Determine the sign of fx and fy at each indicated point using the contour diagram of f shown below

Suppose we take out an $18,000 car loan at interest rate $r$ and we agree to pay off the loan in $t$ years. The monthly payment, in dollars, is

\begin{equation*} M(r,t) = \frac{1500r}{1-\left(1+\frac{r}{12}\right)^{-12t}}. \end{equation*}

What is the monthly payment if the interest rate is $3\%$ so that $r = 0.03\text{,}$ and we pay the loan off in $t=4$ years?
Suppose the interest rate is fixed at $3\%\text{.}$ Express $M$ as a function $f$ of $t$ alone using $r=0.03\text{.}$ That is, let $f(t) = M(0.03, t)\text{.}$ Sketch the graph of $f$ on the left of Figure 10.2.1. Explain the meaning of the function $f\text{.}$
Figure 10.2.1. Left: Graphs $f(t)= M(0.03, t)\text{.}$ Right: Graph $g(r) = M(r,4)\text{.}$
Find the instantaneous rate of change $f'(4)$ and state the units on this quantity. What information does $f'(4)$ tell us about our car loan? What information does $f'(4)$ tell us about the graph you sketched in (b)?
Express $M$ as a function of $r$ alone, using a fixed time of $t=4\text{.}$ That is, let $g(r) = M(r, 4)\text{.}$ Sketch the graph of $g$ on the right of Figure 10.2.1. Explain the meaning of the function $g\text{.}$
Find the instantaneous rate of change $g'(0.03)$ and state the units on this quantity. What information does $g'(0.03)$ tell us about our car loan? What information does $g'(0.03)$ tell us about the graph you sketched in (d)?

Activity 10.2.5.

The wind chill, as frequently reported, is a measure of how cold it feels outside when the wind is blowing. In Table 10.2.7, the wind chill $w\text{,}$ measured in degrees Fahrenheit, is a function of the wind speed $v\text{,}$ measured in miles per hour, and the ambient air temperature $T\text{,}$ also measured in degrees Fahrenheit. We thus view $w$ as being of the form $w = w(v, T)\text{.}$

Table 10.2.7. Wind chill as a function of wind speed and temperature.

$v \backslash T$$-30$$-25$$-20$$-15$$-10$$-5$$0$$5$$10$$15$$20$$5$$-46$$-40$$-34$$-28$$-22$$-16$$-11$$-5$$1$$7$$13$$10$$-53$$-47$$-41$$-35$$-28$$-22$$-16$$-10$$-4$$3$$9$$15$$-58$$-51$$-45$$-39$$-32$$-26$$-19$$-13$$-7$$0$$6$$20$$-61$$-55$$-48$$-42$$-35$$-29$$-22$$-15$$-9$$-2$$4$$25$$-64$$-58$$-51$$-44$$-37$$-31$$-24$$-17$$-11$$-4$$3$$30$$-67$$-60$$-53$$-46$$-39$$-33$$-26$$-19$$-12$$-5$$1$$35$$-69$$-62$$-55$$-48$$-41$$-34$$-27$$-21$$-14$$-7$$0$$40$$-71$$-64$$-57$$-50$$-43$$-36$$-29$$-22$$-15$$-8$$-1$

Estimate the partial derivative $w_v(20,-10)\text{.}$ What are the units on this quantity and what does it mean? (Recall that we can estimate a partial derivative of a single variable function $f$ using the symmetric difference quotient $\frac{f(x+h)-f(x-h)}{2h}$ for small values of $h\text{.}$ A partial derivative is a derivative of an appropriate trace.)
Estimate the partial derivative $w_T(20,-10)\text{.}$ What are the units on this quantity and what does it mean?
Use your results to estimate the wind chill $w(18, -10)\text{.}$ (Recall from single variable calculus that for a function $f$ of $x\text{,}$ $f(x+h) \approx f(x) + hf'(x)\text{.}$)
Use your results to estimate the wind chill $w(20, -12)\text{.}$
Consider how you might combine your previous results to estimate the wind chill $w(18, -12)\text{.}$ Explain your process.

Activity 10.2.6.

Shown below in Figure 10.2.8 is a contour plot of a function $f\text{.}$ The values of the function on a few of the contours are indicated to the left of the figure.

Figure 10.2.8. A contour plot of $f\text{.}$

Estimate the partial derivative $f_x(-2,-1)\text{.}$ (Hint: How can you find values of $f$ that are of the form $f(-2+h)$ and $f(-2-h)$ so that you can use a symmetric difference quotient?)
Estimate the partial derivative $f_y(-2,-1)\text{.}$
Estimate the partial derivatives $f_x(-1,2)$ and $f_y(-1,2)\text{.}$
Locate, if possible, one point $(x,y)$ where $f_x(x,y)= 0\text{.}$
Locate, if possible, one point $(x,y)$ where $f_x(x,y)\lt 0\text{.}$
Locate, if possible, one point $(x,y)$ where $f_y(x,y)>0\text{.}$
Suppose you have a different function $g\text{,}$ and you know that $g(2,2) = 4\text{,}$ $g_x(2,2) > 0\text{,}$ and $g_y(2,2) > 0\text{.}$ Using this information, sketch a possibility for the contour $g(x,y)=4$ passing through $(2,2)$ on the left side of Figure 10.2.9. Then include possible contours $g(x,y) = 3$ and $g(x,y) = 5\text{.}$
Figure 10.2.9. Plots for contours of $g$ and $h\text{.}$
Suppose you have yet another function $h\text{,}$ and you know that $h(2,2) = 4\text{,}$ $h_x(2,2) \lt 0\text{,}$ and $h_y(2,2) > 0\text{.}$ Using this information, sketch a possible contour $h(x,y)=4$ passing through $(2,2)$ on the right side of Figure 10.2.9. Then include possible contours $h(x,y) = 3$ and $h(x,y) = 5\text{.}$

Determine the sign of $f_x$ and $f_y$ at each indicated point using the contour diagram of $f$ shown below. (The point $P$ is that in the first quadrant, at a positive $x$ and $y$ value; $Q$ through $T$ are located clockwise from $P\text{,}$ so that $Q$ is at a positive $x$ value and negative $y\text{,}$ etc.)

(a) At point $P\text{,}$

$f_x$ is

positive
negative

and

$f_y$ is

positive
negative

(b) At point $Q\text{,}$

$f_x$ is

positive
negative

and

$f_y$ is

positive
negative

$f_x$ is

positive
negative

and

$f_y$ is

positive
negative

Answer. 1

$\text{negative}$

Answer. 2

$\text{negative}$

Answer. 3

$\text{negative}$

Answer. 4

$\text{positive}$

Answer. 5

$\text{positive}$

Answer. 6

$\text{negative}$

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10.

Your monthly car payment in dollars is $P = f(P_0,t,r)\text{,}$ where $$P_0$ is the amount you borrowed, $t$ is the number of months it takes to pay off the loan, and $r$ percent is the interest rate.

(a) Is $\partial P /\partial t$ positive or negative?

positive
negative

Suppose that your bank tells you that the magnitude of $\partial P /\partial t$ is 15.

What are the units of this value?

(For this problem, write our your units in full, writing dollars for $, months for months, percent for %, etc. Note that fractional units generally have a plural numerator and singular denominator.)

(b) Is $\partial P /\partial r$ positive or negative?

positive
negative

Suppose that your bank tells you that the magnitude of $\partial P /\partial r$ is 25.

What are the units of this value?

(For this problem, write our your units in full, writing dollars for $, months for months, percent for %, etc. Note that fractional units generally have a plural numerator and singular denominator.)

For both parts of this problem, be sure you can explain what the practical meanings of the partial derivatives are.

Answer. 1

$\text{negative}$

Answer. 2

${\text{dollars/month}}$

Answer. 3

$\text{positive}$

Answer. 4

${\text{dollars/percent}}$

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11.

An experiment to measure the toxicity of formaldehyde yielded the data in the table below. The values show the percent, $P=f(t,c)\text{,}$ of rats surviving an exposure to formaldehyde at a concentration of $c$ (in parts per million, ppm) after $t$ months.

$t=14$$t=16$$t=18$$t=20$$t=22$$t=24$$c=0$100100100999795$c=2$1009998979592$c=6$969593908680$c=15$969382705836

(a) Estimate $f_t(18,6)\text{:}$

$f_t(18, 6)\approx$

(b) Estimate $f_c(18,6)\text{:}$

$f_c(18, 6)\approx$

(Be sure that you can give the practical meaning of these two values in terms of formaldehyde toxicity.)

Answer. 1

$\frac{90-95}{20-16}$

Answer. 2

$\frac{82-98}{15-2}$

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12.

An airport can be cleared of fog by heating the air. The amount of heat required depends on the air temperature and the wetness of the fog. The figure below shows the heat $H(T,w)$ required (in calories per cubic meter of fog) as a function of the temperature $T$ (in degrees Celsius) and the water content $w$ (in grams per cubic meter of fog). Note that this figure is not a contour diagram, but shows cross-sections of $H$ with $w$ fixed at $0.1\text{,}$ $0.2\text{,}$ $0.3\text{,}$ and $0.4\text{.}$

(a) Estimate $H_T(10, 0.2)\text{:}$

$H_T(10,0.2) \approx$

(Be sure you can interpret this partial derivative in practical terms.)

(b) Make a table of values for $H(T,w)$ from the figure, and use it to estimate $H_T(T,w)$ for each of the following:

$T = 10, w = 0.2$ : $H_T(T,w) \approx$

$T = 20, w = 0.2$ : $H_T(T,w) \approx$

$T = 10, w = 0.3$ : $H_T(T,w) \approx$

$T = 20, w = 0.3$ : $H_T(T,w) \approx$

$T = 10, w = 0.2$ : $H_w(T,w) \approx$

$T = 20, w = 0.2$ : $H_w(T,w) \approx$

$T = 10, w = 0.3$ : $H_w(T,w) \approx$

$T = 20, w = 0.3$ : $H_w(T,w) \approx$

(Be sure you can interpret this partial derivative in practical terms.)

Answer. 1

$\frac{190-240}{10}$

Answer. 2

$\frac{190-240}{10}$

Answer. 3

$\frac{160-190}{10}$

Answer. 4

$\frac{260-330}{10}$

Answer. 5

$\frac{220-260}{10}$

Answer. 6

$\frac{330-240}{0.1}$

Answer. 7

$\frac{260-190}{0.1}$

Answer. 8

$\frac{450-330}{0.1}$

Answer. 9

$\frac{350-260}{0.1}$

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13.

The Heat Index, $I\text{,}$ (measured in apparent degrees F) is a function of the actual temperature $T$ outside (in degrees F) and the relative humidity $H$ (measured as a percentage). A portion of the table which gives values for this function, $I=I(T,H)\text{,}$ is reproduced in Table 10.2.10.

Table 10.2.10. A portion of the wind chill data.

T $\downarrow \backslash$ H $\rightarrow$7075808590106109112115921121151191239411812212713296125130135141

State the limit definition of the value $I_T(94,75)\text{.}$ Then, estimate $I_T(94,75)\text{,}$ and write one complete sentence that carefully explains the meaning of this value, including its units.
State the limit definition of the value $I_H(94,75)\text{.}$ Then, estimate $I_H(94,75)\text{,}$ and write one complete sentence that carefully explains the meaning of this value, including its units.
Suppose you are given that $I_T(92,80) = 3.75$ and $I_H(92,80) = 0.8\text{.}$ Estimate the values of $I(91,80)$ and $I(92,78)\text{.}$ Explain how the partial derivatives are relevant to your thinking.
On a certain day, at 1 p.m. the temperature is 92 degrees and the relative humidity is 85%. At 3 p.m., the temperature is 96 degrees and the relative humidity 75%. What is the average rate of change of the heat index over this time period, and what are the units on your answer? Write a sentence to explain your thinking.

14.

Let $f(x,y) = \frac{1}{2}xy^2$ represent the kinetic energy in Joules of an object of mass $x$ in kilograms with velocity $y$ in meters per second. Let $(a,b)$ be the point $(4,5)$ in the domain of $f\text{.}$

Calculate $f_x(a,b)\text{.}$
Explain as best you can in the context of kinetic energy what the partial derivative
\begin{equation*} f_x(a,b) = \lim_{h \to 0} \frac{f(a+h,b) - f(a,b)}{h} \end{equation*}
tells us about kinetic energy.
Calculate $f_y(a,b)\text{.}$
Explain as best you can in the context of kinetic energy what the partial derivative
\begin{equation*} f_y(a,b) = \lim_{h \to 0} \frac{f(a,b+h) - f(a,b)}{h} \end{equation*}
tells us about kinetic energy.
Often we are given certain graphical information about a function instead of a rule. We can use that information to approximate partial derivatives. For example, suppose that we are given a contour plot of the kinetic energy function (as in Figure 10.2.11) instead of a formula. Use this contour plot to approximate $f_x(4,5)$ and $f_y(4,5)$ as best you can. Compare to your calculations from earlier parts of this exercise.
Figure 10.2.11. The graph of $f(x,y) = \frac{1}{2}xy^2\text{.}$

15.

The temperature on an unevenly heated metal plate positioned in the first quadrant of the $xy$-plane is given by

\begin{equation*} C(x,y) = \frac{25xy+25}{(x-1)^2 + (y-1)^2 + 1}. \end{equation*}

Assume that temperature is measured in degrees Celsius and that $x$ and $y$ are each measured in inches. (Note: At no point in the following questions should you expand the denominator of $C(x,y)\text{.}$)

Determine $\frac{\partial C}{\partial x}|_{(x,y)}$ and $\frac{\partial C}{\partial y}|_{(x,y)}\text{.}$
If an ant is on the metal plate, standing at the point $(2,3)\text{,}$ and starts walking in the direction parallel to the positive $y$ axis, at what rate will the temperature the ant is experiencing change? Explain, and include appropriate units.
If an ant is walking along the line $y = 3$ in the positive $x$ direction, at what instantaneous rate will the temperature the ant is experiencing change when the ant passes the point $(1,3)\text{?}$
Now suppose the ant is stationed at the point $(6,3)$ and walks in a straight line towards the point $(2,0)\text{.}$ Determine the average rate of change in temperature (per unit distance traveled) the ant encounters in moving between these two points. Explain your reasoning carefully. What are the units on your answer?

Partial derivative calculator

Determine the sign of fx and fy at each indicated point using the contour diagram of f shown below

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