Calculus for Life Sciences MAT 1332 C Winter 2010 Jing Li - - PowerPoint PPT Presentation

Calculus for Life Sciences

MAT 1332 C Winter 2010 Jing Li

Department of Mathematics and Statistics University of Ottawa

March 17, 2010

Jing Li (UofO) MAT 1332 C March 17, 2010 1 / 27

Outline

1

Introductory Example

2

Functions of two or more independent variables

3

The level set

Jing Li (UofO) MAT 1332 C March 17, 2010 2 / 27

Introductory Example

Introductory Example

To survive in cold temperatures, humans must maintain a sufficiently high metabolic rate, or regulate heat loss by covering their skin with insulating material. There is a functional relationship that gives the lowest temperature for survival (Te) as a function

• f metabolic heat production (M) and whole-body thermal conductance (gHb).

The metabolic heat production depends on the type of activity; some values for humans are summarized in the following table: Activity M in Wm−2 Sleeping 50 Working at a desk 95 Level walking at 2.5 mph 180 Level walking at 3.5 mph with a 40-lb pack 350

Jing Li (UofO) MAT 1332 C March 17, 2010 3 / 27

Introductory Example

The whole body thermal conductance gHb describes how quickly heat is lost. The value of gHb depends on the type of protection; for instance,

gHb = 0.45 mol m−2 s−1 without clothing, gHb = 0.14 mol m−2 s−1 for a wool suit, gHb = 0.04 mol m−2 s−1 for warm sleeping bag.

That is, the smaller gHb, the better protection from the cold the material provides.

Jing Li (UofO) MAT 1332 C March 17, 2010 4 / 27

Introductory Example

The relationship between Te, M and gHb is given by [see Campbell(1986)] Te = 36 − (0.9M − 12)(gHb + 0.95) 27.8gHb where M is measured Wm−2; gHb is measured in mol m−2 s−1 Te is measured in degree Celsius. The temperature Te is a function of two variables, namely M and gHb; to meet the required temperature, we can change either M (by starting to move when we get cold)

• r gHb (by putting on more clothes when we get cold).

Jing Li (UofO) MAT 1332 C March 17, 2010 5 / 27

Introductory Example

We can plot Te as a function of M for different values gHb,

50 100 150 200 250 300 350 −50 −40 −30 −20 −10 10 20 30 40 M T

e

gHb=0.14 gHb=0.45

Figure: The graph of Te as a function of M for various values of gHb.

Jing Li (UofO) MAT 1332 C March 17, 2010 6 / 27

Introductory Example

plot Te as a function of gHb for different values of M,

0.1 0.2 0.3 0.4 0.5 −500 −400 −300 −200 −100 100 gHb T

e

M=50 M=180

Figure: The graph of Te as a function of gHb for various values of M.

Jing Li (UofO) MAT 1332 C March 17, 2010 7 / 27

Introductory Example

The goal of this chapter: to discuss functions of two or more independent variables, develop the theory of differential calculus for such functions, and discuss a number of application.

Jing Li (UofO) MAT 1332 C March 17, 2010 8 / 27

Functions of two or more independent variables

For the functions of one variable, for example, let f : [0, 4] − → R x − → √ x

• 1
• 0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

• 1.6
• 0.8

0.8 1.6 2.4 3.2

Figure: The graph of f(x) = √x.

The function y = f(x) has the domain, [0, 4]; the range, [0, 2]

Jing Li (UofO) MAT 1332 C March 17, 2010 9 / 27

Functions of two or more independent variables

Now, we consider functions for which the domain consists of pairs of real numbers (x, y) with x, y ∈ R, or more generally, of n-tuples of real numbers (x1, x2, . . . , xn) with x1, x2, . . . , xn ∈ R. We also call n-tuples points. We use the notation Rn to denote the set of all n-tuples (x1, x2, . . . , xn) with x1, x2, . . . , xn ∈ R, Rn = {(x1, x2, . . . , xn) : x1, x2, . . . , xn ∈ R} Note: For n = 1, R1 = R, which is the set of all real numbers. For n = 2, R2 is the set of all points in the plane. We will use notation (x, y) for points in R2. For n = 3, R3 is the set of all points in the space. We will use notation (x, y, z) for points in R3. n-tuples are ordered; for instance, (2, 3) = (3, 2)

Jing Li (UofO) MAT 1332 C March 17, 2010 10 / 27

Functions of two or more independent variables

The functions of several variables

Definition Suppose D ⊂ Rn. A real-valued function f on D assigns a real number to each element in D and we write f : D − → R (x1, x2, . . . , xn) − → f(x1, x2, . . . , xn) The set D is the domain of the function f; The set {w ∈ R : f(x1, x2, . . . , xn) = w for some (x1, x2, . . . , xn) ∈ D} is the range of the function f.

Jing Li (UofO) MAT 1332 C March 17, 2010 11 / 27

Functions of two or more independent variables

The functions of several variables

Definition Suppose D ⊂ Rn. A real-valued function f on D assigns a real number to each element in D and we write f : D − → R (x1, x2, . . . , xn) − → f(x1, x2, . . . , xn) The set D is the domain of the function f; The set {w ∈ R : f(x1, x2, . . . , xn) = w for some (x1, x2, . . . , xn) ∈ D} is the range of the function f. Remark: If a function f depends on just two independent variables, write f(x, y). In the case of three variables, write f(x, y, z). If f is a function of more than three independent variables, it is more convenient to use subscripts to label the variables, for example, f(x1, x2, x3, x4).

Jing Li (UofO) MAT 1332 C March 17, 2010 11 / 27

Functions of two or more independent variables

Example 1.

When n = 2,D = R2, f(x, y) = x2 + y 2, evaluate function f at points (0, 0), (1, 0), (0, 1), (2, 4). Solution:

1

f(0, 0) = 0

2

f(1, 0) = 1

3

f(0, 1) = 1

4

f(2, 4) = 22 + 42 = 20

Jing Li (UofO) MAT 1332 C March 17, 2010 12 / 27

Functions of two or more independent variables

Example 1.

When n = 2,D = R2, f(x, y) = x2 + y 2, evaluate function f at points (0, 0), (1, 0), (0, 1), (2, 4). Solution:

1

f(0, 0) = 0

2

f(1, 0) = 1

3

f(0, 1) = 1

4

f(2, 4) = 22 + 42 = 20 Fix y = 0, then we have a function of a single variable f(x, 0) = x2;

Jing Li (UofO) MAT 1332 C March 17, 2010 12 / 27

Functions of two or more independent variables

Example 1.

When n = 2,D = R2, f(x, y) = x2 + y 2, evaluate function f at points (0, 0), (1, 0), (0, 1), (2, 4). Solution:

1

f(0, 0) = 0

2

f(1, 0) = 1

3

f(0, 1) = 1

4

f(2, 4) = 22 + 42 = 20 Fix y = 0, then we have a function of a single variable f(x, 0) = x2; Fix x = 0, then we have a function of a single variable f(0, y) = y 2.

Jing Li (UofO) MAT 1332 C March 17, 2010 12 / 27

Functions of two or more independent variables

function f(x, y) = x2 + y2

Figure: The graph of function f(x, y) = x2 + y2.

Jing Li (UofO) MAT 1332 C March 17, 2010 13 / 27

Functions of two or more independent variables

Example 2.

For n = 2, let D = {(x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} and f(x, y) = x + y Graph the domain of f in the x − y plane and determine the range of f. Solution: The domain of f is the set D:

Jing Li (UofO) MAT 1332 C March 17, 2010 14 / 27

Functions of two or more independent variables

Example 2.

For n = 2, let D = {(x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} and f(x, y) = x + y Graph the domain of f in the x − y plane and determine the range of f. Solution: The domain of f is the set D:

• 1
• 0.5

0.5 1 1.5 2 0.5 1

(1,1)

D Figure: The domain of the function in Example 2.

Jing Li (UofO) MAT 1332 C March 17, 2010 14 / 27

Functions of two or more independent variables

f(x, y) = x + y, D = {(x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

To find the range of f, we need to determine what values f can take when we plug in points (x, y) from the domain D.

Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27

Functions of two or more independent variables

f(x, y) = x + y, D = {(x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

To find the range of f, we need to determine what values f can take when we plug in points (x, y) from the domain D. The function z = f(x, y) is smallest when (x, y) = (0, 0), namely, f(0, 0) = 0;

Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27

Functions of two or more independent variables

f(x, y) = x + y, D = {(x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

To find the range of f, we need to determine what values f can take when we plug in points (x, y) from the domain D. The function z = f(x, y) is smallest when (x, y) = (0, 0), namely, f(0, 0) = 0; The function z = f(x, y) is largest when (x, y) = (1, 1), namely, f(1, 1) = 2;

Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27

Functions of two or more independent variables

f(x, y) = x + y, D = {(x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

To find the range of f, we need to determine what values f can take when we plug in points (x, y) from the domain D. The function z = f(x, y) is smallest when (x, y) = (0, 0), namely, f(0, 0) = 0; The function z = f(x, y) is largest when (x, y) = (1, 1), namely, f(1, 1) = 2; The range of f is the set {z : 0 ≤ z ≤ 2}.

Jing Li (UofO) MAT 1332 C March 17, 2010 15 / 27

Functions of two or more independent variables

f(x, y) = x + y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 x y z

Figure: The graph of function f = x + y.

Jing Li (UofO) MAT 1332 C March 17, 2010 16 / 27

Functions of two or more independent variables

Example 3.

Find the largest possible domain for the function f(x, y) =

• y 2 − x

Solution: The square root is real only if y 2 − x ≥ 0, i.e., y 2 ≥ x.

Jing Li (UofO) MAT 1332 C March 17, 2010 17 / 27

Functions of two or more independent variables

Example 3.

Find the largest possible domain for the function f(x, y) =

• y 2 − x

Solution: The square root is real only if y 2 − x ≥ 0, i.e., y 2 ≥ x. This gives two conditions: y ≥ √ x

• r

y ≤ − √ x

Jing Li (UofO) MAT 1332 C March 17, 2010 17 / 27

Functions of two or more independent variables

Example 3.

Find the largest possible domain for the function f(x, y) =

• y 2 − x

Solution: The square root is real only if y 2 − x ≥ 0, i.e., y 2 ≥ x. This gives two conditions: y ≥ √ x

• r

y ≤ − √ x So the domain is D = {(x, y) ∈ R2 : y ≥ √ x

• r

y ≤ − √ x}

Jing Li (UofO) MAT 1332 C March 17, 2010 17 / 27

Functions of two or more independent variables

D = {(x, y) ∈ R2 : y ≥ √x

• r

y ≤ −√x}

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −1.5 −1 −0.5 0.5 1 1.5 x y

Figure: The white area is the maximal domain of definition of the function f(x, y) =

• y2 − x. In the shaded area, the square root is not real.

Jing Li (UofO) MAT 1332 C March 17, 2010 18 / 27

Functions of two or more independent variables

f(x, y) =

• y2 − x

Figure: The graph of the function f(x, y) =

• y2 − x.

Jing Li (UofO) MAT 1332 C March 17, 2010 19 / 27

Functions of two or more independent variables

f(x, y) =

• y2 − x

Figure: The graph of the function f(x, y) =

• y2 − x.

Jing Li (UofO) MAT 1332 C March 17, 2010 20 / 27

Functions of two or more independent variables

f(x, y) =

• y2 − x

Figure: The graph of the function f(x, y) =

• y2 − x.

Jing Li (UofO) MAT 1332 C March 17, 2010 21 / 27

Functions of two or more independent variables

f(x, y) =

• y2 − x

Figure: The graph of the function f(x, y) =

• y2 − x.

Jing Li (UofO) MAT 1332 C March 17, 2010 22 / 27

The level set

The level set

Definition The level set, Lc, or contour line of a function f(x, y) is the set of all points (x, y) ∈ D where f has a given value c; i.e., Lc = {(x, y) ∈ D : f(x, y) = c}

Jing Li (UofO) MAT 1332 C March 17, 2010 23 / 27

The level set

Example 1 revisited: f(x, y) = x2 + y2

Pick some value c, then f(x, y) = c ⇐ ⇒ x2 + y 2 = c ⇐ ⇒ y = ±

• c − x2.

Jing Li (UofO) MAT 1332 C March 17, 2010 24 / 27

The level set

Example 1 revisited: f(x, y) = x2 + y2

Pick some value c, then f(x, y) = c ⇐ ⇒ x2 + y 2 = c ⇐ ⇒ y = ±

• c − x2.

x y −1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

Figure: Level sets of the function f(x, y) = x2 + y2.

Jing Li (UofO) MAT 1332 C March 17, 2010 24 / 27

The level set

Example 2 revisited: f(x, y) = x + y

Pick some value c, then f(x, y) = c ⇐ ⇒ x + y = c ⇐ ⇒ y = c − x.

Jing Li (UofO) MAT 1332 C March 17, 2010 25 / 27

The level set

Example 2 revisited: f(x, y) = x + y

Pick some value c, then f(x, y) = c ⇐ ⇒ x + y = c ⇐ ⇒ y = c − x.

x y 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure: Level sets of the function f(x, y) = x + y.

Jing Li (UofO) MAT 1332 C March 17, 2010 25 / 27

The level set

Example 3 revisited: f(x, y) =

• y2 − x

Pick some value c, then f(x, y) = c ⇐ ⇒ y 2 − x = c2 ⇐ ⇒ y = ±

• x + c2.

Jing Li (UofO) MAT 1332 C March 17, 2010 26 / 27

The level set

Example 3 revisited: f(x, y) =

• y2 − x

Pick some value c, then f(x, y) = c ⇐ ⇒ y 2 − x = c2 ⇐ ⇒ y = ±

• x + c2.

x y −3 −2 −1 1 2 3 −3 −2 −1 1 2 3

Figure: Level sets of the function f(x, y) =

• y2 − x.

Jing Li (UofO) MAT 1332 C March 17, 2010 26 / 27

The level set

Thank you! I will greatly appreciate it if you can tell me the typo that you find.

Jing Li (UofO) MAT 1332 C March 17, 2010 27 / 27