Math 130-03, Lab 11


Exercise 1


We have related the sign of the derivative of a function to whether that function is increasing or decreasing, and we have related the sign of the second derivative to whether the function is concave up or concave down. It's easy to distinguish increasing from decreasing and concave up from concave down by looking at a graph. The following applet has five example functions. You can switch from one example to another using the menu at the top of the applet. For each of the examples, answer the following questions:


(a) Where is the function increasing, and where is it decreasing?

(b) Where is the function concave up, and where is it concave down?

(c) At which values of $ x$ is the derivative of the function equal to zero?

Your answers will be approximate, since you can only read approximate values from the graph.


Exercise 2


This exercise is pretty much the reverse of the previous exercise: Given some information about a function, its derivative, and its second derivative, sketch a graph that agrees with all the information. In this case, there are usually many different right answers (but any answer is unambiguously either right or wrong).


(a) Draw a graph of a function that satisfies all of the following conditions:

        (1) $ f(-2)=f(0)=0$

        (2) $ f'(x)>0$ for $ x<-1$, and $ f'(x)<0$ for $ x>-1$

        (3) $ f''(x)<0$ for $ -2<x<1$, and $ f''(x)>0$ for $ x<-2$ and for $ x>1$


(b) Draw a graph of a function that satisfies all of the following conditions:

        (1) $ f$ has a vertical asymptote at $ x=1$

        (2) $ f$ is always concave up, at all points where it is defined


(c) Draw a graph of a function that satisfies all of the following conditions:

        (1) $ f'(x)>0$ for $ x>0$ and for $ x<0$

        (2) $ f''(x)>0$ for $ x>0$, and $ f''(x)<0$ for $ x<0$

        (3) $ f(1)=f(-1)=0$


(d) Try to explain why it is not possible to draw a graph that satisfies all of the following conditions:

        (1) $ f'(0)=0$

        (2) $ f(x)>0$ for all $ x>0$

        (3) $ f''(x)<0$ for all $ x>0$


(e) Draw a graph of a function that satisfies all of the following conditions:

        (1) $ f(0)=1$

        (2) $ f'(0)=0$

        (3) $ f'(0)>0$ for $ x<0$, and $ f'(x)<0$ for $ x>0$.

        (4) $ f(x)>0$ for all $ x$.

What can you say about the second derivative of this function? Why?



Exercise 3


Finally, we are at the point where there are no more derivative formulas to learn! Here is one final set of derivatives to work on:

$\displaystyle \textbf{(a)\ \ }$ $\displaystyle \displaystyle \frac{d}{dx}\, \big(\pi x + 2^x\arcsin(x)\big)$ $\displaystyle \textbf{(b)\ \ }$ $\displaystyle \displaystyle \frac{d}{dt}\, \sin(e^t)\arctan(\sqrt{t\,})$    
$\displaystyle \noalign{\medskip } \textbf{(c)\ \ }$ $\displaystyle \displaystyle \frac{d}{dx}\, \left(\frac{\arccos{x}}{1-\arctan{x}}\right)$ $\displaystyle \textbf{(d)\ \ }$ $\displaystyle \displaystyle \frac{d}{d\theta}\, \arctan(\cos(\theta)+\sin(2\theta))$    
$\displaystyle \ $    

David Eck, April 2001