# When an electric current passes through two resistors with a resistance r1 and r2. connected in parallel, the combined resistance, R, can be calculated from the equation: frac{1}{R}=frac{1}{2r1}+frac{1}{3r2} where R, r1 & r2 are greater than 0. Assume that r2 is constant. Show that RR is an increasing function of r1.

Question
Functions
When an electric current passes through two resistors with a resistance r1 and r2. connected in parallel, the combined resistance, R, can be calculated from the equation:
$$\displaystyle{\frac{{{1}}}{{{R}}}}={\frac{{{1}}}{{{2}{r}{1}}}}+{\frac{{{1}}}{{{3}{r}{2}}}}$$
where R, r1 & r2 are greater than 0. Assume that r2 is constant. Show that RR is an increasing function of r1.

2021-01-17
Firstly, let's invert this equation so we don't have to do implicit differentiation.
$$\displaystyle{R}={\frac{{{1}}}{{{\frac{{{2}}}{{{r}{1}}}}+{\frac{{{1}}}{{{3}{r}{2}}}}}}}={\frac{{{1}}}{{{\frac{{{6}{r}{2}+{r}{1}}}{{{3}{r}{1}{r}{2}}}}}}}={\frac{{{3}{r}{1}{r}{2}}}{{{6}{r}{2}+{r}{1}}}}$$
To see if this function is increasing in r1, we need to take its derivative with respect to r1 and check that it is nonnegative for every value of r1.
So let's do that. Remember to use the quotient rule.
$$\displaystyle{R}'{\left({r}{1}\right)}=\frac{{{\left({3}{r}{2}\right)}{\left({6}{r}{2}+{r}{1}\right)}-{3}{r}{1}{r}{2}{\left({1}\right)}}}{{\left({1}\right)}^{{2}}}={\left({18}{r}{2}^{{2}}\right)}+{3}{r}{2}{r}{1}-{3}{r}{2}{r}{1}={18}{r}{2}^{{2}}$$
Because r2>0, we see that R'(r1) > 0 everywhere. Hence R is an increasing function of r1 Q.E.D.

### Relevant Questions

When an electric current passes through two resistors with a resistance r1 and r2. connected in parallel, the combined resistance, R, can be calculated from the equation:
$$\frac{1}{R}=\frac{1}{2r1}+\frac{1}{3r2}$$
where R, r1 & r2 are greater than 0. Assume that r2 is constant. Show that RR is an increasing function of r1.
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on the x-axis with a $$\neq \pm r,$$ use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]
An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by
$$\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.$$
$$\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(A) Complete the table below.
$$\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(Round to one decimal place as​ needed.)
$$A. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
$$B. 20602060xf(x)$$
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
$$C. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
$$D.20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ and for 35
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$
The mileage for the tire pressure $$\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ is
The mileage for the tire pressure $$\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is
(Round to two decimal places as​ needed.)
(D) Write a brief description of the relationship between tire pressure and mileage.
A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase.
B. As tire pressure​ increases, mileage decreases.
C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.
D. As tire pressure​ increases, mileage increases.