Calculus for the Life Sciences: A Modeling Approach

James L. Cornette

Chapter 13 Two Variable Calculus and Diffusion - all with Video Answers

Educators

Section 1

Partial derivatives of functions of two variables

Problem 1

Draw three dimensional graphs of
a. $F(x, y)=2 \quad$ b. $\quad F(x, y)=x$
C. $F(x, y)=x^{2} \quad$ d. $\quad F(x, y)=(x+y) / 2$
e. $F(x, y)=0.2 x+0.3 y \quad$ f. $\quad F(x, y)=\left(x^{2}+y^{2}\right) / 4$
g. $F(x, y)=0.5 x e^{-y} \quad$ h. $\quad F(x, y)=\sin y$
i. $\quad F(x, y)=0.5 x+\sin y \quad$ j. $\quad F(x, y)=x \sin y$
$\mathrm{k}$. $F(x, y)=\sqrt{x^{2}+y^{2}} \quad$ 1. $\quad F(x, y)=x y$
m. $\quad F(x, y)=\frac{1}{0.4+x^{2}+y^{2}} \quad$ n. $\quad F(x, y)=e^{\left(-x^{2}-y^{2}\right)}$
O. $F(x, y)=|x y| \quad$ p. $\quad F(x, y)=\sin \left(x^{2}+y^{2}\right)$

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Problem 2

Find the partial derivatives, $F_{1}, F_{2}, F_{1,1}, F_{1,2}, F_{2,1}$ and $F_{2,2}$ of the following functions.
a. $\quad F(x, y)=3 x-5 y+7$
b. $\quad F(x, y)=x^{2}+4 x y+3 y^{2}$
c. $F(x, y)=x^{3} y^{5} \quad$ d. $\quad F(x, y)=\sqrt{x y}$
e. $\quad F(x, y)=\ln (x \cdot y) \quad$ f. $\quad F(x, y)=\frac{x}{y}$
g. $\quad F(x, y)=e^{x+y}$
h. $\quad F(x, y)=x^{2} e^{-y}$
i. $\quad F(x, y)=\sin (2 x+3 y)$
$F(x, y)=e^{-x} \cos y$

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Problem 3

Is the plane $z=0$ a tangent plane to the graph of $F(x, y)=\sqrt{x^{2}+y^{2}}$ shown in Figure 13.2D.

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Problem 4

Find $F_{1}(a, b)$ and $F_{2}(a, b)$ for
a. $\quad F(x, y)=\frac{x}{1+y^{2}} \quad(a, b)=(1,0)$
b. $\quad F(x, y)=\sqrt{x^{2}+y^{2}} \quad(a, b)=(1,2)$
c. $F(x, y)=e^{-x y} \quad(a, b)=(0,0)$
d. $\quad F(x, y)=\sin x \cos y \quad(a, b)=(\pi / 2, \pi)$

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Problem 5

Find the local linear approximation, $L(x, y),$ to $F(x, y)$ at the point $(\mathrm{a}, \mathrm{b})$. For each case, compute
$$\frac{|F(x, y)-L(x, y)|}{\sqrt{(x-a)^{2}+(y-b)^{2}}} \quad \text { for } \quad(x, y)=(a+0.1, b), \quad \text { and } \quad(x, y)=(a+0.01, b+0.01) .$$
a. $\quad F(x, y)=4 x+7 y-16 \quad(a, b)=$
b. $\quad F(x, y)=x y \quad(a, b)=$
c. $\quad F(x, y)=\frac{x}{y+1} \quad(a, b)=\quad(1,0)$
d. $\quad F(x, y)=x e^{-y} \quad(a, b)=\quad(1,0)$
e. $F(x, y)=\sin \pi(x+y) \quad(a, b)=(1 / 2,1 / 4)$

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Problem 6

For $P=n R T / V$, find $\frac{\partial}{\partial V} P$ and $\frac{\partial}{\partial T} P$. For fixed $T$, how does $P$ change as $V$ increases? For fixed $V,$ how does $P$ change as $T$ increases?

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Problem 7

Draw the graph of $F(x, y)$ and the graph of the plane tangent to the graph of $F$ at the point $(\mathrm{a}, \mathrm{b})$. A MATLAB program to solve $\mathrm{c}$. is shown in the answer to this exercise, Exercise 13.1.7.
a. $F(x, y)=0.5 x+y+1 \quad(a, b)=(0.5,1)$
b. $\quad F(x, y)=\left(x^{2}-y^{2}\right) / 2 \quad(a, b)=(1,1)$
c. $F(x, y)=\left(x^{2}+y^{2}\right) / 2 \quad(a, b)=(1,0.5)$
d. $F(x, y)=e^{x y / 2} \quad(a, b)=(1,0.2)$
e. $F(x, y)=\sqrt{9-x^{2}-y^{2}} \quad(a, b)=(1,0.2)$
f. $\quad F(x, y)=4-x^{2}-y^{2} \quad(a, b)=(1,0.2)$
g. $\quad F(x, y)=2 /\left(1+x^{2}+y^{2}\right) \quad(a, b)=(1,1)$
h. $\quad F(x, y)=e^{-x^{2}-y^{2}}=e^{-x^{2}} e^{-y^{2}} \quad(a, b)=(1,0.5)$

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Problem 8

Let $F$ be defined by
$$\begin{aligned}F(x, y) &=x^{2} \quad \text { for } \quad y>0 \\
&=0 \quad \text { for } \quad y \leq 0\end{aligned}$$
1. Sketch a graph of $F$ in three dimensional space.
2. Is $F_{1}(x, y)$ continuous on the interior of a circle with center (0,0)$?$
3. Let $L(x, y)=0$ for all $(x, y)$. Is it true that
$$\lim _{(x, y) \rightarrow(0,0)} \frac{F(x, y)-L(x, y)}{\sqrt{x^{2}+y^{2}}}=0 \quad ?$$
4. Are you willing to call the plane $z=0$ a tangent plane to the graph of $F ?$

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