If you are told that f(x)
11.2
LIMITS AND CONTINUITY
INTRODUCTION: Consider functions of one variable y = f(x). If you are told that f(x) is continuous at x = a, explain what the graph looks like near x = a.
Formal definition of continuity at a point x = a: A function f(x) is said to be continuous at x = a if... This definition implies three facts: • • • More formally, here is what we mean by the existence of a limit: The function f has a limit L at x = a if |f(x) – L| can be made arbitrarily small by making |x – a| sufficiently small, that is, by bringing x sufficiently close to, but not equal to a. y Note that continuity at a point depends on the behavior of the function .... a
x
Example 1: Using graphical, numerical and analytical methods, investigate the following limits at the given values of a. Is the function continuous at a? a) lim( x 3 − 2) x →1
b) lim x →1
1 x −1
x2 −1 , x ≠1 . c) lim f ( x) where f(x) = x − 1 x →1 2, x =1 | x − 1| , x ≠1 at x = 1. d) lim h( x) , where h( x) = x − 1 x →1 1, x =1 1
FUNCTIONS OF TWO VARIABLES
As with functions of one variable, for a function of two variables, f(x, y) to be continuous at a point (a, b) it must be defined there and the limit of f(x, y) must exist as (x, y) approaches (a, b). Furthermore the limit must be equal to the function value. i.e. lim f ( x, y ) = L = f (a, b) ( x , y )→( a ,b )
In how many directions can (x, y) “approach” (a, b)? (a, b)
We investigate the following limits at (0, 0) informally, using Excel spreadsheets continuity.xls and Maple plots (limits_112.mws). 2 xy 2 a) lim ( x , y )→(0,0) x 2 + y 2 x2 b) lim ( x , y )→(0,0) x 4 + y 2
Which of these functions could be made continuous at x = 0 by “plugging a hole”: 2 xy 2 f ( x, y ) = 2 x + y2
x2 g ( x, y ) = 4 ? x + y2
or
To define continuity at a point for functions of two variables, we need to use some precise mathematical language. Here’s how we define the limit at a point of a function of two variables (compare with definition on page 760). The function f has a limit L at the point (a, b) written
lim
( x , y )→( a ,b )
f ( x, y ) = L
if the difference |f(x, y) – L| can be made arbitrarily small by making the distance between the point (x, y) and the point (a, b) sufficiently small, but not zero.
(a, b) And here is the definition of continuity at a point for a function of two variables (page 763). A function f is continuous at the point (a, b) if
lim
( x , y )→( a ,b )
f ( x, y ) = f ( a, b) .
A function is said to be continuous if it is continuous at every point in its domain. 2
In practice, it is quite difficult (but NOT impossible) to use the definition to show continuity (or discontinuity) at a point. Practically, we can show discontinuity by approaching our limit point on different paths. If one of the limits doesn’t exist, or if the limits along different paths exist but are not the same, we have shown discontinuity. Why can’t we show continuity by this method?
x2 , ( x, y ) ≠ (0, 0) does not have a limit at x4 + y2 (0, 0) by examining the limits of f as (x, y) → (0, 0) along the line y = x and along the parabola y = x2 .
Example 2.
Show that the function g ( x, y ) =
Example 3. Show that
lim
( x , y )→(0,0)
x+ y , does not exist. x− y
Here’s an approach that enables us to actually show that a limit exists. It’s a good idea to use technology to get an idea of the value of the limit ahead of time! 2 xy 2 =0 Example 4. Use the definition to show that lim ( x , y )→(0,0) x 2 + y 2
Example 4. Compute
Example 5. Show that
lim e − x− y . Notice that this is the composition of continuous functions.
( x , y )→(0,0)
xy doesn’t exist. ( x , y )→(0,0) x + y 2 lim
2
3
11.3
PARTIAL DERIVATIVES
KEY POINTS: ♦ Know the definition of partial derivatives. ♦ Estimate partial derivatives graphically from graphs of surfaces and contour diagrams and numerically from tables. ♦ Understand the units associated with partial derivatives
Recall the definition of the derivative of a function of one variable, f(x) at x = a:
DEFINITION and NOTATION: Partial Derivatives with respect to x and y Suppose that f is a function of two variables x and y; we define the partial derivatives of f with respect to x and y at the point (a, b) as follows:
Note that the “∂” in the alternative notation is not a regular “d”!! This distinguishes partial derivatives from regular derivatives of functions of one variable.
Example 1. Consider the function f(x, y) represented in the following table: x / y 1.1 1.2 1.3 11.06 12.06 13.07 3.2 11.75 12.82 13.89 3.4 12.44 13.57 13.89 3.6 13.13 14.33 15.52 3.8 13.82 15.08 16.34 4.0
1.4 14.07 14.95 14.95 16.71 17.59
Is the function linear? Use the table to estimate fx(3.6, 1.2) and fy(3.6, 1.2).
4
Example 2. The figure at right shows the contour diagram for the temperature H(x, t) (in °C) in a room as a function of the distance x in meters from a heater and time t (in minutes) after the heater has been turned on. What are the signs of Hx(10, 15) and Ht(10, 15)? What are the units for these quantities? Estimate these quantities and explain the answers in practical terms.
Example. 3. Consider the graph of f(x, y) = 16 – x2 – y2 shown in the Maple surface plot. What . is f(1, 3)? f(1, 3) = Look at the point (1, 3, ) on the surface. ( The diagrams below will be helpful.)
What is the sign of fx(1, 3)?
... of fy(1, 3)?
Which of these two quantities has the larger magnitude? Estimate the values of fx(1, 3) and fy(1, 3) using the definition and suitable small values of ∆x and ∆y.
5
Example 4. Suppose you borrow $A at an interest rate of r% (per month) and pay it off over t months by making monthly payments of $P as determined by the function P = g(A, r, t). In financial terms, what do the following statements tell you? (a) g(8000, 1, 24) = 376.59
(b)
∂g (8000, 1, 24 ) = 0.047 ∂A
(c)
∂g (8000, 1, 24 ) = 44.83 ∂r
6
COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY KEY POINT: Know how to compute partial derivatives of functions of two or more variables algebraically, by treating all other variables as “constants”.
Example 1: Compute the partial derivatives fx(x, y) and fy(x, y) algebraically for the function f(x, y) = 16 – x2 – y2. Evaluate them at the point (1, 3).
Example 2. Compute the first order partial derivatives of f(x, y, z) = xy4z3
Second order partial derivatives and notation How many second order derivatives does a function f(x, y) have?
(fx)x = fxx = f11 = (fx)y = fxy = f12 = (Note: First take the derivative with respect to x, then with respect to y. (fy)x = fyx = f21 = (Note: First take the derivative with respect to y, then with respect to y. (fy)y = fyy = f22 = This may be generalized to functions of more than 2 variables. Example 3: Find the second order derivatives of f(x, y, z) = xy4z3.
Example 4: Find the first and second order partial derivatives of the following functions. Use correct notation. Check your answer with a calculator C.A.S. 2
(a)
z = yex
(b)
g(x, y) = ln x 2 + y 2
7
Example 3. In 1928, Cobb and Douglas came up with a function to model the production of the entire US economy between the years 1899 and 1922. They found that the total yearly production P was related to the total capital investment, K, and the total labor force, L by the model P = 1.01L0.75K 0.25 . Similar functions are used to model production in industry; you should read about them in sections 11.1 and 11.3. Suppose that in a certain industry the production was related to labor and capital investment by P = 100L0.6 K 0.4 Find ∂P/∂L, the marginal productivity of labor, and ∂P/∂K, the marginal productivity of capital. Show that this Cobb-Douglas production unction satisfies the differential equation: ∂P ∂P L +K =P ∂L ∂K
Example 4.
Evaluate ∂z/∂x and ∂z/∂y for z = arctan(y/x), at (x, y) = (2, –2).
) Example 5. (QUIPP: due Recall the stadium wave function (from section 11.1): h(x, t) = 5 + cos(0.5x – t), where x is the seat number of a person in a given row, and t is the time in seconds after the wave is initiated. Compute hx(8, 5) and ht(8, 5), and interpret these quantities .
8
Section 11.4 TANGENT PLANES AND LINEAR APPROXIMATIONS (Omit Tangent Planes to Parametric Surfaces, page 787) LOCAL LINEARITY Consider the function f(x) = 7 – x2, at x = 1. Use your calculator to graph this function in a suitable window, then zoom in close to the point where x = 1. What does the graph look like?
Find the equation of the tangent line to f(x) at x = 1
Graph it together with f(x). Zoom in until the two graphs are indistinguishable.
Consider the Maple plots of the 3D surface for the function f(x, y) = 16 – x2 – y2. We look at the surface in neighborhoods close to (x, y) = (1, 3). What do you notice? We also look at the contour plots for these regions. What do these contour plots look like very close to (1, 3)? (Maple worksheet locallin.mws is in the MTH253 folder on the ST server.)
We find the equation of the tangent plane to the surface z = 16 – x2 – y2 at (x, y) = (1, 3)
We say that a function f(x, y) is differentiable at a point (x0, y0) if we are able to approximate it locally by a tangent plane. Describe a method for finding the equation of a tangent plane to the surface of a function at a general point (x0, y0).
9
Definition: The Tangent Plane to the surface z = f(x, y) at (x0, y0).
Assuming that f(x, y) is differentiable at the point (x0, y0), the equation of the tangent plane is given by
Example 1a.
b.
Determine the equation of the tangent plane to the surface z = y – x2 at (4, 3).
Use the tangent plane to approximate 2.996 – 4.0132 . (How close is your estimate?)
Example 2. given point.
Find the equation of the tangent plane to the surface z = x/y, at (3, –2) at the
THE DIFFERENTIAL Returning to single-variable functions, recall the tangent line approximation to a function value near a given point. Example 3: For the function f(x) = 7 – x2, use the linear approximation at x = 1 to find f(1.02).
The change in the function value, when we change x by a small amount dx is given by the expression f ′ (x) dx. This is called the differential of f. 10
For functions of two variables we define the differential in a similar manner. To approximate the value of a function f close to any point (x, y) close to a fixed point (a, b), we could use the tangent plane approximation:
Let ∆x = x – a and ∆y = y – b, and define ∆f = f(x, y) – f(a, b), Then ∆f ≈ fx(a, b) ∆x + fy(a, b) ∆y. Replacing ∆x with dx and ∆y with dy, we define the differential of f… The Differential of a function z = f(x, y) The differential df (or dz) at a point (a, b) is the linear function of dx and dy given by
The differential at a general point is often written df = fx dx + fy dy. Example 4: Find the differential of f (x, y) = f(1.04, 1.98).
x 2 + y 3 at the point (1. 2). Use it to estimate
Applications of the differential. Note: In calculating maximum possible error in a computed value, you need to take into account that the given error in measurements may be either positive or negative.
Example 5: The base radius and height of a circular cone are measured as 10 cm and 25 cm respectively, with a possible error in measurement of as much as 0.1 cm each. (That is, error is ±0.1 cm.) Use differentials to estimate the maximum possible error in the calculated volume of the cone.
Example 6. Power: Electrical power P is given by P = E2/R, where E is voltage (volts), and R is resistance (ohms). Approximate the maximum percentage error in calculating power if 200 volts is applied to a 400 ohm resistor and the possible percent errors in measuring E and R are 2% and 3%. 11
LIMITS AND CONTINUITY
INTRODUCTION: Consider functions of one variable y = f(x). If you are told that f(x) is continuous at x = a, explain what the graph looks like near x = a.
Formal definition of continuity at a point x = a: A function f(x) is said to be continuous at x = a if... This definition implies three facts: • • • More formally, here is what we mean by the existence of a limit: The function f has a limit L at x = a if |f(x) – L| can be made arbitrarily small by making |x – a| sufficiently small, that is, by bringing x sufficiently close to, but not equal to a. y Note that continuity at a point depends on the behavior of the function .... a
x
Example 1: Using graphical, numerical and analytical methods, investigate the following limits at the given values of a. Is the function continuous at a? a) lim( x 3 − 2) x →1
b) lim x →1
1 x −1
x2 −1 , x ≠1 . c) lim f ( x) where f(x) = x − 1 x →1 2, x =1 | x − 1| , x ≠1 at x = 1. d) lim h( x) , where h( x) = x − 1 x →1 1, x =1 1
FUNCTIONS OF TWO VARIABLES
As with functions of one variable, for a function of two variables, f(x, y) to be continuous at a point (a, b) it must be defined there and the limit of f(x, y) must exist as (x, y) approaches (a, b). Furthermore the limit must be equal to the function value. i.e. lim f ( x, y ) = L = f (a, b) ( x , y )→( a ,b )
In how many directions can (x, y) “approach” (a, b)? (a, b)
We investigate the following limits at (0, 0) informally, using Excel spreadsheets continuity.xls and Maple plots (limits_112.mws). 2 xy 2 a) lim ( x , y )→(0,0) x 2 + y 2 x2 b) lim ( x , y )→(0,0) x 4 + y 2
Which of these functions could be made continuous at x = 0 by “plugging a hole”: 2 xy 2 f ( x, y ) = 2 x + y2
x2 g ( x, y ) = 4 ? x + y2
or
To define continuity at a point for functions of two variables, we need to use some precise mathematical language. Here’s how we define the limit at a point of a function of two variables (compare with definition on page 760). The function f has a limit L at the point (a, b) written
lim
( x , y )→( a ,b )
f ( x, y ) = L
if the difference |f(x, y) – L| can be made arbitrarily small by making the distance between the point (x, y) and the point (a, b) sufficiently small, but not zero.
(a, b) And here is the definition of continuity at a point for a function of two variables (page 763). A function f is continuous at the point (a, b) if
lim
( x , y )→( a ,b )
f ( x, y ) = f ( a, b) .
A function is said to be continuous if it is continuous at every point in its domain. 2
In practice, it is quite difficult (but NOT impossible) to use the definition to show continuity (or discontinuity) at a point. Practically, we can show discontinuity by approaching our limit point on different paths. If one of the limits doesn’t exist, or if the limits along different paths exist but are not the same, we have shown discontinuity. Why can’t we show continuity by this method?
x2 , ( x, y ) ≠ (0, 0) does not have a limit at x4 + y2 (0, 0) by examining the limits of f as (x, y) → (0, 0) along the line y = x and along the parabola y = x2 .
Example 2.
Show that the function g ( x, y ) =
Example 3. Show that
lim
( x , y )→(0,0)
x+ y , does not exist. x− y
Here’s an approach that enables us to actually show that a limit exists. It’s a good idea to use technology to get an idea of the value of the limit ahead of time! 2 xy 2 =0 Example 4. Use the definition to show that lim ( x , y )→(0,0) x 2 + y 2
Example 4. Compute
Example 5. Show that
lim e − x− y . Notice that this is the composition of continuous functions.
( x , y )→(0,0)
xy doesn’t exist. ( x , y )→(0,0) x + y 2 lim
2
3
11.3
PARTIAL DERIVATIVES
KEY POINTS: ♦ Know the definition of partial derivatives. ♦ Estimate partial derivatives graphically from graphs of surfaces and contour diagrams and numerically from tables. ♦ Understand the units associated with partial derivatives
Recall the definition of the derivative of a function of one variable, f(x) at x = a:
DEFINITION and NOTATION: Partial Derivatives with respect to x and y Suppose that f is a function of two variables x and y; we define the partial derivatives of f with respect to x and y at the point (a, b) as follows:
Note that the “∂” in the alternative notation is not a regular “d”!! This distinguishes partial derivatives from regular derivatives of functions of one variable.
Example 1. Consider the function f(x, y) represented in the following table: x / y 1.1 1.2 1.3 11.06 12.06 13.07 3.2 11.75 12.82 13.89 3.4 12.44 13.57 13.89 3.6 13.13 14.33 15.52 3.8 13.82 15.08 16.34 4.0
1.4 14.07 14.95 14.95 16.71 17.59
Is the function linear? Use the table to estimate fx(3.6, 1.2) and fy(3.6, 1.2).
4
Example 2. The figure at right shows the contour diagram for the temperature H(x, t) (in °C) in a room as a function of the distance x in meters from a heater and time t (in minutes) after the heater has been turned on. What are the signs of Hx(10, 15) and Ht(10, 15)? What are the units for these quantities? Estimate these quantities and explain the answers in practical terms.
Example. 3. Consider the graph of f(x, y) = 16 – x2 – y2 shown in the Maple surface plot. What . is f(1, 3)? f(1, 3) = Look at the point (1, 3, ) on the surface. ( The diagrams below will be helpful.)
What is the sign of fx(1, 3)?
... of fy(1, 3)?
Which of these two quantities has the larger magnitude? Estimate the values of fx(1, 3) and fy(1, 3) using the definition and suitable small values of ∆x and ∆y.
5
Example 4. Suppose you borrow $A at an interest rate of r% (per month) and pay it off over t months by making monthly payments of $P as determined by the function P = g(A, r, t). In financial terms, what do the following statements tell you? (a) g(8000, 1, 24) = 376.59
(b)
∂g (8000, 1, 24 ) = 0.047 ∂A
(c)
∂g (8000, 1, 24 ) = 44.83 ∂r
6
COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY KEY POINT: Know how to compute partial derivatives of functions of two or more variables algebraically, by treating all other variables as “constants”.
Example 1: Compute the partial derivatives fx(x, y) and fy(x, y) algebraically for the function f(x, y) = 16 – x2 – y2. Evaluate them at the point (1, 3).
Example 2. Compute the first order partial derivatives of f(x, y, z) = xy4z3
Second order partial derivatives and notation How many second order derivatives does a function f(x, y) have?
(fx)x = fxx = f11 = (fx)y = fxy = f12 = (Note: First take the derivative with respect to x, then with respect to y. (fy)x = fyx = f21 = (Note: First take the derivative with respect to y, then with respect to y. (fy)y = fyy = f22 = This may be generalized to functions of more than 2 variables. Example 3: Find the second order derivatives of f(x, y, z) = xy4z3.
Example 4: Find the first and second order partial derivatives of the following functions. Use correct notation. Check your answer with a calculator C.A.S. 2
(a)
z = yex
(b)
g(x, y) = ln x 2 + y 2
7
Example 3. In 1928, Cobb and Douglas came up with a function to model the production of the entire US economy between the years 1899 and 1922. They found that the total yearly production P was related to the total capital investment, K, and the total labor force, L by the model P = 1.01L0.75K 0.25 . Similar functions are used to model production in industry; you should read about them in sections 11.1 and 11.3. Suppose that in a certain industry the production was related to labor and capital investment by P = 100L0.6 K 0.4 Find ∂P/∂L, the marginal productivity of labor, and ∂P/∂K, the marginal productivity of capital. Show that this Cobb-Douglas production unction satisfies the differential equation: ∂P ∂P L +K =P ∂L ∂K
Example 4.
Evaluate ∂z/∂x and ∂z/∂y for z = arctan(y/x), at (x, y) = (2, –2).
) Example 5. (QUIPP: due Recall the stadium wave function (from section 11.1): h(x, t) = 5 + cos(0.5x – t), where x is the seat number of a person in a given row, and t is the time in seconds after the wave is initiated. Compute hx(8, 5) and ht(8, 5), and interpret these quantities .
8
Section 11.4 TANGENT PLANES AND LINEAR APPROXIMATIONS (Omit Tangent Planes to Parametric Surfaces, page 787) LOCAL LINEARITY Consider the function f(x) = 7 – x2, at x = 1. Use your calculator to graph this function in a suitable window, then zoom in close to the point where x = 1. What does the graph look like?
Find the equation of the tangent line to f(x) at x = 1
Graph it together with f(x). Zoom in until the two graphs are indistinguishable.
Consider the Maple plots of the 3D surface for the function f(x, y) = 16 – x2 – y2. We look at the surface in neighborhoods close to (x, y) = (1, 3). What do you notice? We also look at the contour plots for these regions. What do these contour plots look like very close to (1, 3)? (Maple worksheet locallin.mws is in the MTH253 folder on the ST server.)
We find the equation of the tangent plane to the surface z = 16 – x2 – y2 at (x, y) = (1, 3)
We say that a function f(x, y) is differentiable at a point (x0, y0) if we are able to approximate it locally by a tangent plane. Describe a method for finding the equation of a tangent plane to the surface of a function at a general point (x0, y0).
9
Definition: The Tangent Plane to the surface z = f(x, y) at (x0, y0).
Assuming that f(x, y) is differentiable at the point (x0, y0), the equation of the tangent plane is given by
Example 1a.
b.
Determine the equation of the tangent plane to the surface z = y – x2 at (4, 3).
Use the tangent plane to approximate 2.996 – 4.0132 . (How close is your estimate?)
Example 2. given point.
Find the equation of the tangent plane to the surface z = x/y, at (3, –2) at the
THE DIFFERENTIAL Returning to single-variable functions, recall the tangent line approximation to a function value near a given point. Example 3: For the function f(x) = 7 – x2, use the linear approximation at x = 1 to find f(1.02).
The change in the function value, when we change x by a small amount dx is given by the expression f ′ (x) dx. This is called the differential of f. 10
For functions of two variables we define the differential in a similar manner. To approximate the value of a function f close to any point (x, y) close to a fixed point (a, b), we could use the tangent plane approximation:
Let ∆x = x – a and ∆y = y – b, and define ∆f = f(x, y) – f(a, b), Then ∆f ≈ fx(a, b) ∆x + fy(a, b) ∆y. Replacing ∆x with dx and ∆y with dy, we define the differential of f… The Differential of a function z = f(x, y) The differential df (or dz) at a point (a, b) is the linear function of dx and dy given by
The differential at a general point is often written df = fx dx + fy dy. Example 4: Find the differential of f (x, y) = f(1.04, 1.98).
x 2 + y 3 at the point (1. 2). Use it to estimate
Applications of the differential. Note: In calculating maximum possible error in a computed value, you need to take into account that the given error in measurements may be either positive or negative.
Example 5: The base radius and height of a circular cone are measured as 10 cm and 25 cm respectively, with a possible error in measurement of as much as 0.1 cm each. (That is, error is ±0.1 cm.) Use differentials to estimate the maximum possible error in the calculated volume of the cone.
Example 6. Power: Electrical power P is given by P = E2/R, where E is voltage (volts), and R is resistance (ohms). Approximate the maximum percentage error in calculating power if 200 volts is applied to a 400 ohm resistor and the possible percent errors in measuring E and R are 2% and 3%. 11
