Day 7, Mat128: Calculus A

• Zoom

  Here's a menu of our past zooms...

• You have a new assignment.

• Reminder: I've also assigned a couple of homework problems from section 1.3, which are due Monday.

• Your first exam is next Wednesday; it will cover material through section 1.4: the derivative function.

• One thing that we're asked to do in this activity is look for a pattern. This is a very important activity in mathematics!
  

Definition 1.4.2. Let $f$ be a function and $x$ a value in the function's domain. We define the derivative of f, a new function called $f'$ by the formula \[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \] provided this limit exists.

• Let's talk more about our worksheet:
  • #8: One student contacted me, asking about Problem 8 on the worksheet. I replied:

    "I suggest you try this exercise: you're given a graph, and told to try to guess the derivative function. That's exactly what you're doing in this case!"

    "For example, you can see a place where the derivative function must be equal to zero (the tangent line is horizontal). You can draw in a few tangent lines, and estimate their slopes; those estimates become derivative function value estimates. In the end, you "connect the dots". Also use symmetry: the function given is even (has reflective symmetry across the y-axis). What does that tell you about the derivative function's symmetry (think about how the slopes are related left and right)."

  • #1: Several students continue to be troubled by the approximations to the derivative, offered by forward differences (say).

    A general comment: many students get the idea that there is always a "right answer" in math. In fact, that's the reason some people like math -- they have the sense that it's certain, unlike the troublesome "subjectiveness" of an essay in English, say. The good news is that there are often many answers -- we're just trying to get a good estimate.

• Let's take a look at a couple of the activities in section 1.4:
  • Activity 1.4.2: these are a little troublesome because we don't have scales for the "change in x" and the "change in y"....
    
    
    

  • 1.4.3 Exercises, #3 (this is an important one!)
  • 1.4.3 Exercises, #5 -- some algebra. Let's make it a little harder, however: let's consider \[ f(x)=\frac{1}{(x-4)^2} \]

    Notice the symmetry between the function and its derivative function!

• Let's take a look at a couple of useful animations, referenced in section 1.4:
  • Graphing the derivative function
  • The Derivative as a Function

• Then I'll have you give it a try, using this worksheet:
  • (pdf)

• I really like this exercise, referenced in the text. Fun and challenging!
• Limit laws you can believe in!