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

> A study guide for MATH 1225 at Virginia Tech: limits, derivative rules, curve sketching, optimization and basic integration, mapped to practice.

Source: https://practicecalculus.com/courses/virginia-tech-math-1225
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 1225
- **Credits:** 4
- **Prerequisites:** A score of 80+ on the MATH 1225 Placement Assessment, a C- or better in MATH 1214, or a qualifying AP, CLEP or IB score
- **Covers:** Limits, continuity, derivative rules, related rates, the Mean Value Theorem, curve sketching, optimization, the definite integral, the Fundamental Theorem, substitution, linear approximation and Newton's method
- **Next course:** MATH 1226 (Calculus of a Single Variable)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 1225 is the 4-credit first-semester calculus course at Virginia Tech and part of the Pathways Quantitative and Computational Thinking curriculum.
- The department organizes it in five units: limits and derivatives, derivative techniques, maxima, minima and curve sketching, basic integration, and applications.
- Entry is by the MATH 1225 Placement Assessment, MATH 1214, or a qualifying AP, CLEP or IB score.
- The next course, MATH 1226, requires MATH 1225 with at least a C-.

**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

| MATH 1225 topic | Practice | Read |
| --- | --- | --- |
| Limits and limit laws | [Direct substitution](/learn/calculus-1/limits/direct-substitution) | [Limit laws](/rules/limit-laws), [how to evaluate limits](/blog/how-to-evaluate-limits) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity) |
| Limits at infinity and asymptotes | [Limits at infinity](/learn/calculus-1/limits/limits-at-infinity) | [Limits at infinity](/blog/limits-at-infinity) |
| The derivative as a function | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Product, quotient and trig rules | [The quotient rule](/learn/calculus-1/derivative-rules/quotient-rule) | [Product and quotient rules](/blog/product-and-quotient-rules), [trig derivatives](/blog/derivatives-of-trig-functions) |
| Chain rule | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/blog/chain-rule) |
| Implicit and logarithmic differentiation | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Logarithmic differentiation](/rules/logarithmic-differentiation) |
| Related rates | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates](/blog/related-rates) |
| Extreme values and the Mean Value Theorem | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Mean Value Theorem](/theorems/mean-value-theorem), [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Curve sketching | [Increasing and decreasing](/learn/calculus-1/applications-of-derivatives/increasing-and-decreasing) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| Definite integral and the Fundamental Theorem | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |
| Linear approximation | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |

## The Virginia Tech calculus sequence

| Course | Title | Prerequisite |
| --- | --- | --- |
| MATH 1225 | Calculus of a Single Variable | Placement, MATH 1214, or AP/CLEP/IB |
| MATH 1226 | Calculus of a Single Variable | MATH 1225, C- or better |
| MATH 2204 | Introduction to Multivariable Calculus | Passing MATH 1226 |

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](/courses/virginia-tech-math-1226) and the [MATH 2204 guide](/courses/virginia-tech-math-2204).

## 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 $x^2$:

$$\lim_{x \to \infty} \frac{3 + 1/x^2}{1 - 4/x^2} = 3$$

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

$$y' = x^x(\ln x + 1)$$

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

$$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1 - \frac{-1}{2} = 1.5, \qquad x_2 = 1.5 - \frac{0.25}{3} \approx 1.4167$$

## 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](/learn/calculus-1/derivative-rules/derivative-rules-practice) lesson before Unit 3.
- **Fix the small errors.** Our [common calculus mistakes](/blog/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](/formulas).
- **Practice under time pressure** with [practice exams](/practice), following [how to prepare for a calculus exam](/blog/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](/learn/calculus-1/applications-of-derivatives/second-derivative) lesson walks through the routine.

## Frequently asked questions

### What is MATH 1225 at Virginia Tech?

MATH 1225, Calculus of a Single Variable, is Virginia Tech's first-semester calculus course. It covers limits, continuity, differentiation and integration, from limit laws through the Fundamental Theorem of Calculus and substitution.

### How do I qualify for MATH 1225?

The math department lists three routes: a score of 80 or higher on the MATH 1225 Placement Assessment, a C- or better in MATH 1214, or a qualifying AP, CLEP or IB exam score that appears on your Virginia Tech record.

### What comes after MATH 1225?

MATH 1226, the second semester of Calculus of a Single Variable. It requires MATH 1225 with a grade of C- or better.

### Is MATH 1225 hard?

It covers all of first-semester calculus at a steady pace, and the department runs several common exams during the term. Students who keep up with practice each week, instead of cramming before each exam, tend to do much better.

### How should I prepare for MATH 1225?

Review functions, exponents, logarithms and trig values, then work through limits. The free functions warm-up and limits units on PracticeCalculus follow the same order as Unit 1.