Author: Suman Ganguli (Page 2 of 3)

Reminder – HW#1 & Quiz #1 Wednesday:

HW#1 will be due Wednesday (Sept 11). Please write out the solutions to the following exercises from the OpenStax textbook on paper and hand them in on Wednesday. Also we will take a short quiz at the end of class Wednesday, with one or two exercises similar to the HW exercises. This quiz will be open book–you can use your HW and class notes.

We went through the concepts and techniques for the exercises in HW#1:

• simplifying f(a+h) for a given function f(x) — with the followup of simplifying f(a+h) – f(a), and then [f(a+h) – f(a)]/h, for the definition of the derivative!
• equations of lines, in particular the point-slope form (since this is what we will use to write down equations of tangent lines
• finding slopes of secant lines

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).