Advanced Level Pure Mathematics s1

Applications of Differential Calculus Advanced Level Pure Mathematics

5.1 L’Hospital’s Rule 1 5.4 Monotonic Functions 2 Proving Inequalities by Using Differential Calculus 2 5.5 Maxima and Minima 3 5.7 Curve Sketching 8

5.1 L' Hospital's Rule

Theorem The limit of a non-constant function f ( x ) as x tends to x0 is said to be an indeterminate of the form .

0 g1( x ) 1 x (i) if f ( x )  , where lim g1( x )  lim g2( x )  0, e.g. lim . x x0 x x0 x1 2 0 g2( x ) 2  2x

g ( x ) tan3x  1 lim g (x)  lim g (x)  , lim . (ii) if f ( x )  , where 1 2 e.g.  g ( x ) x x0 x x0 x tan x  2 2

x 0 g 2 ( x ) lim g ( x )  lim g ( x )  0, lim x . (iii) 0 if f ( x )  [ g ( x )] , where 1 2 e.g.  1 xx0 xx0 x0 (iv)  0 (v) 1 (vi) 0

Remark   ,    and  are not indeterminate forms because none of limits of these forms can exist.

Theorem L'Hospital's Rule g ( x ) 0  g (x) g ' ( x ) If lim 1 is an indeterminate of the form or , then lim 1  lim 1 . xx xx xx 0 g 2 ( x ) 0  0 g 2 (x) 0 g 2' ( x )

ln x ln x Example (a) Evaluate lim (b) Evaluate lim . x1 x 1 x0 cot x

Example ( Other Indeterminate Forms )

 lim( x  )tan x. Evaluate  x 2 2

It is important that before applying L'Hospital's rule, we should check each step whether the limit under consideration is an indeterminate or not, if the limit is not an indeterminate or is an indeterminate but not of 0  the form or , L'Hospital's rule is not applicable. 0 

In the following, which step is wrong?

Applications of Differential Calculus Advanced Level Pure Mathematics e x  ex e x  e x lim = lijm x0 sin x x0 cos x e x  ex = lim x0  sin x e x  e x = lim x0  cos x e0  e0 = =  2  cos0

5.2 Monotonic Functions Theorem Let f ( x ) be continuous on [a, b] and differentiable on (a, b). f ( x ) is a constant function if and only if f '( x )  0 for all x ( a,b )

Definition A function f ( x ) is said to be monotonic increasing ( resp. monotonic decreasing ) or simply

increasing ( resp. decreasing ) on an interval I if and only if x1 ,x2  I , if x1  x2 then

f ( x1 )  f ( x2 ) (resp. x1 ,x2  I , if x1  x2 , then f ( x1 )  f ( x2 ) ).

Definition A function f ( x ) is said to be strictly increasing ( resp. strictly decreasing ) on an interval I if

and only if x1 ,x2  I , if x1  x2 then f (x1 )  f (x2 ) (resp. x1 ,x2  I , if x1  x2 , then

f (x1 )  f (x2 ) ).

Theorem Let f ( x ) be continuous on [a, b] and differentiable on (a, b). Then (a) if f '(x)  0, x ( a,b ), f ( x ) is strictly increasing on [a, b]; and (b) if f '(x)  0, x ( a,b ), f ( x ) is strictly decreasing on [a, b].

Example Prove that f ( x )  x 3 is strictly increasing on R .

Proving Inequalities by Using Differential Calculus

In Practical problems, we always encounter inequalities within a certain range such as: f ( x )  g( x ) a  x  b Usually, it is transformed to F( x )  f ( x )  g( x )  0 a  x  b Based on the properties of increasing function and decreasing function, we can establish inequalities and the method is outlined in the following

Making Use of Strictly Increasing or Decreasing Functions

Want to prove that f ( x )  g( x )

(1) Consider F( x )  f ( x )  g( x ) Try to prove that F' ( x )  0 a  x  b Hence, we have F( x ) is strictly increasing on a,b (2) Try to prove that F( x ) is continuous on a,b Then we have F( x )  f ( x )  g( x )  F( a )

Applications of Differential Calculus Advanced Level Pure Mathematics (3) Try to prove that F( a )  0 Then we can conclude that f ( x )  g( x ) a  x  b

Making Use of The Greatest and Least Values of a Function

Want to prove that f ( x )  g( x )

(1) Consider F( x )  f ( x )  g( x ) Try to prove that F( c ) is the least value in a,b

(2) Try to prove that F( x ) is continuous on a,b Then we have F( x )  f ( x )  g( x )  F( c ) x  c

(3) Try to prove that F( c )  0 Then we can conclude that f ( x )  g( x ) a  x  b,x  c

Example Show that sin x  x for all x  0.

Example Prove e y  ea  ea ( y  a ) .

x 2 x 2 x3 Example Prove if x  0, x   ln (1 x )  x   2 2 3

Example Suppose f ( x ) satisfy (i) f ( x ) is continuous for x  0 . (ii) f ' ( x ) exists for x  0 . (iii) f ( 0 )  0 (iv) f ' ( x ) is increasing on 0, f ( x ) Let g( x )  , show g is increasing. x

5.5 Maxima and Minima

Definition A neighborhood of a point x0 is an open interval containing x0 , i.e. (x0 δ ,x0 δ ) is

a neighborhood of x0 for some δ  0 .

Definition A function f(x) is said to attain a relative maximum ( minimum ) at a point x0 if f(x)  f(x0 )

(f(x)  f(x0 ) ) in a certain neighborhood of x0 , i.e. δ  0 such that f(x)  f(x0 ) (

f(x)  f(x0 ) ) for x  x0  δ

Theorem Fermat Theorem

Given f(x) is a point defined on (a,b) and differentiable at a point x0 if f(x) has an extreme

value ( max. or min ) x0 , then f '(x0 )  0 .

Note f '(x0 )  0  f(x) has maximum or minimum at x0 .

Definition (a) A turning point is a maximum or minimum point. (b) If f '(x)  0 , then x is called a critical or stationary value and its corresponding point on

Applications of Differential Calculus Advanced Level Pure Mathematics the graph y  f(x) is called stationary point.

Notes 1. turning point stationary point 2. turning point + differentiable  stationary point 3. Stationary point turning point

 Therefore, in searching extreme value of a function, we have to investigate not only the stationary points , but also the points where the functions is not differentiable.

Theorem Suppose that the function f(x) has a continuous derivatives f '(x)  0 which vanishes only at

a finite no. of points, then the function has maximum (minimum) at point x0 if and only if

f '(x) is  ve (ve) at points immediately to the left of x0 and  ve (+ve) immediately to

the right of x0 .

Theorem f(x) is a function of x , if f '(x0 )  0 and f ''(x0 ) exists such that

(i) f ''(x0 )  0 , then f(x) attains minimum at x  x0 .

(ii) f ''(x0 )  0 , then f(x) attains maximum at x  x0 .

2 Example Find the max. or min. points of y  ( x 1)x 3  3

2 1 Example Find the local extreme of the function f ( x )  x 3 ( x 1)3

Definition Given that f(x) is continuous on [a,b], if any x1 ,x2  (a,b) such that  x  x  f(x )  f(x ) (i) f 1 2   1 2  2  2

Concave Downward

 x  x  f(x )  f(x ) (ii) f 1 2   1 2  2  2

Concave Upward

*Theorem If f(x) is a function on [a,b] such that f(x) is second differentiable on (a,b) then (i) f ''(x)  0 iff f(x) is concave upward on (a,b) (ii) f ''(x)  0 iff f(x) is concave downward on (a,b) .

Applications of Differential Calculus Advanced Level Pure Mathematics Point of Inflection

Definition Let f(x) be a continuous function. A point (c,f(c)) on the graph of f is a point of inflexion (point of inflection) if the graph on one side of this point is concave downward and concave upward on the other side. That is, the graph changes concavity at x  c .

Note A point of inflexion of a curve y  f(x) must be a continuous point but need not be differentiable there. In Figure (c), R is a point of inflexion of the curve but the

function is not differentiable at x0 .

Theorem If f(x) is second differentiable function and attains a point of inflexion at x  c , then f ''(c)  0 .

Note: (i) max. or min. point but not derivative.

(ii) point of inflexion may not be obtained by solving f ''(x)  0 where f ' (c)   and

f' (c)   such that f ' (c)f ' (c)  0.

(iii) Let f(x) be a function which is second differentiable in a neighborhood of a point of inflexion iff f '(x) does not change sign as x increases through (sign gradient test)

 if f '(c)  0 and f ' (c)f ' (c)  0, then f(x) attains a relative max. or relative min.

 if f '(c)  0 and f ' (c)f ' (c)  0 , then f(x) attains an inflexion point at c .

Example Find the points of inflexion of the curve y  x 4  6x 2  8x 10 .

y' 4x3  12x  8 y'' 12x2  12  12(x  1)(x  1) When x  1, y'' 0 .

Applications of Differential Calculus Advanced Level Pure Mathematics x x  1 x  1  1  x  1 x  1 x  1 y'' y concave pt of inflexion concave pt of inflexion concave upward downward upward

The curve has points of inflexion at x  1. These two points are (1,3) , (1,13) .

Example Find the points of inflexion of the curve y  3x5  5x 4  4 .

y' 15x4  20x3 y'' 60x3  60x2  60x2 (x  1) . When x  0 or 1 , y'' 0 .

x x  0 x  0 0  x  1 x  1 x  1 y'' y concave concave pt of inflexion concave downward downward upward

The curve has a point of inflexion at x  1 . This point is (1,2)

Example Find the points of inflexion of the curve y  3  5 ( x  2 )7 .

7 14 y'  5 (x  2)2 , y''  . 5 25 5 (x  2)3

When x  2 , y'' does not exist (i.e. y' is not differentiable there.)

x x  2 x  2 x  2 y'' not exist y concave pt of inflexion concave upward downward

The curve has a point of inflexion at x  2 , which is (2,3)

Asymptotes To A Curve

Definition A straight line is an ASYMPTOTE to a curve if and only if the perpendicular distance from a variable point on the curve to the line approaches to zero as a limit when the point tends to

Applications of Differential Calculus Advanced Level Pure Mathematics infinity along the curve on both sides or one side of the curve. (see figure below.)

Definition (i) the line x  c is said to be vertical asymptote of the curve y  f(x) .

lim f(x)   or lim f(x)   . xc xc (ii) the line y  ax  b is said to be an oblique asymptote of the curve y  f(x) if limf(x)  (ax  b) 0 x .

1 Example (1) The curve y  has two asymptotes x  0 or y  0 . x (2) The curve y  e x has an asymptote y  0 . 1 (3) The curve y  sin x has an asymptote y  0 . x

Theorem The line y  ax  b is an asymptote to the curve y  f(x) if and only if f(x) a  lim and b  limf(x)  ax x x x where both limits are taken under x   , or x   or x   .

Note Equation of an asymptote (other than the vertical asymptote) to a given curve can be found by using Theorem

f(x) Theorem Let F(x)  , where f(x) and g(x) are polynomials in x and the degree of f(x) exceeds g(x) R(x) that of g(x) by one; let F(x) be written in the form F(x)  ax  b  , where a and b are g(x) constants, and R(x) is a polynomial of degree less than that of g(x) . (This can be done by long division.) Then the line y  ax  b is an oblique asymptote to the curve y  F(x).

Applications of Differential Calculus Advanced Level Pure Mathematics

5.7 Curve Sketching

The following information is useful for sketching the graph of y  f(x) (1) The domain of f(x) , i.e. the range of values of x within which y is well-defined.

(2) Determine whether f(x) is periodic, odd or even, so that the graph may be symmetric about the coordinate axes or about the origin. (3) Turning points and monotonicity of f(x) .

(4) Inflexional points and convexity of f(x) .

(5) Asymptotes including horizontal, vertical and oblique ones (if any).

(6) Some special points on the graph, such as intercepts.

8 AL2003II 8) Let f (x)  x2  ( x  1). x 1 a) Find f ‘ (x) and f “ (x). b) Determine the values of x such that (i) f ‘(x) > 0 (ii) f ‘ (x) < 0 (iii) f “(x) > 0 (ii) f “ (x) < 0 c) Find the relative extreme point(s) and point(s) of inflexion of f(x). d) Find the asymptotes of the graph of f(x). e) Sketch the graph of f(x). f) Let g(x) = f( |x| ) ( | x |  1 ) . (i) Is g(x) differentiable at x = 0? Why ? (ii) Sketch the graph of g(x).