Author: Suman Ganguli (Page 8 of 9)

HW#1:

HW#1 will be due next Wednesday (Sept 11). Please write out the solutions to the following exercises from the OpenStax textbook on paper and hand them in on Wednesday:

Topics

We continued discussing the main topic of Calculus 1–how to find the “steepness” or “rate of change” of a nonlinear function (curve) at a given point (a, f(a)). This material is discussed in the first half of Sec 2.1 of the textbook:

https://openstax.org/books/calculus-volume-1/pages/2-1-a-preview-of-calculus

The steepness/rate of change of a function f(x) at (a, f(a)) is the slope of the tangent line to the curve at the point. We discussed how we can approximate the slope of the tangent line as the slope of a “secant line”–a line through the given point (a, f(a)) and a “closeby” point (a+h, f(a+h)). The slope of the secant line is given by a difference quotient (rise over run). This is what’s shown in Figures 2.4 and 2.5 in the textbook:

In order to get the slope of the tangent line, we calculate what happens to this different quotient as h “goes to 0” (i.e., gets closer and closer to 0)–this is called “taking the limit as h goes to 0.”

This quantity is what’s called the derivative of f(x) at x=a, denoted f'(a).

Topics

We went over the Course information sheet (pdf available on OpenLab Files), then gave an overview of the main ideas of precalculus

Course information sheet

Links discussed on Course Information sheet:

• Brightspace: https://brightspace.cuny.edu/d2l/home/430658
• OpenLab — create account, join the course group here: https://openlab.citytech.cuny.edu/groups/mat1475-ganguli-fall2024/
• OpenStax textbook: https://openstax.org/details/books/calculus-volume-1

Overview of precalculus concepts – functions and their graphs!

In particular, review linear function and graphing lines:

OpenStax Ch 1 is a review of precalculus — we did Example 1.1 from Sec 1.1:

Please write out the exercises from Checkpoint 1.1 in Sec 1.1 (similar to Example 1.1) which will be part of HW#1.

Also study Figures 2.2, 2.3, and 2.4 and the discussion of them in Sec 2.1 (“A Preview of Calculus”). Additional HW exercises will be assigned for HW#1 related to these figures and concepts: equations/slopes of lines, the rate of change of a function at a point, “secant lines” and their slopes, and tangent lines. Try to understand these–these are the key ideas in calculus!

Review of point-slope equation of a line

In particular, it will be useful to review the point-slope equation of a line (which we can write down when we know a point on the line and the slope of the line)–this is what we will use to write down the equation of the tangent line.

This video gives a good review–you should understand the concepts and techniques discussed in this: