# Duke MATH 111L Laboratory Calculus I: Topics and Practice

> A study guide for Duke MATH 111L Laboratory Calculus I: derivatives, optimization, Euler's method, the Fundamental Theorem and slope fields, with free practice.

Source: https://practicecalculus.com/courses/duke-math-111l
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 111L
- **Credits:** Not stated on the department course page
- **Offered:** Fall and/or spring
- **Prerequisites:** Not open to students with credit for MATH 105L, 106L or 121; registration depends on a proctored Calculus I Placement Test on the first day of class
- **Covers:** Differentiation, transcendental functions, optimization, differential equations, numerical approximation, Euler's method, the Fundamental Theorem, separation of variables, slope fields and mathematical modeling
- **Next course:** MATH 112L (Laboratory Calculus II)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 111L is Duke's introductory calculus course with a laboratory component.
- It emphasizes lab projects, group work and written reports alongside the calculus.
- Beyond derivatives and the Fundamental Theorem, it includes differential equations, slope fields and Euler's method.
- Registration depends on a proctored Calculus I Placement Test on the first day of class.

**Duke MATH 111L**, Laboratory Calculus I, is the first calculus course for many Duke students. The "L" stands for its laboratory component: alongside lectures, students work on lab projects in groups and write reports. This guide lists what the department says the course covers and links each topic to free practice with instant feedback.

## What MATH 111L covers

The Duke math department describes MATH 111L as introductory calculus with a laboratory component, with an emphasis on laboratory projects, group work and written reports. The listed topics are differentiation, transcendental functions, optimization, differential equations, numerical approximations, Euler's method, the Fundamental Theorem, separation of variables, slope fields, and mathematical modeling.

That is a standard **Calculus 1** core with an early, practical introduction to differential equations. Slope fields, separation of variables and Euler's method usually appear in Calculus 2 elsewhere, so MATH 111L gives you a head start on them.

The course is typically offered in the fall and/or spring. It is not open to students who already have credit for MATH 105L, 106L or 121, and registration depends on a proctored Calculus I Placement Test on the first day of class.

## MATH 111L topic map

| MATH 111L topic | Practice | Read |
| --- | --- | --- |
| Differentiation | [Power rule](/learn/calculus-1/derivative-rules/power-rule) | [Power rule](/rules/power-rule), [chain rule](/rules/chain-rule) |
| Transcendental functions | [Exponential and log derivatives](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Derivatives of exponential and log functions](/blog/derivatives-of-exponential-and-logarithmic-functions) |
| Trig functions | [Trig derivatives](/learn/calculus-1/derivative-rules/trig-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| Numerical approximation | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| The Fundamental Theorem | [Fundamental Theorem, part 2](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Accumulation | [Accumulation functions](/learn/calculus-1/integrals/accumulation-functions) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Slope fields | [Slope fields](/learn/calculus-2/improper-integrals-and-applications/slope-fields) | [Differential equations practice](/practice-problems/differential-equations) |
| Separation of variables | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Antiderivatives](/blog/antiderivatives) |
| Euler's method | [Euler's method](/learn/calculus-2/improper-integrals-and-applications/eulers-method) | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) |
| Mathematical modeling | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) | [Exponential functions](/functions/exponential-functions) |

## What the lab component means for you

The lab format changes how you study. Instead of only solving short exercises, you explain your reasoning in written reports and work with classmates on longer projects, usually about modeling something real. Two habits help: practice writing a clear sentence about what each derivative or integral means in context, and keep your hand computation fast so the lab time goes to ideas, not arithmetic.

## Example problems at the MATH 111L level

**1. A transcendental derivative.** $\frac{d}{dx}\left(x e^{2x}\right) = e^{2x} + 2x e^{2x}$.

**2. Separation of variables.** Solve $\frac{dy}{dt} = 0.3y$ with $y(0) = 50$. Separating gives $\ln|y| = 0.3t + C$, so $y = 50e^{0.3t}$.

**3. Euler's method.** For $y' = y$, $y(0) = 1$, step $0.5$: $y(0.5) \approx 1 + 0.5 \cdot 1 = 1.5$ and $y(1) \approx 1.5 + 0.5 \cdot 1.5 = 2.25$, compared to the exact value $e \approx 2.718$.

## How to do well in MATH 111L

- **Prepare for the placement test.** Review [functions](/learn/calculus-1/functions-warmup/function-notation), [exponentials and logs](/learn/calculus-1/functions-warmup/exponentials-and-logs) and basic limits before the first day.
- **Practice derivative rules daily.** Fast differentiation frees up time for the lab projects.
- **Read slope fields carefully.** A slope field is a picture of a differential equation, so connect each arrow to the value of $\frac{dy}{dx}$.
- **Explain in words.** The written reports reward clear explanation, so practice describing what your answers mean.

When you finish, see the [MATH 112L guide](/courses/duke-math-112l) for Calculus II.

## Frequently asked questions

### What is Duke MATH 111L?

MATH 111L, Laboratory Calculus I, is introductory calculus with a laboratory component. Duke's math department lists differentiation, transcendental functions, optimization, differential equations, numerical approximation, Euler's method, the Fundamental Theorem, separation of variables, slope fields and mathematical modeling.

### What does the "L" in MATH 111L mean?

It marks the laboratory component. The course emphasizes lab projects, group work and written reports.

### Is there a placement test for MATH 111L?

Yes. The department says registration is contingent on a proctored Calculus I Placement Test on the first day of class.

### Who cannot take MATH 111L?

Students who already have credit for MATH 105L, 106L or 121.

### What comes after MATH 111L?

MATH 112L, Laboratory Calculus II, which lists MATH 106L or 111L as its prerequisite.