule and chain rule until you do not have to think about them.
• Draw a picture for every word problem. Related rates and optimization questions get much easier once you label a sketch and write one equation that links the quantities.
• Watch for algebra slips. Most lost points are sign errors and dropped parentheses. Read our list of common calculus mistakes.
• Keep the formulas in one place. The calculus formula sheet has every derivative and basic integral on one page.
• Rehearse under time pressure with practice exams before each midterm, and follow the plan in how to prepare for a calculus exam.

Tip: substitution is the bridge into MATH 1272. If \int f(g(x))g'(x)\,dx already feels routine by the end of MATH 1271, integration techniques next term will feel like a natural next step.