# Rutgers Math 151 (01:640:151) Calculus I: Study Guide and Practice > A study guide for Rutgers Math 151, Calculus I for the Mathematical and Physical Sciences: limits, derivatives, the Mean Value Theorem and integrals. Source: https://practicecalculus.com/courses/rutgers-01-640-151 Updated: 2026-11-03 ## At a glance - **Course code:** 01:640:151 (Math 151) - **Credits:** 4 - **Offered:** Fall, spring and summer - **Prerequisites:** Math 112 or Math 115, or placement - **Covers:** Differential calculus of rational, trigonometric and exponential functions and their inverses, applications through the Mean Value Theorem, an introduction to integral calculus - **Next course:** 01:640:152 Calculus II for the Mathematical and Physical Sciences - **Practice here:** [Calculus 1](/learn/calculus-1) ## Key takeaways - Math 151 is the first course in the three-semester Rutgers New Brunswick calculus sequence for the mathematical and physical sciences. - It is a 4-credit course offered in fall, spring and summer, with two lectures and one workshop per week. - It covers derivatives of the elementary functions and their inverses, applications through the Mean Value Theorem, and an introduction to integrals. - The School of Engineering requires Math 151 and does not accept Math 135 for Calculus I. **Rutgers Math 151** (01:640:151, Calculus I for the Mathematical and Physical Sciences) is the first semester of the three-semester calculus sequence at Rutgers New Brunswick for students in the mathematical and physical sciences, and the School of Engineering requires it. This guide maps each Math 151 topic to a lesson you can practice with instant feedback. ## What Rutgers Math 151 covers The Rutgers math department describes Math 151 as differential calculus of the elementary functions of one real variable (rational, trigonometric and exponential functions and their inverses), various applications through the Mean Value Theorem, and an introduction to integral calculus. Its learning goals include computing limits, derivatives and integrals, understanding continuity and differentiability, using first and second derivatives to find the shape of a graph, and solving optimization problems. Each week has two lectures and a smaller workshop where students solve in-depth problems in groups. The department says workshop problems form the basis for some exam questions. In standard terms, this is **Calculus 1**, and it follows the PracticeCalculus Calculus 1 path closely. ## Math 151 topic map | Math 151 topic | Practice | Read | | --- | --- | --- | | Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits) | | Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity) | | Differentiability | [When derivatives fail](/learn/calculus-1/the-derivative/differentiability) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) | | Derivative rules | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule), [quotient rule](/rules/quotient-rule) | | Trig and exponential derivatives | [Chain rule: e and ln](/learn/calculus-1/derivative-rules/chain-rule-exponentials) | [Derivatives of exponential and log functions](/blog/derivatives-of-exponential-and-logarithmic-functions) | | Inverse functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Inverse function rule](/rules/inverse-function-rule), [arctangent](/functions/arctangent) | | Mean Value Theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem) | | Shape of a graph | [Increasing and decreasing](/learn/calculus-1/applications-of-derivatives/increasing-and-decreasing) | [Curve sketching](/blog/curve-sketching-with-derivatives) | | Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) | | L'Hôpital's rule | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/blog/lhopitals-rule) | | Introduction to integrals | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) | ## Math 151 vs Math 135 | Course | Intended for | What follows | | --- | --- | --- | | Math 151 | Mathematical and physical sciences, School of Engineering | [Math 152](/courses/rutgers-01-640-152), then [Math 251](/courses/rutgers-01-640-251) | | [Math 135](/courses/rutgers-01-640-135) | Many biological and social science majors, Rutgers Business School | Math 136, or Math 152 (not recommended) | Both are 4 credits with the same prerequisite. The department notes that Math 136 does not satisfy the prerequisite for Math 251, so if multivariable calculus is in your future, start with Math 151. ## Example problems at the Math 151 level **1. An inverse trig derivative.** By the chain rule and $\frac{d}{du}\arctan u = \frac{1}{1 + u^2}$: $$\frac{d}{dx}\arctan(x^2) = \frac{2x}{1 + x^4}$$ **2. The Mean Value Theorem.** For $f(x) = x^2$ on $[1, 3]$, the average slope is $\frac{9 - 1}{3 - 1} = 4$. Solving $f'(c) = 2c = 4$ gives $c = 2$, which lies in $(1, 3)$. **3. Shape of a graph.** For $f(x) = x^3 - 3x$: $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$, and $f''(x) = 6x$. So $x = -1$ is a local maximum with $f(-1) = 2$, $x = 1$ is a local minimum with $f(1) = -2$, and the graph changes concavity at $x = 0$. ## How to do well in Math 151 - **Treat workshops as exam previews.** Since workshop problems feed into exams, redo each one alone a few days later. - **Learn inverse function derivatives early.** Logarithms and inverse trig functions are named in the course description. The [inverse function rule](/rules/inverse-function-rule) explains where their derivatives come from. - **Use the formula sheet while you practice**, then without it. The [calculus formula sheet](/formulas) lists every rule. - **Avoid the classic slips** in [common calculus mistakes](/blog/common-calculus-mistakes), and take timed [practice exams](/practice) before each midterm. > **Tip:** to find the shape of a graph, make a sign chart for $f'$ and another for $f''$. Every increasing, decreasing and concavity question reads straight off them. ## Frequently asked questions ### What is Rutgers Math 151? Math 151 (01:640:151) is Calculus I for the Mathematical and Physical Sciences at Rutgers New Brunswick. It covers differential calculus of rational, trigonometric and exponential functions and their inverses, applications via the Mean Value Theorem, and an introduction to integral calculus. ### Should I take Math 151 or Math 135? Math 151 is intended for mathematical and physical science majors and is required by the School of Engineering. Math 135 is aimed at many biological and social science majors. The department strongly recommends Math 151 if you will need Math 152 or Math 251 later. ### What comes after Math 151? Math 152, Calculus II for the Mathematical and Physical Sciences, and then Math 251 Multivariable Calculus. ### Is Math 151 hard? It is a full Calculus 1 course with weekly workshops and quizzes, so it rewards steady work. The department also notes that students who fail Calculus I twice cannot take it again at New Brunswick, which makes keeping up from week one important. ### Is there extra support for Math 151? Yes. The department offers Math 157, a 1-credit practicum taken alongside Math 151 that applies the course's methods to problems.