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

> A study guide for MATH 1271 at the University of Minnesota: derivatives, optimization, related rates and first integrals, mapped to free practice.

Source: https://practicecalculus.com/courses/university-of-minnesota-math-1271
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 1271
- **Credits:** 4
- **Offered:** Every fall, spring and summer
- **Prerequisites:** A satisfactory placement test score, or a grade of at least C- in MATH 1151 or 1155
- **Covers:** Derivatives of polynomial, rational, exponential and trig functions, optimization, related rates, antiderivatives, simple substitution, area, volume and work
- **Next course:** MATH 1272 (Calculus II)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 1271 is the 4-credit Calculus I course at the University of Minnesota Twin Cities, offered every fall, spring and summer.
- It covers differential calculus of one variable with applications, then the first ideas of integration through antiderivatives and simple substitution.
- Area, volume and work appear as applications of the integral, which is a little further than some Calculus 1 courses go.
- After MATH 1271, students continue with MATH 1272 (Calculus II), and then MATH 2263 (Multivariable Calculus).

**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 topic | Practice | Read |
| --- | --- | --- |
| Limits and continuity | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits), [continuity](/blog/continuity) |
| The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Polynomial and rational functions | [Power rule](/learn/calculus-1/derivative-rules/power-rule) | [Power rule](/rules/power-rule), [quotient rule](/rules/quotient-rule) |
| Exponential functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Derivatives of exponential and log functions](/blog/derivatives-of-exponential-and-logarithmic-functions) |
| Trigonometric functions | [Sine, cosine and tangent](/learn/calculus-1/derivative-rules/trig-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions) |
| Chain rule | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Related rates | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates](/blog/related-rates) |
| Antiderivatives | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives), [reverse power rule](/rules/reverse-power-rule) |
| Simple substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |
| The definite integral | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Area and volume | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves), [volumes of revolution](/blog/volumes-of-revolution) |
| Work | [Work](/learn/calculus-2/improper-integrals-and-applications/work) | [Definite integral properties](/rules/definite-integral-properties) |

## MATH 1271 in the Minnesota calculus sequence

| Course | Title | What it adds |
| --- | --- | --- |
| MATH 1271 | Calculus I | Derivatives, optimization, related rates, first integrals |
| MATH 1272 | Calculus II | Integration techniques, transcendental functions, polar coordinates, Taylor polynomials, vectors and curves in space |
| MATH 2263 | Multivariable Calculus | Calculus of several variables, Jacobians, line and surface integrals, Green, Gauss and Stokes |
| MATH 1371 | CSE Calculus I | The College of Science and Engineering version of Calculus I |

All four courses carry 4 credits. MATH 1272 requires MATH 1271 or an equivalent with at least a C-. See the [MATH 1272 guide](/courses/university-of-minnesota-math-1272) and the [MATH 2263 guide](/courses/university-of-minnesota-math-2263) 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:

$$\frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 2\pi(5)(3) = 30\pi \text{ cm}^2/\text{s}$$

**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 $du = 2x\,dx$:

$$\int 2x\cos(x^2)\,dx = \int \cos u\,du = \sin(x^2) + C$$

## 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](/rules/power-rule), [product rule](/rules/product-rule) and [chain rule](/rules/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](/blog/common-calculus-mistakes).
- **Keep the formulas in one place.** The [calculus formula sheet](/formulas) has every derivative and basic integral on one page.
- **Rehearse under time pressure** with [practice exams](/practice) before each midterm, and follow the plan in [how to prepare for a calculus exam](/blog/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.

## Frequently asked questions

### What is MATH 1271 at the University of Minnesota?

MATH 1271 is Calculus I on the Twin Cities campus. It teaches derivatives of polynomial, rational, exponential and trigonometric functions, uses them for optimization and related rates, and introduces integration with antiderivatives and simple substitution.

### What comes after MATH 1271?

The next course is MATH 1272, Calculus II, which needs MATH 1271 (or an equivalent) with a grade of at least C-. After that comes MATH 2263, Multivariable Calculus.

### Is MATH 1271 hard?

It is a full-speed first calculus course, so the pace is steady and new rules arrive every week. Students with solid algebra and trig who practice derivatives a little every day usually find it manageable.

### How should I prepare for MATH 1271?

Review functions, exponent rules, logarithms and the key trig values, then start on limits. The free functions warm-up and limits units on PracticeCalculus cover exactly that ground.

### What is the difference between MATH 1271 and MATH 1371?

MATH 1371 is CSE Calculus I, a version for College of Science and Engineering students. It covers the same core material: differentiation, the basics of integration, and applications such as max-min, related rates, area and curve sketching.