Rutgers Math 151 (01:640:151) Calculus I: Study Guide and Practice

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 vs Math 135

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}:

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 explains where their derivatives come from.
• Use the formula sheet while you practice, then without it. The calculus formula sheet lists every rule.
• Avoid the classic slips in common calculus mistakes, and take timed practice exams 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.