# Michigan Math 115 Calculus I: Topics, Study Guide and Practice

> A study guide for University of Michigan Math 115 (Calculus I): functions, derivatives, applications and a first look at integrals, with practice.

Source: https://practicecalculus.com/courses/university-of-michigan-math-115
Updated: 2026-11-03

## At a glance

- **Course code:** Math 115
- **Credits:** 4 credits (no credit after any Math course numbered 111 or higher)
- **Offered:** Typically every term: Fall, Winter, Spring and Summer
- **Prerequisites:** 3 to 4 years of high school math, including trigonometry
- **Covers:** Functions and graphs, derivatives and their real-world applications, and an introduction to integration
- **Next course:** Math 116 (Calculus II)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- Math 115 is the first course in Michigan's standard calculus sequence: Math 115, Math 116 and Math 215.
- It covers functions and graphs, derivatives and their applications, and a first introduction to integration.
- The course looks at every idea four ways: graphs, tables, formulas and words, so expect to read and explain, not only compute.
- After Math 115 the natural next step is Math 116 (Calculus II); Math 185 is the more theoretical honors alternative.

**Michigan Math 115** is Calculus I at the University of Michigan, the first course in the standard sequence of Math 115, Math 116 and Math 215. It is taken by future math, science and engineering majors and by students heading into many other fields. This guide maps every Math 115 topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What Michigan Math 115 covers

The official course page describes Math 115 as a free-standing course on the ideas of single-variable calculus: functions, how they change, and how change accumulates. Topics include functions and graphs, derivatives with applications to real problems in many fields, and an introduction to integration.

What makes the course distinctive is its approach. Every concept appears from four points of view: geometric (graphs), numeric (tables), symbolic (formulas) and verbal. Modeling uses linear, polynomial, exponential, logarithmic and trigonometric functions, and the emphasis is on understanding and problem solving rather than rote procedure or proof.

Compared with a standard **Calculus 1** course, Math 115 covers the same core of derivatives and their applications, then stops at an introduction to the definite integral. Integration techniques wait for Math 116.

## Math 115 topic map

| Math 115 topic | Practice | Read |
| --- | --- | --- |
| Functions and graphs | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Linear functions](/functions/linear-function), [exponential functions](/functions/exponential-functions) |
| Exponentials, logs and trig models | [Exponentials and logarithms](/learn/calculus-1/functions-warmup/exponentials-and-logs) | [Natural logarithm](/functions/natural-logarithm), [sine function](/functions/sine-function) |
| Rates of change | [Average rate of change](/learn/calculus-1/the-derivative/average-rate-of-change) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Limits and continuity | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [continuity](/blog/continuity) |
| The derivative and its meaning | [Notation and meaning](/learn/calculus-1/the-derivative/derivative-notation-and-meaning) | [Power rule](/blog/power-rule) |
| Differentiation rules | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule), [product and quotient rules](/blog/product-and-quotient-rules) |
| Implicit differentiation | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Implicit differentiation](/blog/implicit-differentiation) |
| Graphs from derivatives | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching](/blog/curve-sketching-with-derivatives) |
| Optimization and related rates | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [related rates](/blog/related-rates) |
| Linear approximation | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| Introduction to integration | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums), [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |

## Math 115 in the Michigan sequence

| Course | Main focus | Honors or alternative |
| --- | --- | --- |
| Math 115 (Calculus I) | Functions, derivatives, introduction to integration | Math 185, Math 275 |
| Math 116 (Calculus II) | Integration techniques and applications, series, Taylor series | Math 186, Math 156 |
| Math 215 (Calculus III) | Multivariable and vector calculus | Math 205, Math 285 |

If your preparation is not yet ready for Math 115, the department points to Math 105 (Data, Functions and Graphs) first. Strong Math 115 students can talk to an honors advisor about Math 176. Continue with the [Michigan Math 116 guide](/courses/university-of-michigan-math-116), then the [Math 215 guide](/courses/university-of-michigan-math-215).

## Example problems at the Math 115 level

**1. A derivative from a table.** A function has $f(1.9) = 7.6$ and $f(2.1) = 8.4$. Estimate $f'(2)$ with a symmetric difference quotient:

$$f'(2) \approx \frac{f(2.1) - f(1.9)}{2.1 - 1.9} = \frac{0.8}{0.2} = 4$$

**2. A chain rule derivative.** Differentiate $y = \ln(x^2 + 1)$:

$$\frac{dy}{dx} = \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}$$

**3. An optimization problem.** A rectangle has perimeter 40. Its area is $A = x(20 - x) = 20x - x^2$, so $A'(x) = 20 - 2x = 0$ gives $x = 10$. Since $A''(x) = -2 < 0$, this is a maximum: a 10 by 10 square with area 100.

## How to do well in Michigan Math 115

- **Practice all four views.** Read derivatives off graphs and tables, not only formulas. Our [derivative lessons](/learn/calculus-1/the-derivative/derivative-notation-and-meaning) mix graphical and symbolic questions.
- **Explain your answers in words.** When you finish a problem, state what the number means with units, for example "the tank is draining at 3 liters per minute."
- **Keep the rules handy.** The [calculus formula sheet](/formulas) lists every derivative rule on one page.
- **Learn from mistakes.** Our guide to [common calculus mistakes](/blog/common-calculus-mistakes) covers the algebra and chain rule slips that cost the most points.
- **Rehearse before exams** with timed [practice exams](/practice), and use the [exam preparation guide](/blog/how-to-prepare-for-a-calculus-exam) to plan your review.

> **Tip:** modeling questions often hinge on exponential growth. If $e^x$ and $\ln x$ feel shaky, start with the [exponentials and logarithms warm-up](/learn/calculus-1/functions-warmup/exponentials-and-logs) before the derivative rules.

## Frequently asked questions

### What is Math 115 at the University of Michigan?

Math 115 is Calculus I, a 4-credit course on single-variable calculus. It covers functions and graphs, derivatives and how they model real situations, and an introduction to integration, and it is usually offered in every term.

### What comes after Math 115?

Math 116 (Calculus II) is the natural successor. A student who has done well in Math 115 can talk to a math honors advisor about entering the honors sequence with Math 176.

### Is Michigan Math 115 hard?

It asks for more than computation. Because the course presents ideas through graphs, tables, formulas and written explanations, students who only memorize procedures tend to struggle. Steady practice with all four views makes it much more manageable.

### How can I prepare for Math 115?

Review functions, exponent rules, logarithms and trig values, since the course models with linear, polynomial, exponential, logarithmic and trigonometric functions. Then work through limits and derivatives with short practice problems.

### What is the difference between Math 115 and Math 185?

Math 185 (Honors Analytic Geometry and Calculus I) is a more theoretical course that covers much of the same material. Math 115 emphasizes conceptual understanding and problem solving rather than proof.