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Read aloud Readability settings Prev Up Next \(\newcommand{\Z}{\mathbb{Z}}
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Activity 7.4 Supplement
Aside: Read the next lab early!
Before coming to the
next lab session,
Lab 8 , read the special note at the beginning of that lab. You are asked to do some reading outside of class before you begin working on the lab in class. More explanation is given in that note.
Exercises Exercises
1.
Find a formula for
\(\lz{y}{x}\) and use that formula to determine the
\(x\) -coordinate at each of the two points where the curve crosses the
\(x\) -axis. (Note: The tangent line to the curve is vertical at each of these points.) Scales have deliberately been omitted in
Figure 7.4.1 .
Figure 7.4.1. \(x\fe{\sin}{xy} = y\)
2. For the curve
\(x\fe{\sin}{xy} = y\) plotted in
Figure 7.4.1 , find the slope of the tangent line at
\(\point{\sqrt{\sfrac{\pi}{3}}}{\sqrt{\sfrac{\pi}{12}}}\text{.}\)
3. Solutions to the equation
\(\fe{\ln}{x^2y^2} = x + y\) are graphed in
Figure 7.4.2 .
Determine the equation of the tangent line to this curve at the point
\(\point{1}{-1}\text{.}\)
Figure 7.4.2. \(\fe{\ln}{x^2y^2} = x + y\)
Hint .
It is easier to differentiate if you first use rules of logarithms to completely expand the logarithmic expression.
Exercise Group.
Use the process of logarithmic differentiation to find a first derivative formula for each of the following functions.
4.
\(y = \dfrac{x\fe{\sin}{x}}{\sqrt{x-1}}\)
5.
\(y = \dfrac{e^{2x}}{\fe{\sin^4}{x}\sqrt[4]{x^5}}\)
6.
\(y = \dfrac{\fe{\ln}{4x^3}}{x^5\fe{\ln}{x}}\)
