Chapter 10

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=x^{2} y+x y^{2}\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(1.0)}\left(x^{2}-3 y^{2}\right)\)

Show that $$ c(x, t)=\frac{1}{\sqrt{8 \pi t}} \exp \left[-\frac{x^{2}}{8 t}\right] $$ solves $$ \frac{\partial c(x, t)}{\partial t}=2 \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$

Body mass index (or BMI) is often used as an indicator for whether a person is over- or underweight. A person's BMI is calculated from their mass (in \(\mathrm{kg}\) ) and their height (in \(\mathrm{m}\) ). To calculate a person's BMI, divide their mass by the square of their height. (a) If a person's mass is \(m\), and their height is \(h\), write down the formula that would be used to calculate their BMI. (b) Jesse is \(1.75 \mathrm{~m}\) tall, and he weighs \(82 \mathrm{~kg}\). What is his BMI? (c) In a particular population, heights range from \(1.50 \mathrm{~m}\) to \(1.90 \mathrm{~m}\), and masses range from \(45 \mathrm{~kg}\) to \(160 \mathrm{~kg} .\) Calculate the maximum possible range of BMI's for this population.

The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=2 x^{3}+y^{2} ;(1,2,6)\)

The functions are defined for all \((x, y) \in R^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). $$ f(x, y)=x^{2}+y^{2}+2 y $$

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=x^{3} y^{2} $$

1\. Let \(f(x, y)=x^{2}+y^{2}\) with \(x(t)=3 t\) and \(y(t)=t^{2}\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=1\).

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\frac{2 x}{y}-\frac{3}{x y^{2}}\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(-1.1)}\left(2 x y+y^{2}\right)\)

Show that $$ c(x, t)=\frac{1}{\sqrt{2 \pi t}} \exp \left[-\frac{x^{2}}{2 t}\right] $$ solves $$ \frac{\partial c(x, t)}{\partial t}=\frac{1}{2} \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$

The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=x^{2}-3 y^{2} ;(1,2,-11)\)

The functions are defined for all \((x, y) \in R^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). $$ f(x, y)=-2 x^{2}-y^{2}+2 x $$

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=\frac{x y}{x^{2}+y^{2}} $$

2\. Let \(f(x, y)=e^{x}\) with \(x(t)=t\) and \(y(t)=t^{3}\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=0\).

Evaluate the Nicholson-Bailey model for the first 10 generations when \(a=0.02, c=3\), and \(b=0.5 .\) For the initial host density, choose \(N_{0}=15\), and for the initial parasitoid density, choose \(P_{0}=0\).

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=(x y)^{3 / 2}-(x y)^{2 / 3}\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(2,-1)}\left(x^{2} y^{3}-x y\right)\)

A solution of $$ \frac{\partial c(x, t)}{\partial t}=D \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$ is the function $$ c(x, t)=\frac{1}{\sqrt{4 \pi D t}} \exp \left[-\frac{x^{2}}{4 D t}\right] $$ for \(x \in \mathbf{R}\) and \(t>0\). (a) Show that, as a function of \(x\) for fixed values of \(t>0, c(x, t)\) is (i) positive for all \(x \in \mathbf{R}\), (ii) is increasing for \(x<0\) and decreasing for \(x>0\), (iii) has a local maximum at \(x=0\), and (iv) has inflection points at \(x=\pm \sqrt{2 D t}\). (b) Graph \(c(x, t)\) as a function of \(x\) when \(D=1\) for \(t=0.01\), \(t=0.1\), and \(t=1\)

In cold weather the temperature that we feel depends both on the actual air temperature and on whether the wind is blowing or not. Strong winds blow heat away from our bodies, increasing their rate of heat loss and making us feel colder, a phenomenon known as wind chill. In cold winters, weather forecasters report both the real temperature and the apparent temperature including wind chill. One widely used formula for the apparent temperature \(W\), when the air temperature (measured in \(\left.{ }^{\circ} \mathrm{F}\right)\) is \(T\), and the wind speed (measured in mph) is \(V\), is: $$W=35.74+0.6215 T-35.75 V^{0.16}+0.4275 T V^{0.16}$$ This formula is only accurate when \(T \leq 50^{\circ} \mathrm{F}\) and \(V \geq 3 \mathrm{mph}\). (a) Use \((10.2)\) to calculate the apparent temperatures felt by a person in each of the three following locations: (i) Boston in January \(\left(T=14^{\circ} \mathrm{F}, V=11 \mathrm{mph}\right)\) (ii) Minneapolis in January \(\left(T=23^{\circ} \mathrm{F}, V=13 \mathrm{mph}\right)\) (iii) Chicago in January \(\left(T=24^{\circ} \mathrm{F}, V=18 \mathrm{mph}\right)\) In which location will a person feel coldest? (b) In this part we will see the danger of trying to evaluate a function outside of its correct domain. Calculate the wind chill for Los Angeles in January \(\left(T=65^{\circ} \mathrm{F}, V=0.8 \mathrm{mph}\right) .\) [Note that these \((T, V)\) values are outside the domain for \(W(T, V) .]\) Does your answer make sense?

The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=x y ;(-1,2,-2)\)

The functions are defined for all \((x, y) \in R^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). $$ f(x, y)=x^{2} y-4 x^{2}-4 y $$

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=\sqrt{x^{3}-3 x y} $$

3\. Let \(f(x, y)=\sqrt{x^{2}+y^{2}}\) with \(x(t)=t\) and \(y(t)=\sin t\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=\pi / 3\).

Show that when the initial parasitoid density is \(P_{0}=0\), the Nicholson- Bailey model reduces to $$ N_{t+1}=b N_{t} $$ With \(N_{0}\) denoting the initial host density, find an expression for \(N_{t}\) in terms of \(N_{0}\) and the parameter \(b\).

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\frac{y^{4}}{x^{3}}-\frac{x^{3}}{y^{4}}\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(1,-1)}\left(2 x^{3}-3 y\right)(x y+1)\)

A solution of $$ \frac{\partial c(x, t)}{\partial t}=D \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$ is the function $$ c(x, t)=\frac{1}{\sqrt{4 \pi D t}} \exp \left[-\frac{x^{2}}{4 D t}\right] $$ for \(x \in \mathbf{R}\) and \(t>0\). (a) Show that a local maximum of \(c(x, t)\) occurs at \(x=0\) for fixed \(t\). (b) Show that \(c(0, t), t>0\), is a decreasing function of \(t\). (c) Find $$ \lim _{t \rightarrow 0^{+}} c(x, t) $$ when \(x=0\) and when \(x \neq 0\) (d) Use the fact that $$ \int_{-\infty}^{\infty} e^{-u^{2} / 2} d u=\sqrt{2 \pi} $$ to show that, for \(t>0\), $$ \int_{-\infty}^{\infty} c(x, t) d x=1 $$ (e) The function \(c(x, t)\) can be interpreted as the concentration of a substance diffusing in space. Explain the meaning of $$ \int_{-\infty}^{\infty} c(x, t) d x=1 $$ and use your results in (c) and (d) to explain why this means that initially (i.e., at \(t=0\) ) the entire amount of the substance was released at the origin.

The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=\sin x \cdot \cos y ;(0,0,0)\)

The functions are defined for all \((x, y) \in R^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). $$ f(x, y)=-x y-2 y^{2} $$

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=x\left(x^{2}-y^{2}\right)^{2 / 3} $$

4\. Let \(f(x, y)=\ln \left(x y-x^{2}\right)\) with \(x(t)=t^{2}\) and \(y(t)=t\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=5\).

When the initial parasitoid density is \(P_{0}=0\), the NicholsonBailey model reduces to $$ N_{t+1}=b N_{t} $$ as shown in the previous problem. For which values of \(b\) is the host density increasing if \(N_{0}>0\) ? For which values of \(b\) is it decreasing? (Assume that \(b>0 .\) )

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\sin (x+y)\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(-1,3)} x^{2}\left(y^{2}-3 x y\right)\)

A chemical diffuses in a container that occupies the interval \(0 \leq x \leq 1\). The concentration of the chemical at time \(t\) and at a point \(x\) is given by the diffusion equation: $$ \frac{\partial c}{\partial t}=D \frac{\partial^{2} c}{\partial x^{2}} $$ (a) Suppose that the chemical is allowed to diffuse through the entire container until the concentration reaches an equilibrium value where \(c\) does not change any more with time, that is, \(\partial c / \partial t=0 .\) Suppose that chemical that touches the walls of the container is removed so that $$ c(0, t)=c(1, t)=0 . $$ The steady state concentration of chemical will be a function \(C(x)\) with $$ 0=D \frac{d^{2} C}{d x^{2}} \text { for } x \in(0,1) $$ and \(C(0)=C(1)=0\). Show that \(C(x)=0\) satisfies this differential equation and the constraints as the points \(x=0\) and \(x=1\). (b) Now suppose that chemical is added to the container by a reaction that occurs at the wall \(x=0 .\) This reaction keeps the concentration of chemical at this wall equal to \(c(0, t)=1 . \mathrm{Un}\) der these conditions the steady state distribution of chemical will obey a differential equation: $$ 0=D \frac{d^{2} C}{d x^{2}} \text { for } x \in(0,1) $$ with \(C(0)=1\) and \(C(1)=0 .\) Show that \(C(x)=1-x\) satisfies both the differential equation and the boundary conditions at \(x=0\) and \(x=1\). (c) Notice that the steady state distributions in (a) and (b) do not depend on \(D .\) Can you explain why?

Cardiac output (CO) is a way of measuring the amount of blood pushed out by a patient's heart. It is calculated as the product of heart rate (HR) and stroke volume (SV). Write cardiac output as a function of heart rate and stroke volume. If heart rate is measured in beats per minute and stroke volume in liters per beat, what is the unit for cardiac output?

The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=\sin (x y) ;(1,0,0)\)

The functions are defined for all \((x, y) \in R^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). $$ f(x, y)=-2 x^{2}+y^{2}-6 y $$

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=\exp \left[\sqrt{x^{2}+y^{2}}\right] $$

5\. Let \(f(x, y)=\frac{1}{x}+\frac{1}{y}\) with \(x(t)=t\) and \(y(t)=1-t .\) Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=1 / 2\).

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\tan (x-2 y)\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(-5,1)} y\left(x y+x^{2} y^{2}\right)\)

Morphogenesis Most embryos start out as lumps of cells. Cells in these lumps are initially undifferentiated-that is, they start out all in the same state. Over time cells then commit to different functions, e.g., to becoming legs, eyes, and so on. To do this chemicals called morphogens are distributed unequally through the embryo, allowing each cell to tell where in the embryo it is located. How are unequal distributions of morphogens achieved? One model for how morphogens can be distributed through the embryo is that morphogens are continuously produced at one end (also called pole) of the embryo. From there they diffuse through the embryo. As the morphogens diffuse, they are constantly broken down by the cells in the embryo. First let's ignore the process of morphogen degradation, and focus only on diffusion. We will assume that the pole at which the morphogen is produced is located at \(x=0 ;\) and for simplicity's sake the cell occupies the interval \(x \geq 0\). Then our partial differential equation model for the distribution of morphogen becomes: $$ \frac{\partial c}{\partial t}=D \frac{\partial^{2} c}{\partial x^{2}} \quad \text { for } \quad x>0 $$ with $$ -D \frac{\partial c}{\partial x}(0, t)=Q \quad \text { and } \quad c(x, t) \rightarrow 0 \text { as } x \rightarrow \infty $$ where \(Q\) is the rate of morphogen production. (a) Let's try to find a steady state distribution of morphogen. That is, we will assume that over time the morphogen concentration reaches some state that does not change with time, i.e., the concentration is given by a function \(C(x) .\) Then \(C(x)\) will satisfy the partial differential equation if and only if: $$ \begin{aligned} 0 &=D \frac{d^{2} C}{d x^{2}} \quad \text { for } \quad x>0 \\ -D C^{\prime}(0) &=Q \quad \text { and } \quad C(x) \rightarrow 0 \text { as } x \rightarrow \infty \end{aligned} $$ Show that there is no function \(C(x)\) that satisfies this differential equation. [Hint: Start by integrating once \((10.48)\) to find \(d C / d x\) and then again to find \(\mathcal{C}(x)\), then try to impose the constraints at \(x=0\), and as \(x \rightarrow \infty\) on your solution.] (b) Now let's incorporate morphogen degradation into our model. We will assume that the breakdown of morphogen has first order kinetics (see Section \(5.9\) for a discussion of the different kinds of kinetics that chemical reactions may have). This means that in one unit of time a fraction \(r\) of the morphogen contained in each region of the embryo is degraded. Then our partial differential equation must be altered to: $$ \frac{\partial c}{\partial t}=D \frac{\partial^{2} c}{\partial x^{2}}-r c \quad \text { for } \quad x>0 $$ with $$ -D \frac{\partial c}{\partial x}(0, t)=Q \quad \text { and } \quad c(x, t) \rightarrow 0 \text { as } x \rightarrow \infty $$ Show that this partial differential equation does have a steady state solution of the form: $$ C(x)=Q \sqrt{\frac{1}{D r}} \exp \left(-\sqrt{\frac{r}{D}} x\right) $$ That is, check that this function \(C(x)\) satisfies both the steady state form of \((10.49)\) as well as the constraints at \(x=0\) and as \(x \rightarrow \infty\).

Mean arterial blood pressure (MAP) is a function of systolic blood pressure (SP) (that is, the pressure during the heart beat) and diastolic blood pressure (DP) (that is, the pressure between heart beats). At a resting heart rate, $$\mathrm{MAP} \approx \mathrm{DP}+\frac{1}{3}(\mathrm{SP}-\mathrm{DP})$$ If systolic pressure is greater than diastolic pressure and both are nonnegative, what is the range of the function describing mean arterial pressure?

The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=e^{x-y} ;(1,1,1)\)

The functions are defined for all \((x, y) \in R^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). $$ f(x, y)=x(1+x-y) $$

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=\tan \frac{x-y}{x+y} $$

6\. Let \(f(x, y)=x e^{y}\) with \(x(t)=e^{t}\) and \(y(t)=t^{2}\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=0\).

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\cos ^{2}\left(x^{2}-2 y\right)\)