Quiz 6 may include topics through §3.6.
Extra Credit 3. (Due Mar 2.) The curves \(xy=1\) and \(y=x^3\) cross in the first quadrant. What is the angle between their tangent lines where they cross?
Derivatives of the inverse trigonometric functions and \(\ln x\) were derived (§3.5, §3.6).
The scores on Quiz 5 are 30,29,29,27,26,26,25,25,24,23,20,19,17,17,16,15,14,14,12,8,7,6,2,0 with μ=18.0 and σ=8.8. Individual scores are on Blackboard.
More was done with implicit differentiation and the inverse trigonometric functions were reviewed.
Here is a solution for Extra Credit 2.
From the chain rule and the product rule \begin{align*} \frac{d}{dx}(F^3+&G^3+H^3-3FGH)\\ &= 3F^2F'+3G^2G'+3H^2H'-3(F'GH+FG'H+FGH')\\ &= 3F^2G+3G^2H+3H^2F-3(G^2H+FH^2+F^2H)\\ &= 3F^2G+3G^2H+3H^2F-3G^2H-3FH^2-3F^2H\\ &= 0. \end{align*}
Quiz 5 may include topics through §3.4
The main topic of the day was the chain rule. Implicit differentiation was introduced near the end of the class period.
After Quiz 4, the Chain Rule was introduced.
The scores on Quiz 4 are 30,30,30,30,30,29,29,28,28,27,27,25,25,24,24,22,20,19,18,10,10,9,8,5 with μ=22.4 and σ=8.1. Individual scores are on Blackboard.
Extra Credit 2. (Due Feb 19.)
Given functions \(F\), \(G\) and \(H\) satisfying
\[
F'=G,\quad G'=H,\quad H'=F
\]
determine
\[
\frac{d}{dx}(F^3+G^3+H^3-3FGH).
\]
Here is a solution for Extra Credit 1.
This equation can be written as one quadratic in form as follows. \begin{align*} 5^x+2=3(5^{-x})\\ 5^{2x}+2(5^x)=3\\ 5^{2x}+2(5^x)-3=0 \tag{1}\label{p1:line3}\\ \left(5^x+3\right)\left(5^x-1\right)=0\\ 5^x=-3,1 \end{align*} Since \(5^x>0\) for all \(x\), the first is clearly an extraneous solution, so \(5^x=1\) is the only solution to the quadratic. This only happens when \(x=0\). Therefore, \(x=0\) is the only solution to the original equation.
Another way to see this is the only solution is to look at the left-hand side of \eqref{p1:line3} and let \(f(x)=5^{2x}+2(5^x)-3\). The solutions of the equation are where the graph of \(f\) crosses the \(x\)-axis. Examining the graph, given below, shows the only solution is \(x=0\).
Quiz 4 may include topics through §3.3.
The limits \[ \lim_{\theta\to0}\frac{\sin\theta}{\theta}=1 \text{ and } \lim_{\theta\to0}\frac{\cos\theta-1}{\theta}=0 \] were found and used to find the derivatives of the trigonometric functions (§3.3).
Things accomplished included more examples using the product rule, the quotient rule with examples and a few examples showing all the rules mixed together (§3.2). On Friday the trigonometric functions will be differentiated (§3.3).
The scores on Quiz 3 are 30,30,30,30,30,30,30,28,26,24,23,21,20,20,19,13,13,10,6,3,3,2,2,0 with μ=18.5 and σ=11.0. Individual scores are on Blackboard.
The number \(e\) was introduced. It was shown that if \(f(x)=e^x\), then \(f'(x)=f(x)\). After that, the product rule was derived.
Extra Credit 1. (Due Feb 12.) Find all solutions of the equation \(5^x+2=3(5^{-x})\). Show your work.
Quiz 3 will be given on Wednesday, as scheduled. It may include topics through §3.1.
The problems on Quiz 2 are exact copies or very similar clones of the following.
The scores on Quiz 2 are 30,30,30,30,30,30,28,27,26,26,26,20,20,20,16,16,15,10,10,8,8,3,3,0 with μ=19.3 and σ=10.1. Individual scores are on Blackboard.
Please no texting during class!
After Quiz 2, the constant multiple and sum rules for derivatives were derived (§3.1). On Friday, derivatives of the exponential function will be considered along with the product rule (§3.2).
Quiz 2 may include topics through §2.7.
A few more examples of limits at infinity were presented (§2.6). Then the definition of the derivative was presented (§2.7).
The scores on Quiz 1 are 30,30,30,30,30,30,30,30,28,28,27,26,26,25,21,20,20,19,19,18,10,8,0,0 with μ=22.3 and σ=9.3. Individual scores are on Blackboard.
The Spring schedule for the Math Resource Center has been announced:
Monday-Thursday 9:00 AM–7:00 PM
Friday 9:00 AM–7:00 PM
The MRC also has online tutoring through GoBoard. Sessions can be scheduled Monday-Saturday. To set up a tutoring session, complete the online request form.
Some examples of the \(\e\)-\(\d\) definition of limit were presented (§2.4). Continuity was defined (§2.5)
Quiz 1 on Wednesday may include topics through §2.3.
More examples of calculating limits were done and the \(\e\)-\(\d\) definition of limit was presented (§2.3–2.4).
The basic properties of limits were presented (§2.2–2.3).
Please don’t text during class!
Because of all the weather complications, the first quiz will be delayed a week to Wednesday, January 22. In the meantime, please read §2.2–2.3.
The tangent line and velocity problems from §2.1 were presented.
Most of the WebAssign problems correspond to problems in the text. For example, consider the following screen shot from the §2.2 problems in WebAssign.
The SCalcET8 is the code for the textbook, Stewart Calculus, Early Transcendentals. The numbers 2.2.004 mean it is Exercise 4 from §2.2 in the text. When you ask about a problem in class, make note of its coordinates in the text.
Read §2.1–2.2
.During the weather-shortened class period, all that was accomplished was a look at the syllabus.
1. Read the syllabus. It is on Blackboard and can be found elsewhere on this Web site.
2. Get squared away with WebAssign. You will need a WebAssign account and you must purchase access to the online textbook. The access code for this course is louisville 3434 0175. If you are new to WebAssign, then you should watch some of the introductory videos they provide.
3. Chapter 1 in the text is prerequisite material. Look through that chapter to make sure you are familiar with it. In particular, trigonometry with radian measure is important; degree measure is almost never used in calculus. Understanding the inverse trigonometric functions and logarithms will also be necessary. There are some sugggested homework problems from the text on the class Web site. If you find there is a lot of material in Chapter 1 with which you are unfamiliar, then you’re in the wrong course!