Virginia Tech MATH 1225 Calculus 1: Topics, Study Guide, Practice

MATH 1225 is Calculus of a Single Variable at Virginia Tech, the first-semester calculus course for engineering, science and math students. It is a 4-credit course in the Pathways curriculum for Quantitative and Computational Thinking, and it is taught with common exams across sections. This guide maps every MATH 1225 topic to free practice, so you can learn each idea by solving problems with instant feedback.

What MATH 1225 covers

The Virginia Tech math department describes MATH 1225 as a course in limits, continuity, differentiation and integration, and lays it out in five units:

1. Tangents, limits and derivatives: limit laws, continuity, limits at infinity and horizontal asymptotes, the derivative as a rate of change and as a function.
2. Techniques for derivatives: polynomial and exponential derivatives, product and quotient rules, trig derivatives, the chain rule, implicit and logarithmic differentiation, related rates.
3. Maxima, minima and curve sketching: extreme values, the Mean Value Theorem, how derivatives shape a graph, optimization.
4. Basic integration: antiderivatives, areas and distances, the definite integral, the Fundamental Theorem, net change, substitution.
5. Applications: linear approximation, Newton's method, rates of change in the sciences such as particle motion.

That is a complete standard Calculus 1 course.

MATH 1225 topic map

The Virginia Tech calculus sequence

MATH 1225 and MATH 1226 share a title because they are two halves of one single-variable course. The department also lists MATH 1025 and MATH 1026, Elementary Calculus, as a separate track. See the MATH 1226 guide and the MATH 2204 guide.

Example problems at the MATH 1225 level

1. A limit at infinity. Find the horizontal asymptote of f(x) = \frac{3x^2 + 1}{x^2 - 4}. Divide top and bottom by x^2:

so the asymptote is y = 3.

2. Logarithmic differentiation. Differentiate y = x^x for x > 0. Take logs: \ln y = x\ln x. Differentiate: \frac{y'}{y} = \ln x + 1, so

3. Newton's method. Approximate \sqrt{2} as the root of f(x) = x^2 - 2 starting from x_0 = 1:

How to do well in MATH 1225

• Plan around the common exams. MATH 1225 has several unit exams in the evening outside class time, so treat each unit as its own mini course and review as you go.
• Make derivative rules automatic with the mixed derivative rules practice lesson before Unit 3.
• Fix the small errors. Our common calculus mistakes guide covers the sign and chain rule slips that cost the most on multiple choice.
• Use one formula sheet for review: the calculus formula sheet.
• Practice under time pressure with practice exams, following how to prepare for a calculus exam.

Tip: when a curve sketching question looks long, split it into a sign chart for f' and a sign chart for f''. The concavity lesson walks through the routine.