Print preview
Handout Wednesday, 9/3
Learning target quizzes discussion. Reminder: you try this again so that you can demonstrate mastery. Your next opportunity is next Tuesday. Let’s talk a little bit about instantaneous velocity.
Let’s continue investigating the derivative of a function at a point.
-
Look at the interactive applet the textbook links to. Compare this to the definition of the derivative we discussed in class and the idea of the instantaneous velocity.
-
We can always approximate a derivative at a point by approximating the limit of the average rate of change. In other words, approximating the difference quotient at that point.
-
Sometimes, we can find the derivative exactly by computing that limit exactly.
-
Let’s think of this as two distinct steps: First, we want to find an expression whose limit is the derivative at a point. Second, we want to evaluate that limit.
-
Here are some examples. For each function below, write a limit that represents the derivative at the given point.
-
Let \(f(x) = \sqrt{x+1}\text{.}\) Find an expression for \(f'(3)\text{,}\) the derivative at \(a = 3\text{.}\) Then repeat for \(f'(8)\text{.}\)
-
Let \(f(x) = x^2 + 5x\text{.}\) Find an expression for \(f'(2)\text{,}\) the derivative at \(a = 2\text{.}\)
-
Let \(f(x) = x^3 - 2x + 1\text{.}\) Find an expression for \(f'(1)\text{,}\) the derivative at \(a = 1\text{.}\)
-
Let \(f(x) = \sin(x) + \cos(x)\text{.}\) Find an expression for \(f'(0)\text{,}\) the derivative at \(a = 0\text{.}\)
Pay attention to how the substitution of the function and the difference quotient interact.
Now, evaluate some of the limits we found. Note that the last one... no clue what to do there.
