Math 111 - Calculus I

This section contains some resources for the Fall 2022 Math 111 Section 23 at NJIT. Hopefully, the notes, problems, and supplementary plots/videos/animations can help deepen understanding or act as a reference for review.

Quiz Solutions

A Couple of Notes

Limits and Continuity

Motivation: People were interested in the speed and direction of planets at any moment in time. So, they wanted to know if they could calculate that by measuring the path of the planets. We can use a concept called “limits” to find this speed and direction.

  • Calculus relates the rate of changes of things. i.e. how quickly does water empty out of a barrel is a ratio of the rate of change of the water relative to the rate of change of time.

Average and Instantaneous Rates of Change

  • Average speed: Divide the distance traveled by the time elapsed

  • Instantaneous speed: The average speed over an infinitely small amount of time

  • Secant line: A line connecting two points on a curve

  • As the two points of a secant line get closer and closer, the slope of the secant approaches the instantaneous velocity

  • Tangent line: A line that only touches one point on a line

  • The slope of a curve at a point \(x\) is the slope of the tangent line that passes through only \(x\)

  • For a curve that represents a quantity (y-axis) over time (x-axis), the slope of the tangent line at a point \(t\) in time is the instantaneous velocity (or rate of change of time) at \(t\)

  • Formula for secant line / average velocity from time \(t_1\) to \(t_2\) where function \(f(t)\) is the position at time \(t\) and \(\Delta\) represents “change in …”: \[ \boxed{\frac{\Delta f}{\Delta t}} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} = \frac{f(t+h) - f(t)}{t+h - t} = \boxed{\frac{f(t+h) - f(t)}{h}} \]

Helpful animation

Animation of Figure 2.6 from the book.