Month: September 2023 (Page 2 of 3)

Class Info

• Date: Mon, Sept 18
• Meeting Info: 12p-1:40p, N700

Announcement

We will have our first quiz at the end of class on Wednesday. It will consist of an exercise similar to the exercises in “Derivatives – Limit Definition”, i.e, calculating the derivative of a given function at a given point using the limit definition of the derivative.

You don’t need to memorize the limit definition–it will be provided.

Topics

Went through an example from “Derivatives – Functions”, continuation of the example we started at the end of Class 5–finding the derivative f'(x) for f(x) = 5/x:

We used f'(x) to find the slope and then the equation of the tangent line at x = 7 (using the point-slope form):

We went one step further and wrote down the equation of the tangent line for this function at an arbitrary point (a, f(a)):

We also solved the x-values where the slope of the tangent line is 1.

See this Desmos graph, which shows y = 5/x together with the tangent line at (a, f(a)) for various values of x = a.

We similarly calculated the derivative of f(x) = x^2:

Next time we will generalize this to find “the Power Rule”–the derivative of any power function f(x) = x^n (See Sec 3.3 – Differentiation Rules)

To-Do After Class

• Work on “Derivatives – Functions” (due Thurs Sept 21)

Class Info

• Date: Wed, Sept 13
• Meeting Info: 12p-1:40p, N700

Topics

Introduced alternative notation for “the limit definition of the derivative” using “h” in place of “x – a”:

Figure and definition from textbook (Sec 3.1: Defining the Derivative):

Example from WebWork “Derivatives – Limit Definition” (Problem 3) using the new “h” notation:

Started next topic: Sec 3.2 – The Derivative as a Function

with example from next WebWork set “Derivatives – Functions”:

To-Do After Class

• Finish “Derivatives – Limit Definition” (due Monday, Sept 18)
• Start next WebWork set “Derivatives – Functions” (due Thurs Sept 21)

Class Info

• Date: Mon, Sept 11
• Meeting Info: 12p-1:40p, N700

Topics

• Introduced “delta y over delta x” notation for slope of the secant line:
• Examples from WebWork “Derivatives – Limit Definition”:
  • Went through guiding text in WebWork exercises:
  • Note: what in this WebWork set is called “instantaneous slope” is also called the “instantaneous rate of change” and is “the slope of the tangent line at x=a”
    • this is what we will call the derivative of f at x = a, for which we will use the notation f'(a)
  • Revisited this figure and discussed in context of new notation:
  • Went through Derivatives – Limit Definition: Problem 1:
  • 
  • We went one step further and found the equation of the tangent line to f(x) at that point – using the point-slope form (since we were given the point and calculated the slope!)
  • See the Desmos graph illustrating this example here: https://www.desmos.com/calculator/kfllhpu38j
  • Derivatives – Limit Definition: Problem 2:
  • 

To-Do After Class

• Work through “Derivatives – Limit Definition” (due Monday, Sept 18)
• Read OpenStax Sec 3.1: Defining the Derivative