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Transcript of 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets...
![Page 1: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/1.jpg)
3B MAS
4. Functions
![Page 2: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/2.jpg)
Limit of a FunctionGraphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is obtained by moving along the curve from both sides of 'a' as x moves toward 'a'.
The limiting value of f(x) as x gets closer and closer to 'a' is denoted by
x alim f (x)
![Page 3: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/3.jpg)
Right/Left Hand LimitsAs x moves towards 'a' from right (left) hand side, the limiting value of f(x) is denoted by
)x(flimand)x(flimaxax
![Page 4: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/4.jpg)
Limiting and Functional ValueIf both sides limits are equal,
Otherwise, does not exist.
Note that may not equal to f(a)
)x(flim)x(flim)x(flimaxaxax
)x(flimax
)x(flimax
![Page 5: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/5.jpg)
a
f(x)
x
y
f(a)
)()(lim afxfax
Limiting Value
![Page 6: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/6.jpg)
Example 1Find the limit of the f(x) as x approaches a for the following functions.
(a)
a
![Page 7: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/7.jpg)
Example 1Find the limit of the f(x) as x approaches a for the following functions.
(a)
a
The limit does not exist as the function is not defined 'near' a.
![Page 8: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/8.jpg)
Example 1 (cont'd)
(b)
a
![Page 9: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/9.jpg)
Example 1 (cont'd)
(b)
a
The limit does not exist as the left side limit is not the same as right side limit.
![Page 10: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/10.jpg)
Example 1 (cont'd)
(c)
f(a)
a
![Page 11: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/11.jpg)
Example 1 (cont'd)
(c)
f(a)
a
The limit exists but it does not equal to f(a).
![Page 12: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/12.jpg)
Example 1 (cont'd)
(d)
a
f(a)
![Page 13: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/13.jpg)
Example 1 (cont'd)
(d)The limit exists and it equals to f(a).
a
f(a) )a(flimax
![Page 14: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/14.jpg)
Evaluating LimitsIf f(x) is not broken at 'a', use direct substitution to evaluate its limit as x approaches 'a'
Otherwise, find the left side and right side limits and check if they are equal.
![Page 15: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/15.jpg)
Example 2Evaluate the following limits if they exist.
(a) f(x) = 2x – 5 as x 1
f(x) is not broken at x = 1, so use direct sub.352)5x2(lim
1x
![Page 16: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/16.jpg)
Example 2 (cont'd)(b) f(x) = ln x as x 0
f(0) is not defined. So consider limit from both sides.
But f(x) is not defined for x < 0.So the limit does not exist.
xlnlim0x
![Page 17: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/17.jpg)
Example 2(c) f(x) = 1/(x – 2) as x 2
f(2) is not defined. So consider limit from both sides.
Since the left side limit does not equal to the right side limit, the limit of the f(x) as x approaches 2 does not exist.
)2x/(1limand)2x/(1lim2x2x
![Page 18: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/18.jpg)
Example 2 (cont'd)(d) f(x) = (x – 1)/(x2 – 1) as x 1
f(x) = (x – 1)/(x + 1)(x – 1) = 1/(x + 1)1/(x + 1) is not broken at x = 1, so use direct sub.
5.01x
1xlim
21x
![Page 19: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/19.jpg)
Example 2 (cont'd)
2)x(flimSo
2)7x(lim)x(flim
2)1x(lim)x(flim
3xif7x
3xif1x)x(f)e(
3x
2
3x3x
3x3x
2
![Page 20: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/20.jpg)
Limits to InfinityIf f(x) = x + c, f(x) as x (note that x is the dominant term)
If f(x) = 1/x, f(x) 0 as x If f(x) = ax2 + bx + c, ax2 is the dominant term as x
)x
c
x
ba(xcbxax
222
![Page 21: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/21.jpg)
Example 3Find the limit of f(x) as x (if they exist) for:
4x5x
2x7x3)x(f)d(
1xx
4x2x)x(f)c(
3x2x
x71)x(f)b(
1x4
3x2)x(f)a(
2
23
2
2
2
![Page 22: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/22.jpg)
03x2x
x71limSo
xasx
7
x
x7
3x2x
x71)x(f)b(
2
1
1x4
3x2limSo
xas2
1
x4
x2
1x4
3x2)x(f)a(
2x
22
x
Example 3 (cont'd)
![Page 23: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/23.jpg)
)x(flimand)x(flimSo
xasx3x
x3
4x5x
2x7x3)x(f)d(
11xx
4x2xlimSo
xas1x
x
1xx
4x2x)x(f)c(
xx
2
3
2
23
2
2
x
2
2
2
2
Example 3 (cont'd)
![Page 24: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/24.jpg)
Trigonometric Limits
0xtanlim
0xsinlim
1xcoslim
0x
0x
0x
![Page 25: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/25.jpg)
Example 4Find the following limits.
x
x4tanlim)d(
x
xcos1lim)c(
x
xtanlim)b(
x
xsinlim)a(
0x
0x
0x
0x
![Page 26: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/26.jpg)
x
C
B
AO r
Consider the relationship between the areas OAC, sector OAC , and OAB
0
sinlimx
xInvestigating
x
![Page 27: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/27.jpg)
x
C
B
AO r
Area of OAC = r2 sin x / 2Area of sector OAC = r2 x / 2Area of OAB = r2 tan x / 2
Example 4 (cont'd)
![Page 28: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/28.jpg)
x
C
B
AO r
Area of OAC = r2 sin x / 2Area of sector OAC = r2 x / 2Area of OAB = r2 tan x / 2
So (size of areas)
r2 sin x / 2 < r2 x / 2 < r2 tan x / 2 sin x < x < tan x1 < x / sin x < 1 / cos x1 > sin x / x > cos x
Take limit as x 0 to get
That means sin x x as x 0
1x
xsinlim
0x
![Page 29: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/29.jpg)
4x4
x4tanlim4
x4
x4tanlim4
x
x4tanlim)d(
0
01xcos1
xsin
x
xsinlim
)xcos1(x
xcos1lim
xcos1
xcos1
x
xcos1lim
x
xcos1lim)c(
1xcos
1
x
xsinlim
x
xtanlim)b(
0x40x0x
0x
2
0x0x
0x
0x0x
Example 4 (cont'd)
![Page 30: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/30.jpg)
ContinuityGraphically a graph is continuous at x = a if it is not broken (disconnected) at that point.
Algebraically the limit of the function from both sides of 'a' must equal to f(a).
)a(f)x(flim)x(flimaxax
![Page 31: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/31.jpg)
Example 5The following functions are not continuous at x = a.
a
af(a)
a
Why?
![Page 32: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/32.jpg)
Example 6The following functions are continuous at x = a.
a
f(a)
a
![Page 33: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/33.jpg)
Example 7Determine if the given function is continuous at the given point.
(a) f(x) = | x – 2 | at x = 2
(b) f(x) = x at x = 0
(c) f(x) = 1 / (x + 3) at x = -3
![Page 34: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/34.jpg)
.3xatcontinuousnotis)3x/(1So3x
1limbut
3x
1lim)c(
.0xatcontinuousnotisxSo
0xfordefinednotisx)b(
.2xatcontinuousis|2x|So
)2(f0|2x|lim|2x|lim)a(
3x3x
2x2x
Example 7 (cont'd)
![Page 35: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/35.jpg)
Example 8Given that f(x) is continuous over the set of all real numbers, find the values of a and b.
2xax2
2x16bx
1xax
)x(f
2
![Page 36: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/36.jpg)
Only need to consider the junctions (x = -1 and x = 2)
1band4a
a46b2So
a4)x(flimand)x(flim6b2)2(f
6ba1So
6b)x(flimand)x(flima1)1(f
2x2x
1x1x
Example 8 (cont'd)
![Page 37: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/37.jpg)
DifferentiabilityGraphical approach: A function f(x) is said to be differentiable at x = a if there is no 'corner' or 'vertical tangency' at that point.
A function must be continuous (but not sufficient) in order that it may be differentiable at that point.
![Page 38: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/38.jpg)
Example 9The following functions are not differentiable at x = a.(a)
a
f(a)
Corner at x = a
![Page 39: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/39.jpg)
Example 9 (cont'd)
(b)
y
x1 2 3 4 5 – 1 – 2 – 3 – 4
1
2
– 1
– 2
Vertical tangency at x = 1
![Page 40: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/40.jpg)
Example 9 (cont'd)
(c)
y
x1 – 1 – 2 – 3 – 4 – 5
2
4
6
8
– 2
– 4
– 6
– 8
Not continuous (not even defined) at x = -2
![Page 41: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/41.jpg)
Example 10The following functions are differentiable everywhere.
(a)
![Page 42: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/42.jpg)
Example 10 (cont'd)
(b)
![Page 43: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/43.jpg)
Example 10 (cont'd)
(c)
![Page 44: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/44.jpg)
Derivative of a FunctionA function is differentiable at a point if it is continuous (not broken), smooth (no corner) and not vertical (no vertical tangency) at that point.
Its derivative is given by (First Principle)
h
xfhxfh
)()(lim
0
![Page 45: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/45.jpg)
f(x)
f(x+h)
x x+h
P
Q
Differentiability (cont'd)
![Page 46: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/46.jpg)
Differentiability (cont'd)The gradient of PQ is given by
As Q moves closer and closer to P (i.e. as h tends to 0), the limiting value of the gradient of PQ (i.e. the derivative of f(x) at x) becomes the tangent at P.
h
xfhxf )()(
![Page 47: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/47.jpg)
Differentiability (cont'd)The derivative of a function y = f(x) is denoted by
It also represents the rate of change of y with respect to x.
dy dfor y or or f
dx dx
![Page 48: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/48.jpg)
Example 11(a) Find the gradient function of y = 2x2 using first
principle. Find also the gradient at the point (3, 18).
(b)Use the definition (first principle) to find the derivative of ln x and hence find the derivative of ex.
![Page 49: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/49.jpg)
1234dx
dy
x4
)h2x4(limh
h2hx4lim
h
x2)hhx2x(2lim
h
x2)hx(2lim
dx
dy)a(
3x
0h
2
0h
222
0h
22
0h
Example 11 (cont'd)
![Page 50: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/50.jpg)
Example 11 (cont'd)
x
1
elnx
1
)h
xmwhere(])
m
11(lim[ln
x
1
)]x
h1(
h
xlim[ln
x
1)
x
h1(ln
h
1lim
hx
hxln
limh
xln)hx(lnlimxln
dx
d
m
m
0h0h
0h0h
![Page 51: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/51.jpg)
Example 11 (cont'd)
xx
x
eedx
d
ydx
dy1
dx
dy
y
1
1ydx
dyln
dy
d1yln
dx
d
xylney
![Page 52: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/52.jpg)
Example 12Find the derivative of the following functions from first principles.
(a) f(x) = 1/x(b) f(x) = x(c) f(x) = xn
![Page 53: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/53.jpg)
Example 12 (cont'd)
2
0h
0h
0h
x
1
)hx(x
1lim
h)hx(xhxx
lim
hx1
hx1
lim)x
1(
dx
d)a(
![Page 54: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/54.jpg)
Example 12 (cont'd)
x2
1xhx
1lim
)xhx(h
xhxlim
xhx
xhx
h
xhxlim
h
xhxlimx
dx
d)b(
0h
0h
0h
0h
![Page 55: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/55.jpg)
Example 12 (cont'd)
1n
1n2n2n
1n
0h
n22n2n
1n
0h
nn
0h
n
nx
)h......hxCnx(lim
h
h......hxChnxlim
h
x)hx(limx
dx
d)c(
![Page 56: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/56.jpg)
ConcavityIf f(x) opens downward, it is said to be concave down
If f(x) opens upwards, it is concave up
concave down concave up
f '(x): + 0 - - 0 +
![Page 57: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/57.jpg)
Point of InflectionPoints of inflection: points where the curve changes from concave up to concave down or concave down to concave up
point of inflection
f '(x): - - - + + + maximum
![Page 58: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/58.jpg)
Horizontal InflectionHorizontal inflection: a point of inflection where the graph is momentarily horizontal, dy/dx = 0
horizontal inflection
-ve
-ve+ve
+ve
![Page 59: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/59.jpg)
Stationary PointsTurning points: max and min
Stationary points: max, min and horizontal inflection
dy/dx = 0
![Page 60: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/60.jpg)
Example 13Given the f(x) graph below draw f '(x).
y
x
![Page 61: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/61.jpg)
y
x
f '(x)f (x)
a b c
f '(x) changes as below:
x < a: +ve, but
x = a: f '(a) = 0 local max
a < x < b: -ve, , then (less –ve)point of inflection
Example 13 (cont'd)
![Page 62: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/62.jpg)
y
x
f '(x)f (x)
a b c
x = b: f '(b) = 0 local min
b < x < c: +ve, , then point of inflection
x = c: f '(c) = 0 global max
x > c: -ve,
Example 13 (cont'd)
![Page 63: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/63.jpg)
Noteworthy FeaturesMin TP: dy/dx = 0, sign change –ve, 0, +ve
Max TP: dy/dx = 0, sign change +ve, 0, –ve
Horizontal inflection: dy/dx = 0, +ve, 0, +ve or –ve, 0, –ve, (i.e. no sign change)
Point of inflection: d2y/dx2 = 0, (dy/dx is a max/min),
![Page 64: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/64.jpg)
Example 14Graph the following function and its derivative.
Use your graphs to locate the stationary points and points of inflection on
y = x4/4 – 4x3/3 – 7x2/2 + 10x + 5and determine the nature of each.
![Page 65: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/65.jpg)
y = x4/4 – 4x3/3 – 7x2/2 + 10x + 5dy/dx = x3 – 4x2 – 7x + 10
TP dy/dx = 0So x3 – 4x2 – 7x + 10 = 0i.e. (x + 2)(x – 1)(x – 5) = 0 x = -2, 1 or 5
When x = -2, y = -43/3When x = 1, y = 125/12When x = 5, y = -515/12
Example 14 (cont'd)
![Page 66: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/66.jpg)
Example 14 (cont'd)
x-4 -2 2 4 6 8
y
-40
-30
-20
-10
10
20
f(x)
f '(x)
min
max
min PoI
PoI
![Page 67: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/67.jpg)
Piecewise Defined FunctionsA piecewise defined function has different formulas for different parts of its domain.
At junction a filled circle indicates that a point actually exists there, whereas an empty circle shows a discontinuous point.
![Page 68: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/68.jpg)
Example 15Given the function below
(a) Find f(-2), f(1) and f(2)
(b)Graph f and determine whether f is continuous at x = 0 and x = 2.
2xfor1x
2x0forx
0xfor1x/1
)x(f 2
![Page 69: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/69.jpg)
2xfor1x
2x0forx
0xfor1x/1
)x(f 2
(a) f(-2) = 1/(-2) – 1 = -3/2f(1) = 12 = 1f(2) = 2 + 1 = 3
Example 15 (cont'd)
![Page 70: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/70.jpg)
y
x1 2 3 4 – 1 – 2 – 3 – 4
1
2
3
4
5
6
– 1
– 2
– 3
– 4
Example 15 (cont'd)
2xatcontinuousnotis)x(f
)x(flim)x(flim
0xatcontinuousnotis)x(f
)x(flim)x(flim
2x2x
0x0x
![Page 71: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/71.jpg)
Example 16Graph y = | x2 – 6x + 8 | and determine whether the function is continuous at x = 2 and x = 4.
![Page 72: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/72.jpg)
Example 16 (cont'd)
x-1 1 2 3 4 5 6 7
y
-2
2
4
6
8
From graph, the function is continuous at x = 2 and x = 4.
![Page 73: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/73.jpg)
The Sign FunctionIt can be considered as a logical function (especially in computer science)
It extracts the sign of the function
It returns 1 if f(x) is positive, 0 if f(x) equals to 0 and –1 if f(x) is negative.
0)x(f1
0)x(f0
0)x(f1
)x(sgn
![Page 74: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/74.jpg)
Sign Function (cont'd)
0)x(f1
0)x(f0
0)x(f1
)x(sgn
x-3 -2 -1 1 2 3
y
-2
-1
1
2
![Page 75: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/75.jpg)
Example 17Graph
(a) y = sgn (x/|x|)
(b)y = sgn (x2 – 1)
![Page 76: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/76.jpg)
y
x1 2 3 – 1 – 2 – 3
1
2
– 1
– 2
(a) y = sgn (x/|x|)
Example 17 (cont'd)
Not continuous at x = 0
![Page 77: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/77.jpg)
(b) y = sgn (x2 – 1)y
x1 2 – 1 – 2
1
2
– 1
– 2
Example 17 (cont'd)
Not continuous at x = -1 and x = 1
![Page 78: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/78.jpg)
Greatest Integer FunctionAlso known as floor functionDefined as the greatest integer less than or equal to the numberThat is, it rounds any number down to the nearest integerSymbol: int [x] orint [4.2] = = 4 int [-2.1] = = -3
x
1.2 2.4
![Page 79: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/79.jpg)
Greatest Integer Function (cont'd)y
x1 2 3 4 – 1 – 2 – 3
1
2
3
– 1
– 2
– 3
Not continuous at all integers.
![Page 80: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/80.jpg)
Example 18Graph the following functions:
(a) int [2x – 1]
(b) int [x2]
![Page 81: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/81.jpg)
(a) int [2x –1]
Consider 2x – 1 = n where n is an integerx = (n + 1) / 2So the 'breaking points' are steps of half of an integer
y
x1
2
1 3
2
2 5
2 –
1
2
– 1 –
3
2
– 2
1
2
3
– 1
– 2
– 3
– 4
Example 18 (cont'd)
![Page 82: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/82.jpg)
(b) int [x2]
Consider x2 = n where n is a positive integerx = nSo the breaking points are square root of +ve integers
y
x1
2
1 3
2
2 5
2 –
1
2
– 1 –
3
2
– 2 –
5
2
1
2
3
4 Example 18 (cont'd)
![Page 83: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/83.jpg)
Rules of Differentiation (Review)
2
1nn
gdxdg
fg)dxdf
()
g
f(
dx
d
dx
dgfg)
dx
df()gf(
dx
d
gdx
dfg
dg
d)]x(g[f
dx
d
gdx
df
dx
d)gf(
dx
d
xnxdx
d
![Page 84: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/84.jpg)
Example 19Find the derivative of the following functions:
1xx3
)4x9(y)d(
)3x2(8x
5y)c(
)1x3x(
4y)b(
x
3x7y)a(
3
2
4
22
![Page 85: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/85.jpg)
32
32'
22
22
)1x3x(
)3x2(8
)3x2()1x3x()2(4y
)1x3x(4y)b(
x
37x)1(37'y
x
3x7y)a(
Example 19 (cont'd)
![Page 86: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/86.jpg)
2
3
3
3
2
4
3142
414
)8x(
)67x6()3x2(5
]8x
3x28[
8x
)3x2(5
8x
)3x2(40
)8x(
)3x2(5
)2()3x2(4)8x(5)3x2()8x)(1(5'y
)3x2()8x(5)3x2(8x
5y)c(
Example 19 (cont'd)
![Page 87: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/87.jpg)
23
223
23
223
3
2
)1xx3(
)1x9()4x9()1xx3)(4x9(18
dx
dy
)1xx3(
)1x9()4x9()1xx3)(9)(4x9(2
dx
dy
1xx3
)4x9(y)d(
Example 19 (cont'd)
![Page 88: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/88.jpg)
Differentiating ef(x) and ln f(x)
)x(f
)x(f)x(fln
dx
d
e)x(fedx
d
'
)x(f')x(f
![Page 89: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/89.jpg)
Example 20Differentiate the following with respect to x:
x31x2
x
e2
1e2y)b(
ee3y)a(
![Page 90: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/90.jpg)
x31x2
x31x2
x31x2x3
1x2
x
x
e2
3e4
dx
dy
e)2
3(e)2(2
dx
dy
e2
1e2
e2
1e2y)b(
e3dx
dy
ee3y)a(
Example 20 (cont'd)
![Page 91: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/91.jpg)
Example 21Differentiate
x
2x
2
e1
exlney)b(
)3x7x(lne
1y)a(
![Page 92: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/92.jpg)
]e1
e
ex
x2)
e1
ex(ln2[e
dx
dy
]e1
e
ex
x2[e)]e1(ln)ex([lne2
)]e1(ln)ex([lnee1
exlney)b(
)3x7x
7x2(
e
1
dx
dy
)3x7x(lne
1y)a(
x
x
2x
2x2
x
x
2x2x2x2
x2x2x
2x2
2
2
Example 21 (cont'd)
![Page 93: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/93.jpg)
Differentiability (Revisit)Graphical approach: continuous, no corner, no vertical tangency
Algebraical approach:
)gencytanverticalno(finiteis)a(f
)cornerno()a(f)x(flim)x(flim
)continuous()a(f)x(flim)x(flim
'
''
ax
'
ax
axax
![Page 94: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/94.jpg)
Example 22Determine if the following functions are differentiable at the indicated points.
(a) y = 1 / (x + 1) at x = -1
(b) y = | x + 1 | at x = -1
(c) f(x) = -6x + 5 for x < 3 = -x2 – 4 for x 3 at x = 3
![Page 95: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/95.jpg)
(a) Let f(x) = 1 / (x + 1)
f(-1) is not definedf(x) is not continuous at x = -1f(x) is not differentiable at x = -1
Example 22 (cont'd)
![Page 96: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/96.jpg)
1xatabledifferentinotis)x(f
1)x('flim1)x('flim
1x1
1x1)x('f
1xatcontinuousis)x(f
0)1(f)x(flim)x(flim
1x1x
1x1x)x(f
|1x|)x(fLet)b(
1x1x
1x1x
Example 22 (cont'd)
![Page 97: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/97.jpg)
3xatabledifferentiis)x(f
6)x('flim)x('flim
3xx2
3x6)x('f
3xatcontinuousis)x(f
13)3(f)x(flim)x(flim
3x4x
3x5x6)x(f)c(
3x3x
3x3x
2
Example 22 (cont'd)
![Page 98: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/98.jpg)
Example 23Find the value of a and b so that
f(x) = 3x + 1 for x < 1 = x2 + ax + b for x 1
is continuous and differentiable everywhere.
![Page 99: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/99.jpg)
Possible discontinuity and non-differentiability at x = 1
continuous at x = 1 if 4 = 1 + a + bi.e. a + b = 3
f ’(x) = 3 for x < 1 = 2x + a for x > 1differentiable at x = 1 if 3 = 2 + a
So a = 1 and b = 2
Example 23 (cont'd)
![Page 100: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/100.jpg)
Riemann SumsTo find an approximate area under a curve between two x values [a, b]
The area is divided into n rectangles of equal width
So the width x = (b – a)/n
There are many ways to find the height h of each rectangle (see later)
Then the required area A = hx over the interval [a, b]
![Page 101: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/101.jpg)
Example 24The shaded area below shows the exact area under the curve f(x) = x3 – 3x2 + 8 in the interval [0, 3]
x0.5 1 1.5 2 2.5 3
y
1
2
3
4
5
6
7
8
9
y = x3 – 3x2 + 8
Actual area = 17.25
![Page 102: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/102.jpg)
Example 25(Example 24) Consider n = 5, x = (3 – 0)/5 = 0.6 and h = left endpoint
x = (3 – 0)/5 = 0.6
Left endpoint for the 3rd rectangle
![Page 103: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/103.jpg)
Example 25 (cont'd)f(x1)= (0)3 – 3(0)2 + 8 = 8f(x2)= (0.6)3 – 3(0.6)2 + 8 = 7.136f(x3)= (1.2)3 – 3(1.2)2 + 8 = 5.408f(x4)= (1.8)3 – 3(1.8)2 + 8 = 4.112f(x5)= (2.4)3 – 3(2.4)2 + 8 = 4.544
52.17
)544.4112.4408.5136.78(6.0
x)x(fA5
1ii
![Page 104: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/104.jpg)
Example 26(Example 24) Consider n = 5, x = (3 – 0)/5 = 0.6 and h = right endpoint
Right endpoint for the 3rd rectangle
![Page 105: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/105.jpg)
Example 26 (cont'd)f(x1)= (0.6)3 – 3(0.6)2 + 8 = 7.136f(x2)= (1.2)3 – 3(1.2)2 + 8 = 5.408f(x3)= (1.8)3 – 3(1.8)2 + 8 = 4.112f(x4)= (2.4)3 – 3(2.4)2 + 8 = 4.544f(x5)= (3.0)3 – 3(3.0)2 + 8 = 8
52.17
)8544.4112.4408.5136.7(6.0
x)x(fA5
1ii
![Page 106: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/106.jpg)
Example 27(Example 24) Consider n = 5, x = (3 – 0)/5 = 0.6 and h = minimum point
Minimum point for the 4th rectangle
![Page 107: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/107.jpg)
Example 27 (cont'd)f(x1)= (0.6)3 – 3(0.6)2 + 8 = 7.136f(x2)= (1.2)3 – 3(1.2)2 + 8 = 5.408f(x3)= (1.8)3 – 3(1.8)2 + 8 = 4.112 f(x4)= (2.0)3 – 3(2.0)2 + 8 = 4 f(x5)= (2.4)3 – 3(2.4)2 + 8 = 4.544
12.15
)544.44112.4408.5136.7(6.0
x)x(fA5
1ii
![Page 108: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/108.jpg)
Example 28(Example 24) Consider n = 5, x = (3 – 0)/5 = 0.6 and h = maximum point
Maximum point for the 4th rectangle
![Page 109: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/109.jpg)
Example 28 (cont'd)f(x1)= (0)3 – 3(0)2 + 8 = 8 f(x2)= (0.6)3 – 3(0.6)2 + 8 = 7.136f(x3)= (1.2)3 – 3(1.2)2 + 8 = 5.408f(x4)= (2.4)3 – 3(2.4)2 + 8 = 4.544f(x5)= (3.0)3 – 3(3.0)2 + 8 = 8
8528.19
)8544.4408.5136.78(6.0
x)x(fA5
1ii
![Page 110: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/110.jpg)
Example 29(Example 24) Consider n = 5, x = (3 – 0)/5 = 0.6 and h = midpoint
Midpoint for the 3rd rectangle
![Page 111: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/111.jpg)
Example 29 (cont'd)f(x1)= (0.3)3 – 3(0.3)2 + 8 = 7.757f(x2)= (0.9)3 – 3(0.9)2 + 8 = 6.299f(x3)= (1.5)3 – 3(1.5)2 + 8 = 4.625f(x4)= (2.1)3 – 3(2.1)2 + 8 = 4.031f(x5)= (2.7)3 – 3(2.7)2 + 8 = 5.813
115.17
)813.5031.4625.4299.6757.7(6.0
x)x(fA5
1ii
![Page 112: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/112.jpg)
A Better ApproximationDue to the use of h (left, right, mid, min and max), the rectangles do not truly represent the area under the curve for each strip
If n (number of rectangles) increases, the error decreases
![Page 113: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/113.jpg)
Example 30n = 20, x = (3 – 0)/20 = 0.15, h = midpoint
Actual area = 17.25
![Page 114: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/114.jpg)
Example 31n = 100, x = (3 – 0)/100 = 0.03, h = midpoint
Actual area = 17.25
![Page 115: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/115.jpg)
Limit of a SumThe more rectangles, the greater accuracy
So the actual area A is given by
This is written as
x)x(flimAn
egrationintcalledisprocessabovetheand
sumaofitlimtherepresentswhere
dx)x(fx)x(flimAn
![Page 116: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/116.jpg)
Integration f(x)dx represents the area under the curve
x1 2 3 4
y
1
2
3
4
5
6
7
8
4
1dx)x(f
f(x)
![Page 117: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/117.jpg)
Example 32
b 3 4 5 6 7 x
Area
Evaluate the area under the curve y = 3 from x = 2 to x = b by completing the following table.
Hence give an answer for where k is a constant
x
akdx
![Page 118: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/118.jpg)
Example 32 (cont'd)
b 3 4 5 6 7 b
Area 3 6 9 12 15 3(b – 2)
x1 2 3 4 5 6 7 8 9 10
y
1
2
3
4
)ab(kkdxb
a
![Page 119: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/119.jpg)
Evaluate the area under the curve y = 2x from x = 0 to x = b by completing the following table.
Hence give an answer for where k is a constant
Example 33
b 0 1 2 3 4 b
Area
b
0kxdx
![Page 120: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/120.jpg)
Example 33 (cont'd)
b 0 1 2 3 4 x
Area 0 1 4 9 16b2b/2 =
b2
x1 2 3 4 5 6
y
2
4
6
8
10
12
14
2
kbkxdx
2b
0
![Page 121: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/121.jpg)
Given the areas under the curve y = x2 from x = 0 to x = 4 in the following table, find the area when x = b
Hence give an answer for
Example 34
b 0 1 2 3 4 b
Area 0 1/3 8/3 9 64/3
b
0
2dxx
![Page 122: 3B MAS 4. Functions. Limit of a Function Graphically the limiting value of a function f(x) as x gets closer and closer to a certain value (say 'a') is.](https://reader035.fdocuments.in/reader035/viewer/2022070412/56649e3f5503460f94b2fc0e/html5/thumbnails/122.jpg)
Example 34 (cont'd)
b 0 1 2 3 4 b
Area 0 1/3 8/3 9 64/3 b3/3
x1 2 3 4 5
y
5
10
15
20
25
30
3
bdxx
3b
0
2