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

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 in the Michigan sequence

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, then the Math 215 guide.

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:

2. A chain rule derivative. Differentiate y = \ln(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 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 lists every derivative rule on one page.
• Learn from mistakes. Our guide to common calculus mistakes covers the algebra and chain rule slips that cost the most points.
• Rehearse before exams with timed practice exams, and use the exam preparation guide 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 before the derivative rules.