Again, we started with 2 questions. Here we go...

**1. On what intervals is the function, f(x) = x^4 - 4x^3 + 10, both concave down and decreasing?**

*Solution:*

First, we have to find the first and second derivatives of the given function.

Note: Remember, the first derivative determines the graph of f if its increasing or decreasing. The second derivative determines the concavity of the parent function, f. By simplifying the first and second derivatives, we can determine the roots of each equation.

Using these roots, we can use the number line test to find which interval is the graph of "f" concave down and decreasing by substituting values on each interval.

Since the interval (0,2) agrees on both number lines, we can say that on the interval (0,2), the graph of the parent function is both concave down and decreasing.

**2. "f" has a domain [-5,5]. The second derivative is shown below.**

**a) On what interval(s) is "f" concave up?**

*Solution:*

On the interval (-4,1), the graph of the 2nd derivative is above the x-axis. Since the graph of f" is positive, it means that the graph of the first derivative is increasing. Therefore, the graph of the parent function on the interval (-4,1) is concave up.

**b) Where does "f" have any points of inflection?**

*Solution:*

The roots of the 2nd derivative are x = -4,1. On these points, the graph of the first derivative have either a local min or max. If there is a local min or max, then the parent function must be shifting from either positive to negative or negative to positive.

**c) Suppose that f'(1) = 0. Is "f" increasing or decreasing at x=4? Explain.**

*Solution:*

The parent function is decreasing at x = 4 because the graph of the 2nd derivative starts to decrease at x = 1. If the graph of the 2nd derivative is below the x-axis, then the graph of the first derivative must be negative. If the graph of the first derivative is decreasing, then the parent function must be decreasing. The graph of the 2nd derivative is also shifting from positive to negative.

The face example...

Mr. K asked,

*No, because we're only focusing on his face. We can't tell where he is because we don't pay attention to the surroundings. We can apply this concept on the previous problem. In part (c), could we answer the question if we focus our attention on x=1? If we look around the graph, we could make a better and more accurate conclusion to answer the question.*

**"Can you tell if I'm standing on top of a mountain cliff or on a valley with my face zoomed in the picture??"**Lastly, we talked about 2 new rules: Reciprocal and Quotient Derivative Rules

Mr. K told us something to help us remember the quotient rule.

*[key words:*

**LO DI HI MINUS HI DI LO ALL OVER LO LO****HI**-f(x)

**LO**-g(x)

**DI HI**(LO)- f'(x)/g'(x)]

I guess thats about it. Whew, I almost went crazy cuz blogger suddenly went offline on me. I had to start things over again. T.T Anyways, I hope I mentioned everything we did today. Next scribe will be.... uhh... ANH!

## 3 comments:

You guys are on a roll. This is another scribe worthy of the Hall of Fame. Great detail, images and colour. I had a great time reading it. I hope your classmates do too.

Mr. Harbeck

Sargent Park School

I think this post is definitely worthy of the Hall of Fame. At first I just glimpsed at it, and bam it refreshed everything that happened in class that day. If I were to read this post not having been in class, I'd probably feel if I was really there. It's evident Jann put lots of effort into this post. Good stuff.

Nice post Jann! I skimmed through this last week, but I didn't get a chance to post a comment. Today as I was rewriting my messy notes, I had to look back at this post because I couldn't remember how we got the answer for the first question. Thanks for including the number line on your post, it refreshed my memory. The only thing I want to add is to put the label of scribe post and a title too for future reference, when we have to go back and study for the exam. Other than that, good job! I'd like to see Jann's post on the HALL OF FAME too please =)

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